Materials and Design - Khalifa Universitykshanmugam.faculty.masdar.ac.ae/PDF/JMAD2016.pdf · Multiscale modeling of effective electrical conductivity of short carbon fiber-carbon

Materials and Design 89 (2016) 129–136

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Materials and Design

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Multiscale modeling of effective electrical conductivity of short carbonfiber-carbon nanotube-polymer matrix hybrid composites

G. Pal a,1, S. Kumar a,b,⁎a Institute Center for Energy (iEnergy), Department of Mechanical and Materials Engineering, Masdar Institute of Science and Technology, PO Box 54224, Abu Dhabi, UAEb Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139–4307

⁎ Corresponding author.E-mail addresses: [email protected], kshanmugam@m

1 Currently at the Department of Civil Engineering, Am

http://dx.doi.org/10.1016/j.matdes.2015.09.1050264-1275/© 2015 Elsevier Ltd. All rights reserved.

a b s t r a c t

Article history:Received 28 July 2015Received in revised form 15 September 2015Accepted 16 September 2015Available online 25 September 2015

Keywords:Hybrid compositesCarbon nanotubesShort carbon fibersElectrical conductivityMultiscale modeling

Epoxy matrix reinforced with conventional microscale short carbon fibers (SCFs) and carbon nanotubes (CNTs)form a hybridmaterial systemwhere the characteristic length scales of SCFs and CNTs differ bymultiple orders ofmagnitude. Several recent studies show that the addition of CNTs into a non-conducting polymer matriximproves both structural performance such asmodulus, strength and fracture toughness and functional responsesuch as electrical and thermal conductivities of the resulting nano-composite. In this study, a physics-basedhierarchical multiscale modeling approach is presented to calculate the effective electrical conductivity ofSCF-CNT-polymer hybrid composites. A dual step procedure is adopted to couple the effects of nano- andmicro-scale so as to estimate the effective electrical properties of the composite. First, CNTs are dispersed intothe non-conducting polymer matrix to obtain an electrically conductive CNT-epoxy composite. The effectiveelectrical conductivity of CNT-epoxy composite is modeled using a physics-based formulation for both randomlydistributed and vertically aligned cases of CNTs and the results are verified with the measured data available inthe literature. In the second step, SCFs are randomly distributed in the CNT-epoxy composite and the effectiveelectrical conductivity of the resulting SCF-CNT-epoxy hybrid composite is estimated using a micromechanicsbased self-consistent approach considering SCFs as microscopic inhomogeneities.

© 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Hybrid composites are gaining the attention of the materialsresearch community as they offer a wide range of possibilities to tailortheir properties at various length scales. Hybrid polymer compositesconsisting of randomly distributed micron-size short carbon fibers andCNTs possess excellent specific mechanical properties [1,2] coupledwith inherent multifunctionality [3,4]. Addition of SCFs changes anon-conducting pristine polymer into a conductive polymer composite.As the volume fraction of SCF is increased, the effective electrical con-ductivity of the composite increases. With the addition of a few percentSCF, the effective electrical conductivity of composite approaches alimiting value and remains within the same order of magnitude withfurther addition of SCF into the matrix. The maximum achievable limitof electrical conductivity for these microscale carbon fiber reinforcedcomposites is in the range of 0.1–20 S/m. However, this envelopeof electrical conductivity of SCF-polymer composites can be pushedfurther by the addition of CNTs providing another scale of electricalpathways in the composites. In this way, polymer matrix compositesreinforced with microscopic carbon fibers and CNTs can exhibit

asdar.ac.ae (S. Kumar).ity University, Noida, India.

improved electrical conductivity compared to carbon fiber-polymermatrix composites [5]. Addition of nanoscale fibers triggers themechanisms which are otherwise unavailable in these conventionalcomposites. It enhances the utilization of these composites eitherby expanding the limits of the available properties or by convertingthem into a multifunctional composite by imparting one or more previ-ously unavailable functionality. In case of CNT-composites, several suchexamples exist where addition of CNTs has improved the mechanicalresponse e.g. fracture toughness [6–10] of existing composite systemor has introduced/enhanced the piezoelectric behavior in conventionalcomposites for structural health monitoring [11,12]. Addition of evensmall amount of CNTs into the non-conducting polymer improves theelectrical conductivity considerably due to extremely low percolationthershold (0.05–0.5wt.%) [13,14]. However, the extent of improvementalso depends on physical features of the CNTs (singlewall vs multiwallCNTs, aspect ratio), their dispersion in thematrix and inter-CNT contactresistance. In case of CNT-polymer composites, the two electricalconductivity mechanisms, namely, electron hopping (tunneling) andconductive network, complement each other.

The aim of this study is to develop a physics-based modelingframework capable of predicting the effective electrical conductivity ofSCF-CNT reinforced polymer hybrid composites. However, predictingthe effective electrical conductivity of these hybrid composites remainsa challenging task due to the vast difference between the spatial scales

130 G. Pal, S. Kumar / Materials and Design 89 (2016) 129–136

and electrical conductionmechanisms associatedwith these two sets ofinhomogeneities. Due to this difference, the solution strategy todetermine the effective electrical conductivity of SCF-CNT-polymercomposite is divided into two parts. Fig. 1 shows the two parts withthe schematics of respective microstructure of the hybrid compositeand the flowchart for the modeling strategy. In the first part, CNTs aredispersed into the non-conductive polymer to convert it into a conduc-tive matrix [Fig. 1(c)]. The computational framework presented hereinis based on the assumption that CNT content in the composite isabove the percolation threshold and that the CNT conductive networkhas already been established. Above the percolation threshold, electrontunnelling mechanism dominates. The effective electrical conductivityof resulting CNT-polymer composite is determined in two steps. In thefirst step, the CNTs surrounded by a thin layer of polymer are convertedinto equivalent CNT inclusions and the homogenized electrical conduc-tivity of CNT-polymer composite consisting of these equivalent inclu-sions is calculated using micromechanics based Mori-Tanaka method.In the second part, SCFs are distrubuted in the resulting conductivepolymer matrix [Fig. 1(d)]. The homogenized electrical conductivityof CNT-SCF-polymer composite is determined by a self-consistentapproach. Based on the approach described above, the manuscript isorganized as follows. Section 2 describes the methodology used forcalculating the homogenized electrical conductivity of CNT-polymercomposite i.e. the hostmatrix for SCFs. Following this, Section 3 presentsthe details of self-consistent formulation used in predicting the effectiveelectrical conductivity of SCF-conductive matrix composite. Section 4presents the results of homogenized electrical conductivity of CNT-polymer and CNT-SCF-polymer matrix composites. Section 5 concludeswith the main findings of this research.

2. Effective electrical conductivity of carbonnanotube-epoxy composites

The effective electrical conductivity of CNT-polymer compositeor the ‘conductive matrix’ is largely limited by the interfacial contactresistance between CNTs and thematrix. Several physics-based compu-tational approaches have been developed for calculating the homoge-nized electrical conductivity of CNT-polymer composites [15–21]. InCNT-polymer composites, the CNT-CNT and CNT-polymer interactionsare governed by weak non-bonding interactions. These interactions

Fig. 1. Schematic representation of the computational approach involved in calculating the efsurrounded by the interphase; (b) Homogenized CNT inclusion; (c) Representative volume efor SCF-CNT-polymer hybrid composite.

create a shell around CNTs which is referred to as interphase layer.This interphase layer provides interfacial resistance to CNTs. Theproperties of the interphase layer depends upon CNT concentration,CNT-CNT potential-well depth and the average separation distancebetween the CNTs. The pristine CNTs together with the surroundinginterphase layer is replaced by a homogenized CNT inclusion whoseeffective properties include the effects of the interphase. Fig. 2 showsthe strategy adopted in determining the electrical conductivity of thehomogenized CNT inclusions.

Consider an isolated CNT surrounded by an interphase layer on itslateral surface as shown in Fig. 1(a). Due to the difference in resistivecontributions from the interphase layer at the ends and on the lateralsurface of the CNTs, the electrical conductivity of homogenized inclu-sion becomes transversely isotropic. The effective electrical conductivityof the homogenized inclusion is therefore calculated in two steps. In thefirst step, the CNT is considered to be surrounded by the interphaselayer of uniform thickness along its length only as shown in Fig. 2(a).The longitudinal and transverse electrical conductivities of thisinclusion are calculated separately (see [22]). Consider the intermediateinclusion as shown in Fig. 2(b), subjected to a uniform electrical field, E0,which is directed perpendicular to CNT axis of symmetry [Fig. 2(d)].Considering the length of CNT along the z-axis, the CNT cross sectionthat lies in the x-y plane is shown in Fig. 2(d). Due to uniform thicknessof the interphase layer around the CNT, the two transverse electricalconductivities, σx and σy for the intermediate inclusion can be consid-ered to be equal. In order to calculate the transverse electrical conduc-tivity of the intermediate inclusion, σx(=σy), the Maxwell's equationin cylindrical coordinates is applied to intermediate inclusion[Fig. 2(b)] subjected to uniform electrical field E0 [see Fig. 2(d)].

∇2ϕ ¼ ϕ;rr þ1rϕ;r þ

1r2

ϕ;θθ ¼ 0 ð1Þ

where, ϕ is the electrical field potential. The plane containing coordinate(r, θ) forms the plane of isotropy of the homogenized intermediate inclu-sion. The general solution for the Laplacian given by Eq. (1) is given as:

ϕ ¼X∞n¼1

Anrn þ Bn

rn

� ��Cn cos nθð Þ þ Dn sin nθð Þð Þ ð2Þ

fective electrical conductivity of CNT - SCF - polymer matrix hybrid composites: (a) CNTlement for the conductive matrix (CNT + polymer); (d) Representative volume element

Fig. 2. Schematic diagram of the approach used in determining the electrical conductivity of the homogenized inclusion: (a) CNT with interphase at the outer surface; (b) Intermediateinclusion with interphase; (c) Equivalent CNT inclusion; (d) Intermediate CNT inclusion (CNT with interphase on outer surface) subjected to uniform electrical field, E0.

131G. Pal, S. Kumar / Materials and Design 89 (2016) 129–136

As, r → ∞, ϕm=−E0(rcosθ), therefore term with sin θ as coefficientmust be ignored in the final solution and the applicable final solutiontakes the following form.

ϕ ¼X∞n¼1

Anrn þ Bn

rn

� ��Cn cos nθð Þð Þ ð3Þ

In the absence of surface charge on CNT, the continuity of normalcomponents of electrical potential field and current density acrossCNT-interphase and interphase-matrix interfaces is used to establishthe boundary conditions given below.

ϕc r ¼ 0ð Þ ¼ constant; ϕm r→∞ð Þ ¼ −E0rcosθϕc r ¼ að Þ ¼ ϕint r ¼ að Þ; ϕint r ¼ aþ tð Þ ¼ ϕm r ¼ aþ tð Þσ c

dϕc

drr ¼ að Þ ¼ σ int

dϕint

drr ¼ að Þ;

σ intdϕint

drr ¼ aþ tð Þ ¼ σm

dϕm

drr ¼ aþ tð Þ

ð4Þ

where, a in the average radius of the CNT and t is the thickness of theinterphase region surrounding the CNT in the intermediate inclusionas shown in Fig. 2(d). Imposing the boundary conditions given byEq. (4) yields the electrical potential field in CNT, interphase layer andthe bulk matrix as:

ϕc ¼ P rcos θð Þ 0 ≤ r ≤ a ð5aÞ

ϕint ¼ Qrcos θð Þ þ Trcos θð Þ a ≤ r ≤ aþ t ð5bÞ

ϕm ¼ Hrcos θð Þ− E0rcos θð Þ aþ t ≤ r ≤ ∞ ð5cÞ

where, ϕc, ϕint and ϕm are the electrical field potentials in the CNT,interphase layer and bulk matrix, respectively. P, Q, T and H are con-stants and are determined using appropriate boundary conditions[17]. The average electrical field intensity in CNTs, Ecx, and theinterphase, Eintx, are given, respectively, by

Ecx ¼ −dϕc

dxð6aÞ

Eintx ¼ −dϕint

dxð6bÞ

The in-plane electrical conductivities of intermediate inclusion,σx = σy, can be determined from the average electrical field intensities,Ecx and Eintx, and current densities, jcx and jintx, of the CNT and theinterphase [Fig. 2(a)] as follows.

σ x ¼ σy ¼ b jxc N þ b jintx Nb Exc N þ b Eintx N

ð7aÞ

σx ¼ σy ¼d2σ cσ int þ 2σ int σ c þ σ intð Þ t2 þ 2at

� �d2σ int þ 2σ int σ c þ σ intð Þ t2 þ 2at

� � ð7bÞ

The longitudinal electrical conductivity (σz) of the inclusion shownin Fig. 2(a) canbe calculated from the resistor-typeparallel combinationof interphase layer along length and the CNT as follows.

1Reff

¼Xi

1Ri

ð8Þ

where,

1Ri

¼ σ iAi

lið9Þ

where, Ai and li are the cross-sectional area and length of thecorresponding resistive elements. The expressions for li and Ai ofthe CNT and the interphase (i = 1, 2) can be calculated with thehelp of Fig. 2(a) and (d). The longitudinal electrical conductivity,σz, is given by

σ z ¼σ cd

2 þ σ int 4dt þ t2� �

dþ 2tð Þ2ð10Þ

The overall effective longitudinal and transverse electrical conduc-tivities for the homogenized inclusion can be calculated in terms of σx

and σz by considering series and parallel combinations of intermediateinclusion and interphase layer as shown in Fig. 2(b).

σeffx ¼ σ effy ¼2tσ int þ σ xl

lþ 2tð11Þ

σeffz ¼ lþ 2tð Þ 2tσ int

þ lσ z

� �−1

ð12Þ

The inter-CNT contact conductivity at the CNT-CNT interface, σint, canbe modeled using the resistance to electron tunneling in CNT-polymer-CNT capacitor through Simmon's model as follows [18].

Rint ¼dah

2

ae2 2mλð Þ1=2exp

4πδa2mλð Þ1=2

!ð13Þ

σ int ¼da

aRintð14Þ

where, h is Planck constant, λ is the work function for CNTs, e isthe electric charge on an electron in Coulomb, m is the mass of anelectron in kg. The values of these constants are given in Table 1.The parameters a and δa represent the CNT-CNT contact area and

Table 1Physical constants used in Simmon's model.

Planck's constant, J.s 6.626068×10-34

Mass of an electron, kg 9.10938291×10-31

Electric charge on an electron, coulombs 1.602176565×10-19

CNT-CNT work function, eV 5

132 G. Pal, S. Kumar / Materials and Design 89 (2016) 129–136

the gap between two overlapping CNTs after the formation of con-ductive network, respectively. In the present work, projection ofone CNT crossing over another CNT at right angle is taken as theCNT-CNT contact area which is the minimum contact area betweentwo overlapping CNTs.

In the Eq. (13), Simmon's model for interface resistance, theinter-CNT gap after percolation, δa, is the only unknown parameter.At the percolation limit, electron tunneling through the thin layerof polymer between two CNTs is the primary source of electrical con-ductivity of the composite. The thickness of this thin polymer layer atthe percolation threshold, δc, defines the upper limit of δa. Li et al.[23] calculated the value of δc to be equal to 1.8 nm for overlappingCNTs. As the volume fraction of CNTs is increased above the percola-tion limit, the thickness of inter-CNT layer, δa, decreases while thenumber of inter-CNT contacts increase. The combined effect ofthese two factors results in increased electrical conductivity ofCNT-polymer composite. When CNT volume fraction increases be-yond the percolation limit, the thickness of inter-CNT gap, δa startsdecreasing from 1.8 nm. This drop in δa continues over a range ofCNT volume fractions bringing CNTs closer to each other. Allaouiet al. [24] studied the inter-CNT layer thickness with respect to CNTvolume fraction for eopxy in the vicinity of percolation thresholdusing Simmon's model (Eq. (13)). They reported that δa near thepercolation threshold is approximately equal to 1.2 nm. However,even with increasesd volume fraction of CNTs, δa can not be reducedto zero due to opposing non-bonding interactions. It attains a mini-mum value at higher CNT volume fractions and can not be reducedbelow this minimum value. Ideally, the minimum value of δa couldbe assumed to be close to 3.4Å which is the interlayer gap inmultiwalled CNTs. However, currently there are no models to deter-mine the lower bound for the inter-CNT gap, δa. In this researchwork, the lower bound, δa,min, for the CNT-polymer composite isdetermined by fitting the electrical conductivity of CNT-polymercomposite calculated for different minimum values of δa to theexperimentally determined electrical conductivity values [25].

3. Effective electrical conductivity of SCF - conductive polymermatrix composites

In this section, a self-consistent formulation is presented to estimatethe effective electrical conductivity of SCF-CNT-polymer composite con-sidering the SCFs as microscopic inhomogeneities (see Fig. 1). Consideran RVE of SCF-CNT-polymer hybrid composite with randomly distribut-ed SCFs in the polymer matrix as shown in Fig. 1(d). Dispersion of CNTsinto the pure polymer converts the non-conducting polymer into aconductingmatrix which in turn increases the achievable limit of effec-tive electrical conductivity for carbon fiber-polymer matrix composites.In this study, the effective electrical conductivity of SCF - conductivepolymer matrix composite is estimated by micromechanics basedhomogenization schemes such as differential effective medium (DEM)theory and self-consistent (SC) scheme considering SCFs asmicroscopicinhomogeneities. In the past, self-consistent schemes have been suc-cessfully employed to calculate the effective transport properties ofthe fractured earth media (rock, reservior, etc.) [26–29]. Among otherschemes, self-consistent scheme is considered more appropriate forcomposites with high volume fractions and agglomerates of inclusionswhich may or may not be interconnected [26]. The self-consistentscheme is based on the assumption that the perturbation in the field

variable due to the presence of inhomogeneity disappears when it isaveraged over the inhomogeneities, i.e.

Xnk¼1

ξ kð Þ σ kð Þ − σeff� �

A� kð Þ ¼ 0 ð15Þ

where, σeff is the effective electrical conductivity of the homogenizedcomposite; ξ(k), σ(k) andA∗(k) are the volume fraction, electrical conduc-tivity and electrical field concentration tensor of kth phase in the com-posite, respectively. In the present case, σeff, σ(k) and A∗(k) are secondorder tensors. The electrical field concentration tensor A∗(k) is given by

A� kð Þ ¼ Iþ P� kð Þ σeff� �−1

: σ kð Þ − σeff� �� −1

ð16Þ

where, P∗(k) is second order depolarization or interaction tensor of thekth inclusion and it depends upon the σeff, shape and the orientationof the inclusions. The depolarization tensor is related to the Green'sfunction, G, as the integral of its second derivative over the volume ofthe inclusion [27].

P �ij ¼ −

∂∂z j

ZΩ

∂G z− z0ð Þ∂zi

dΩ ð17Þ

Consider the composite material with ellipsoidal inclusions withelectrical conductivities along principal directions only (i.e. σ(m) is adiagonal matrix) are embedded in an isotropic conductive matrix. Allinclusions are aligned such that their axis of symmetry directed along3rd direction of the matrix coordinate system. For transversely isotropicellipsoidal inclusions, the principal terms of the depolarizationmatrix inthe plane of the cross-section are given by [29].

P�11 ¼ 1

4

Z 1

−1

1− y2

1− by2dy ¼ h1

2ð18aÞ

P �22 ¼ P �

11 ð18bÞ

b ¼ ξ2scf − 1

ξ2scfð18cÞ

where, ξscf is the aspect ratio of the ellipsoidal SCF inclusion; and

h1 ¼

−1b

ffiffiffiffiffiffiffiffi−b

pþ

ffiffiffiffiffiffiffiffi−1b

r" #arctan

ffiffiffiffiffiffiffiffi−b

p− 1

!; ξscf b 1

23; ξscf ¼ 1

1b

1−12

ffiffiffi1b

r−

ffiffiffib

p" #ln

1þffiffiffib

p

1−ffiffiffib

p !

; ξscf N 1

0BBBBBBBB@

ð19Þ

The principal element of the depolarization tensor along thedirection of the axis of ellipsoid is given as [29]

P �33 ¼ 1

2

Z 1

−1

y2

ξ2scf 1− by2ð Þdy ¼ h2 ð20Þ

and

h2 ¼

−1ξ2scfb

1−

ffiffiffiffiffiffiffiffi−1b

rarctan

ffiffiffiffiffiffiffiffi−b

p !; ξscf b 1

13; ξscf ¼ 1

1ξ2scfb

−1þ 12

ffiffiffi1b

rln

1þffiffiffib

p

1−ffiffiffib

p !

; ξscf N 1

0BBBBBBBB@

ð21Þ

Fig. 3. Comparison of calculated electrical conductivity of CNT-polymer composite(CNT electrical conductivity=1000 S/m) for different values of inter-CNT layer thickness,δa with the experimental results reported by Kim et al. [25].

Fig. 4. Comparison of calculated longitudinal electrical conductivity of aligned CNT-epoxycomposite with the experimental results reported by Cebeci et al. [30].

133G. Pal, S. Kumar / Materials and Design 89 (2016) 129–136

Thus, for ellipsoidal inclusionswhose axis of symmetry (long axis) isoriented along the 3rd direction of the isotropic matrix, the depolariza-tion or interaction tensor, P∗(k), is given as:

P� kð Þ ¼

h12

0 0

0h12

0

0 0 h2

26664

37775 ð22Þ

However, for the oriented inclusions whose axis of ellipsoidalsymmetry is along x-axis or y-axis, the P∗(k) matrix is given byEq. (23a) or Eq. (23b), respectively.

P� kxð Þ

h2 0 0

0h12

0

0 0h12

26664

37775 ð23aÞ

P� kyð Þ

h12

0 0

0 h2 0

0 0h12

26664

37775 ð23bÞ

The resulting self-consistent scheme (Eqs. (15) and (16)) containsthe effective electrical conductivity, σ eff, both as a parameter and asan unknown variable. The problem is solved iteratively with initialvalue of σeff=σ (m) as a helpful starting point. During the iterative pro-cess, the value of σeff is updated at the end of each iteration to obtain aconverged value of σ eff for composite with oriented inclusions. Theprincipal components of the converged effective electrical conductivitytensor, σ eff, determine the longitudinal and transverse componentsof effective electrical conductivity of SCF-conductive polymer matrixcomposite. In case of composite with randomly oriented inclusions,the effective electrical conductivity, σ eff ,random is given by the averageof the principal terms of σ eff as follows.

σ eff;random ¼ 13tr σ eff� �

ð24Þ

4. Results and discussion

4.1. Effective electrical conductivity of the conductive matrix orCNT-polymer composite, σcnp

In the literature, the electrical conductivity of pristine CNTs is report-ed to vary from 103 - 106 S/m. In addition, the CNT aspect ratio, ξcnt, mayalso vary from 102 to 104, depending upon the diameter and length ofindividual CNTs. Therefore, in addition to CNTvolume fraction, variationin electrical conductivity and aspect ratio of CNTs affect the effectiveelectrical conductivity of CNT-polymer composites significantly. Fig. 3shows the comparison of electrical conductivity of CNT-polymer matrixcomposite calculated using δa ,min=7Å and δa ,min=7.5Å for σc = 1000S/m with the experimental results reported by Kim et al. [25]. In thesesimulations, the diameter of CNT and ξ cnt are 20 nm and 1000, respec-tively. The electrical conductivity of matrix, σm=2×10–12 S/m. The in-terfacial electrical conductivity at δa ,min=7Å is equal to 7.6×10–3 S/mand the same at δa ,min=7.5Å is equal to 2.4×10–3 S/m. These homoge-nized CNT inclusionswithσeffx=σeffy andσeffz are randomly distributedin the polymer matrix to construct a three dimensional representativevolume element (RVE). The effective isotropic conductivity of RVE iscalculated by employing micromechanics based Mori-Tanaka method.By comparison, the computational curve with δa ,min=7Å agreesreasonably well with the measured data available in the literature.Therefore, δa ,min=7Å is taken as the minimum value of inter-CNT

layer thickness in further calculations. Similarly, Fig. 4 shows goodagreement between the predicted values of longitudinal electricalconductivity of aligned CNT-epoxy composites and those of experi-mentally determined corresponding values reported by Cebeciet al. [30]. Another series of simulations were performed with differ-ent values of σc ranging from 103 S/m to 105 S/m to study its effect onσcnp. The non-conducting polymer matrix is considered isotropicwith electrical conductivity, σm= 2×10–12 S/m. In these simula-tions, the diameter of CNT was taken to be 20 nm; however, thelength of CNT (l) is varied such that l = {0.2, 2, 20}μm in order tomaintain ξcnt = {100, 1000, 10,000}, respectively. Fig. 5(a) - (c)show the effect of electrical conductivity of CNTs, σc, on the upperlimits of σcnp. In these figures, at low aspect ratio (ξcnt=100), theupper limit of σcnp changes slightly (from 0.63 S/m to 0.64 S/m) as σc

is varied (from 103 S/m to 105 S/m). However, at higher aspect ratio(ξ cnt = 10,000), the upper limit of σcnp shows significant variation(from 65.3 S/m to 212 S/m) as σc is varied (from 103 S/m to 105 S/m).Another critical observation from these plots is that in each case, asthe ξ cnt is increased, the limiting value for σcnp increases while thepercolation threshold decreases. This is due to the fact that CNTs withhigher aspect ratio get connected to each other and form a conductivenetwork at lower η cnt. At the same time, the variation in CNT percola-tion threshold with respect to σc is more prominent at lower aspectratios. Fig. 5(d) compares the effect of ξ cnt on the upper limit of σcnp.The physical parameters of the CNTs and the matrix are same as those

Fig. 5. Electrical conductivity of CNT-polymer composite calculated for different values of ξcnt with CNT diameter = 20 nm and σm = 2 × 10–12 S/m: (a) CNT electricalconductivity =1000 S/m; (b) CNT electrical conductivity =10,000 S/m; (c) CNT electrical conductivity =100,000 S/m; (d) Effect of CNT electrical conductivity on effectiveelectrical conductivity of CNT-polymer composite as a function of ξcnt.

Fig. 6. Effect of ξ scf and η scf on the σeff. Electrical conductivity of conductive matrix,σcnf, = 1 S/m.

134 G. Pal, S. Kumar / Materials and Design 89 (2016) 129–136

reported in the previous case. It shows that the CNT-polymer compos-ites containing CNTs with same aspect ratio but different electrical con-ductivities {103, 104, 105} S/m show comparable upper limit value ofσcnp. This observation signifies that ξ cnt has more pronounced effecton σcnp than σc. Fig. 5(c) shows that for σc=105 S/m and η cnt= 30%,the effective electrical electrical conductivity of the composite increases331 times (from 0.64 S/m to 212 S/m) as ξ cnt is changed from 100 to10,000. On the other hand, for a fixed ξ cnt=10000 and ηcnt= 30%,the electrical electrical conductivity of the composite increases only3.2 times asσc is increased from 103 S/m to 105 S/m. At lower aspect ra-tios i.e., ξ cnt =100, the percolation behavior of CNT-polymer compos-ites changes considerably as σc changes from 104 S/m to 105 S/mwhereas no such abrupt change in percolation behavior is observed athigher ξcnt [as shown by dotted lines in Fig. 5(d)].

4.2. Effective electrical conductivity of SCF-conductive matrix composites

From the previous section, it can be seen that as η cnt is increased,σcnp attains a plateau and asymptotes to its upper limit. Further additionof CNT into the matrix does not produce any significant change in σcnp.However, when SCFs are distributed randomly in the conductivematrix(CNT-polymer composite), the electrical conductivity of resulting SCF-CNT-polymer matrix composite, σeff, increases beyond the upper limitof σcnp.

Fig. 6 shows the variation of σeff for SCF-CNT-polymer composite asa function of volume fraction of randomly distributed SCFs for differentξscf with corresponding upper and lower Hashin-Shtrikaman (HS)bounds. The effective electrical conductivity of isoptropic conductivematrix σcnp was set to 1 S/m. The upper and lower HS bounds for σ eff

are added for the validation of predicted σ eff for hybrid SCF-CNT-polymer composites. In these simulations, the electrical resistivity ofSCF, ρSCF is equal to that of HexTow-IM7 carbon fibers (ρSCF = 1.5 ×10–3 Ω-cm). IM7 Hex-Tow fibres are aerospace grade PAN based fibreswith surface treatment from Hexcel Corporation. Typically, the

diameter of individual carbon fibers ranges over 5–8 μm and the lengthof SCF may vary from 50 to 100 μm to a milimeter. Therefore, a series ofcomposites with ξscf = {10, 20, 50, 100, 200} are considered in our sim-ulations to study their influence on σeff. For a given conductive matrix,(σcnp), variation in ξ scf affects the σ eff significantly. For lower aspectratios, i.e., ξscf=10,σeff continues to rise steadily as the volume fractionof the SCF (ηscf) is increased. As ξscf is increased, randomly distributedSCFs start forming conductive network at lower volume fractions. For

Fig. 7. Carbon fiber hybrid composites: Effect of matrix conductivity and ηscf onσeff (SCF electrical resistivity=1.5 × 10–3Ω-cm): (a) ξscf=10; (b) ξscf=20; (c) ξscf=50; (d) ξscf=200.

135G. Pal, S. Kumar / Materials and Design 89 (2016) 129–136

composites with large ξ scf (≥50), even at low ηscf, σ eff is approximatelytwo orders of magnitude higher than that of the composites containingcarbonfiberswith low ξscf (≤10). Fig. 7 presents the effect of hostmatrixconductivity, σcnp, i.e. the advantage of adding CNTs, on σ eff for a rangeof SCF concentrations. At low fiber aspect ratios (ξscf=10), σeff is moresensitive to hostmatrix conductivity than to the SCF content in the com-posites. For a given ηscf, as the ξscf is increased, the percolation range forthe composites reduces. However, at very low ηscf (≤0.1%), irrespectiveof ξscf, the SCF content lies in the percolation range of SCF - conductivematrix composites and therefore σ eff fails to attain plateau. At thesame time, as the ξscf is increased, σ eff attains the respective plateaulevel for all SCF contents except for very low values of ηscf as explainedearlier. The central theme of this work is to increase σ eff for SCF-polymer composites by adding CNTs. Addition of CNTs into non-conducting polymer gives an advantageous starting point for hybridcomposites and comparatively higher σ eff levels can be achieved with

Table 2Effective electrical conductivity of SCF - conductive matrix composites, σeff.

% ηscf Electrical conductivity of host matrix, S/m

1 5 10 25 100

ξscf = 100.1 1.1 5.1 10.2 25.3 101.01 1.2 5.6 11.1 27.7 110.65 2.2 10.9 21.8 54.0 207.410 22.5 102.6 188.2 379.9 900.4

ξscf = 500.1 1.1 5.3 10.5 26.2 104.21 1.9 9.3 18.5 44.9 164.25 791.6 970.8 1049.9 1163.1 139510 2879 3052.3 3114.3 3229.6 3444.7

ξscf = 1000.1 1.3 6 12 29.5 113.21 120.5 169.4 195.8 238.5 364.25 1705 1776.2 1800.4 1852.4 1989.110 3703.2 3805 3852.8 3901.9 4065.8

reduced SCF loadings. The predicted values of σeff for different hostmatrix conductivities, σcnp, and ξscf are summarized in Table 2.

5. Concluding remarks

A physics-based multi-scale hierarchical modeling approach ispresented to predict the effective electrical conductivity of CNT-SCF-polymer matrix composites. The model takes account of not only CNTconcentration, percolation, interfacial contact resistance, aspect ratio,orientation and CNT's anisotropic conductivity but also SCF concentra-tion, SCF conductivity anisotropy, SCF aspect ratio, and orientation.Model predictions agree well with experimental values reported inthe literature. Results indicate that even at low SCF content (ηscf≤ 1%),adding CNTs in the SCF-polymer composite significantly improves theeffective electrical conductivity and electrical percolation threshold ofthe resulting composite. Electrical conductivity of the compositeimproves as the SCF content and host matrix conductivity increases.However, high SCF aspect ratio (ξscf≥ 50) helps in improving the electri-cal conductivity of the composites even at low SCF concentrations.

Acknowledgements

Authors would like to acknowledge the financial support fromthe Lockheed Martin Corporation for this work (Project code:13NZZA1). Authors also thank Dr. Tushar Shah of Lockheed MartinCorporation, USA.

References

[1] P. Upadhyaya, S. Kumar, Micromechanics of stress transfer through the interphasein fiber-reinforced composites, Mech. Mater. 89 (2015) 190–201.

[2] J. Liljenhjerte, S. Kumar, Pull-out performance of 3d printed composites withembedded fins on the fiber, MRS Proceedings, vol. 1800, Cambridge Univ Press,2015 (pages mrss15–2135597).

[3] F. Rezaei, R. Yunus, N.A. Ibrahim, Effect of fiber length on thermomechanicalproperties of short carbon fiber reinforced polypropylene composites, Mater. Des.30 (2) (2009) 260–263.

136 G. Pal, S. Kumar / Materials and Design 89 (2016) 129–136

[4] Y. Cui, S. Kumar, B.K. Rao, D. Van Houcke, Gas barrier properties of polymer/claynanocomposites, RSC Adv. (2015).

[5] I.E. Sawi, P.A. Olivier, P. Demont, H. Bougherara, Processing and electrical character-ization of a unidirectional CFRP composite filled with double walled carbon nano-tubes, Compos. Sci. Technol. 73 (2012) 19–26.

[6] K.L. Kepple, G.P. Sanborn, P.A. Lacasse, K.M. Gruenberg, W.J. Ready, Improved frac-ture toughness of carbon fiber composite functionalized with multi walled carbonnanotubes, Carbon 46 (15) (2008) 2026–2033.

[7] E.J. Garcia, B.L. Wardle, A.J. Hart, N. Yamamoto, Fabrication and multifunctionalproperties of a hybrid laminate with aligned carbon nanotubes grown in situ,Compos. Sci. Technol. 68 (9) (2008) 2034–2041.

[8] J. Blanco, E.J. Garca, R. Guzmn de Villoria, B.L. Wardle, Limiting mechanisms of modeI interlaminar toughening of composites reinforced with aligned carbon nanotubes,J. Compos. Mater. 43 (8) (2009) 825–841.

[9] S.S. Wicks, R. Guzman de Villoria, B.L. Wardle, Interlaminar and intralaminarreinforcement of composite laminates with aligned carbon nanotubes, Compos.Sci. Technol. 70 (1) (2010) 20–28.

[10] R. Hollertz, S. Chatterjee, H. Gutmann, T. Geiger, F.A. Nesch, B.T.T. Chu,Improvement of toughness and electrical properties of epoxy composites withcarbon nanotubes prepared by industrially relevant processes, Nanotechnology22 (12) (2011) 125702.

[11] L. Boger, M.H.G. Wichmann, L.O. Meyer, K. Schulte, Load and health monitoring inglass fibre reinforced composites with an electrically conductive nanocompositeepoxy matrix, Compos. Sci. Technol. 68 (78) (2008) 1886–1894.

[12] V. Kostopoulos, A. Vavouliotis, P. Karapappas, P. Tsotra, A. Paipetis, Damagemonitoring of carbon fiber reinforced laminates using resistance measurements.improving sensitivity using carbon nanotube doped epoxy matrix system, J. Intell.Mater. Syst. Struct. 20 (9) (2009) 1025–1034.

[13] F. Aviles, J.O. Aguilar, J.R. Bautista-Quijano, Influence of carbon nanotube clusteringon the electrical conductivity of polymer composite films, Express Polym. Lett. 4(5) (2010) 292–299.

[14] F.H. Gojny, M.H.G. Wichmann, B. Fiedler, I.A. Kinloch, W. Bauhofer, A.H. Windle, K.Schulte, Evaluation and identification of electrical and thermal conduction mecha-nisms in carbon nanotube/epoxy composites, Polymer 47 (6) (2006) 2036–2045.

[15] T. Takeda, Y. Shindo, Y. Kuronuma, F. Narita, Modeling and characterization of theelectrical conductivity of carbon nanotube-based polymer composites, Polymer 52(17) (2011) 3852–3856.

[16] M. Mohiuddin, S.V. Hoa, Estimation of contact resistance and its effect on electricalconductivity of cnt/peek composites, Compos. Sci. Technol. 79 (2013) 42–48.

[17] K.Y. Yan, Q.Z. Xue, Q.B. Zheng, L.Z. Hao, The interface effect of the effectiveelectrical conductivity of carbon nanotube composites, Nanotechnology 18(25) (2007) 255705.

[18] C. Feng, L. Jiang, Micromechanics modeling of the electrical conductivity ofcarbon nanotube (cnt)polymer nanocomposites, Compos. A: Appl. Sci. Manuf.47 (2013) 143–149.

[19] B. De Vivo, P. Lamberti, G. Spinelli, V. Tucci, Numerical investigation on the influencefactors of the electrical properties of carbon nanotubes-filled composites, J. Appl.Phys. 113 (24) (2013).

[20] D.A. Jack, C.-S. Yeh, Z. Liang, S. Li, J.G. Park, J.C. Fielding, Electrical conductivitymodeling and experimental study of densely packed swcnt networks,Nanotechnology 21 (19) (2010) 195703.

[21] F. Deng, Q.-S. Zheng, An analytical model of effective electrical conductivity ofcarbon nanotube composites, Appl. Phys. Lett. 92 (7) (2008).

[22] Q.Z. Xue, Model for the effective thermal conductivity of carbon nanotube compos-ites, Nanotechnology 17 (6) (2006) 1655.

[23] C. Li, E.T. Thostenson, T.-W. Chou, Effect of nanotube waviness on the electricalconductivity of carbon nanotube-based composites, Compos. Sci. Technol. 68 (6)(2008) 1445–1452.

[24] A. Allaoui, S.V. Hoa, M.D. Pugh, The electronic transport properties andmicrostructure of carbon nanofiber/epoxy composites, Compos. Sci. Technol.68 (2) (2008) 410–416.

[25] Y.J. Kim, T.S. Shin, H.D. Choi, J.H. Kwon, Y.-C. Chung, H.G. Yoon, Electrical conductiv-ity of chemically modified multiwalled carbon nanotube/epoxy composites, Carbon43 (1) (2005) 23–30.

[26] J.G. Berryman, G.M. Hoversten, Modelling electrical conductivity for earthmedia with macroscopic fluid-filled fractures, Geophys. Prospect. 61 (2)(2013) 471–493.

[27] A. Giraud, C. Gruescu, D.P. Do, F. Homand, D. Kondo, Effective thermal conductivityof transversely isotropic media with arbitrary oriented ellipsodal inhomogeneities,Int. J. Solids Struct. 44 (9) (2007) 2627–2647.

[28] J.-F. Barthelemy, Effective permeability of media with a dense network of long andmicro fractures, Transp. Porous Media 76 (1) (2009) 153–178.

[29] P.A. Fokker, General anisotropic effective medium theory for the effectivepermeability of heterogeneous reservoirs, Transp. Porous Media 44 (2) (2001)205–218.

[30] H. Cebeci, R.G. de Villoria, A.J. Hart, B.L. Wardle, Multifunctional properties of highvolume fraction aligned carbon nanotube polymer composites with controlledmorphology, Compos. Sci. Technol. 69 (1516) (2009) 2649–2656.