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Polymer 55 (2014) 5488e5499

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Polymer

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Carbon nanotube agglomeration effect on piezoresistivity of polymernanocomposites

S. Gong a, Z.H. Zhu a, *, S.A. Meguid b

a Department of Earth and Space Science and Engineering, York University 4700 Keele Street, Toronto, ON M3J 1P3, Canadab Department of Mechanical and Industrial Engineering, University of Toronto 5 King's College Road, Toronto, ON M5S 3G8, Canada

a r t i c l e i n f o

Article history:Received 22 June 2014Received in revised form18 August 2014Accepted 24 August 2014Available online 10 September 2014

Keywords:Carbon nanotube agglomerationPiezoresistivityPolymer-matrix composites

* Corresponding author. Tel.: þ1 416 736 2100x777E-mail address: gzhu@yorku.ca (Z.H. Zhu).

http://dx.doi.org/10.1016/j.polymer.2014.08.0540032-3861/© 2014 Elsevier Ltd. All rights reserved.

a b s t r a c t

Carbon nanotube (CNT) agglomeration exists inevitably in all CNT-polymer composites. This paperquantified the effect of CNT agglomeration on the piezoresistivity of CNT-polymer composites. A newmultiscale model of 3-dimensional deformable CNT percolating networks has been developed, where theCNT agglomerates were modeled as second phases embedded randomly in the polymer matrix. Thenewly developed model agrees quantitatively with experimental data. The study found that the CNTagglomeration is responsible for the reduced electrical conductivity and nonlinearity of piezoresistivitywith respect to the zero strain. Its effect can be quantified by the newly developed model. Parametricanalyses were conducted to show the effects of morphology and electrical properties of CNTs, thePoisson's ratio of CNT-polymer composites and the extent, internal density and size of CNT agglomer-ation on the electrical conductivity and piezoresistivity. The current work provides a useful analysis toolfor designing smart sensing and multifunctional polymer composites.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Carbon nanotubes (CNTs) have great potential in fabricatinghigh-performance and multifunctional polymer composites due totheir good electrical conductivity, ultra-small diameter, high aspectratio, and lightweight [1e5]. The performance of these compositesis highly affected by the distribution of CNTs in polymers, where theCNTs tend to attract each other and form large particle clusters oragglomerates [6]. Investigations find that the presence of CNT ag-glomerates will significantly reduce the electrical properties ofCNT-polymer composites compared with theoretical predictionsthat are based on the assumption of uniform distribution of CNTs inpolymers [7e9]. Many efforts have been devoted to disperse CNTsinto polymer matrices uniformly [10e14] in order to reach thetheoretical properties of CNT-polymer composites. Althougheffective, studies founded that even the well-dispersed CNTs inpolymers could re-agglomerate in the subsequently curing process[15]. The situation becomes worse at high CNT loadings. Therefore,it is imperative to quantify the effect of CNT agglomeration on theelectrical properties precisely since the CNT agglomeration isinevitable.

29; fax: þ1 416 736 5817.

Currently, many progresses have been made in this area.Compared with the extensive experimental works [16e18], rela-tively limited analytical and numerical efforts have been devoted tothis field [19]. Analytical studies usually adopt the Mori-Tanakaaveraging method to model the piezoresistivity by the averageelectric field. For instance, Seidel and Lagoudas [20] estimated theelectrical conductivity of CNT-polymer composites by the Mori-Tanaka method in the micromechanics domain. Using the similarapproach, Pham [21] evaluated the effective piezoresistivity of CNT-polymer composites based on the nanoscale Simmons' tunnelingresistance [39] at CNT junctions. Wichmann et al. [22] investigatedthe influence of strain range on the piezoresistivity by a combina-tion of Simmons' tunneling resistance and simplified CNT resistornetworks. Ren and Seidel [9] studied the macroscale piezor-esistivity of CNT-polymer composites based on the nanoscaleSimmons' tunneling resistance and CNT intrinsic piezoresistivity bya computational micromechanics finite element approach.Although successful, these models predict the piezoresistivity ofCNT-polymer composites qualitatively and no CNTagglomeration isconsidered. To provide accurate and quantitative predictions of thepiezoresistivity, numerical approaches have been developed, whichare generally based on the statistical percolating network theory ofrandomly distributed CNTs in polymers and the Monte Carlo sim-ulations. For instance, Theodosiou et al. [24] analyzed the nanoscalepiezoresistive response of CNTs and then derived averaged or

Fig. 1. Schematic of (a) an electrical conductive CNT network in a polymer matrix, (b)detailed CNTs and CNT junctions in network, and (c) electrical resistance of a typicalconductive path (Thick solid, dotted and thin solid lines represent conductive pathsformed by CNTs, CNT junctions and CNT parts that are not involved in a conductivepath, respectively).

S. Gong et al. / Polymer 55 (2014) 5488e5499 5489

effective piezoresistivity of CNT-polymer composites at macroscaleby a numerical CNT percolating network model. Yasuoka et al. [25]showed that the macroscale piezoresistivity is nonlinear due to thetunneling effect at CNT junctions and is inversely proportional toCNT loadings by an electrical circuit analog to percolating networks.In the report by Alamusi et al. [26], the piezoresistivity of CNT-polymer composites was modeled by assuming CNTs translatingand rotating rigidly in polymer matrices when subject to anexternal strain [27]. Li and Chou [28] analyzed the effective elec-trical resistance of the percolating network by considering thenanoscale tunneling and intrinsic resistances of CNTs and thenaveraging through the finite element method of an electrical circuitanalog to percolating networks. Wang et al. [29] studied numeri-cally the dependence of piezoresistivity on the Poisson's ratio ofpolymer matrix by power-law-type resistor networks analog to theCNT percolating networks. Up to date, none of these analytical ornumerical approaches can accurately quantify the significantreduction in electrical conductivity, piezoresistivity, and piezor-esistive sensitivity to CNT loadings of CNT-polymer composites dueto CNT agglomeration.

To address the discrepancy between the experiments and thetheory, a 3D multiscale percolating network model has beendeveloped here to model the electrical conductivity and piezor-esistivity of CNT-polymer composites quantitatively, where theeffects of CNT agglomeration and deformation (wall deformationand tube-bending) are considered. The effect of CNT agglomerationon the electrical properties has been modeled at three differentscales, ranging from the nanoscale, to the mesoscale and to themacroscale. Themodel agrees quantitatively with the experimentaldata [31] of CNT-polymer composites with and without significantCNT agglomeration. The analysis results reveal that the CNTagglomeration is responsible for the reduced electrical conductiv-ity, piezoresistivity, sensitivity of piezoresistivity to CNT loadingsand nonlinearity of piezoresistivity with respect to zero strain. Thestudy also finds that the CNT deformation is one of the majormechanisms that affect the electrical conductivity and piezor-esistivity. Finally, parametric studies show the different de-pendences of piezoresistivity of CNT-polymer composites on thePoisson's ratio of composites, the size of CNT agglomerates, and themorphology and intrinsic resistance of CNTs.

Compared to the existing models, the novelty of the new modelis that the CNTagglomerates are considered andmodeled as secondphases embedded in the 3D CNT percolating network and the CNTbending and wall deformation are modeled at CNT junctions. Thenew model improves the representation of 3D CNT percolatingnetwork for agglomerated CNT-polymer composites and is capableof quantifying the effect of CNT agglomeration on composites'electrical conductivity and piezoresistivity. The outcome of thecurrent work would provide useful information and an optimiza-tion tool for the design of smart sensing and multifunctionalpolymer composites.

2. Multiscale modeling methodology

2.1. CNT percolating network in a polymer matrix

It is well known that the electrical resistance of CNT-polymercomposites can be represented by the resistance of CNT perco-lating networks inside the composites [3,32]. As shown in Fig. 1(a),each CNT in the network is assumed as a straight and caped cyl-inder with length of L and diameter of D initially. The CNTs areassumed deformable (tube-bending or wall flattening) to avoid CNToverlap at crossed CNT junctions in the network [33]. Assume CNTsare randomly distributed in a representative volume cuboid withdimensions of Lx � Ly � Lz. Thus, the coordinates of starting point

and the azimuthal and polar angles (fi, qi) of each CNT are gener-ated randomly [32],

�x0i ; y

0i ; z

0i

� ¼ �Lx � Rand; Ly � Rand; Lz � Rand

�;�

fi; qi� ¼ �

2p� Rand; cos�1�2Rand� 1

�� (1)

where i is the index of i-th CNT and ‘Rand’ denotes uniformlydistributed random numbers in the interval [0, 1].

S. Gong et al. / Polymer 55 (2014) 5488e54995490

Then, the ending point of the CNT can be determined accord-ingly, such that.

�x1i ; y

1i ; z

1i

�¼

�x0i þ li sin qi cos fi; y

0i þ li sin qi sin fi;

z0i þ li cos qi� (2)

where li is the length of i-th CNT, which is assumed to obey aWeibull distribution F(x) ¼ 1 � exp(�(x/a)b) [23].

It should be noted that the CNTs generated by Eq. (2) mightprotrude outside the cuboid and overlap or penetrate each other atCNT junctions, see Fig. 2. In the former case, the protruded parts ofCNTs are truncated by boundary planes of the cuboid and thenrelocated back to the cuboid by the periodic boundary conditions asper Ref. [34] to maintain the required CNT loadings. The latter casewill be muchmore complicated but has to be addressed properly toavoid (i) the incorrect CNT loading calculation and (ii) the over-estimation of the electrical conductivity at CNT junctions [33]. Thefollowing is the approach developed in the present work.

Once the position and orientation of CNTs are determined, theshortest distances (d) between each pair of straight CNTs arecalculated to determine if the pair is in contact or not. If the min-imum distance betweenwalls of the paired i-th and j-th CNTs (dij eD) is less than the maximum effective distance (dcutoff) of tunnelingeffect, the CNT pair is considered in contact, see Fig. 1(b). Among allembedded CNTs, only the segments of contacted CNTs that formconductive paths contribute to the network conductance. Due tothe Van der Waals repellent force, the walls of contacting CNTs, orthe wall of CNT and surrounding polymer matrix, are separated bythe so-called Van der Waals distance dvdw [23]. Accordingly, theinter-tube distance d should be greater than D þ dvdw, or D þ 2dvdwin case there is a polymer film between CNTs. If the distance be-tween walls of paired CNTs is less than the Van der Waals distancedvdw, the walls of paired CNTs deforms or flattens to maintain theVan der Waals distance. This is evident by the experimentalobservation in Ref. [35]. It is further found experimentally that theVan der Waals distance could be reduced if the CNT walls wereflattened [34,36,37]. Accordingly, a 3D deformable CNT percolatingnetworkmodel was introduced in Ref. [33], where thewalls of CNTswere assumed deformable and could be flattened completely bythe Van der Waals force. The relationship between the imposed theinter-tube distance and the wall deformation can be determined byminimizing a pseudo-potential energy at CNT junctions by thecoarse grain atomistic simulation. This model showed that theelectrical conductivity at CNT junctions was significantly reduceddue to CNT wall deformation and simulation results agreed with

Fig. 2. Schematic of deformable CNT mo

the experiments better than the rigid CNT model. However, it isfound that the CNTs cannot be completely flattened in reality andthey bent as a beam after the wall deformation reached certainlimits [35]. Therefore, the straight CNTs at CNT junctions should bereplaced by bent CNTs in the model if the wall distance of twostraight CNTs is less than the allowed minimum Van der Waalsdistance between deformed CNTs. For the sake of simplicity, eachbent CNT is replaced by two straight segments that form an angle atCNT junctions in the present model. Fig. 2 shows the deformableCNT model at a CNT junction. Note that the image of CNTs is forillustrative purpose only and the CNTs may be single- or multi-walltype. This is simply an imposed bending to reflect the experimentalobservation [35] to avoid the CNT overlap at CNT junctions. Oncethe CNT percolating network is revised after incorporating all CNTdeformation at junctions, the resistance of conductive paths iscalculated in the following.

2.2. Electrical resistance of CNT percolating network

2.2.1. Electrical resistance of undeformed CNT percolating networkThe electrical resistance of CNT percolating network consists of

two parts, namely, (i) the intrinsic resistance of CNTs and (ii) thecontact or tunneling resistance resulting from electron tunneling atcrossed CNT junctions [34]. As shown in Fig. 1(c), the intrinsicresistance of a CNT between contact points 1 and 2 can be evaluatedas [34],

Rint ¼4l12

pscntD2 (3)

where l12 is the length of the CNT segment between points 1 and 2and scnt is the CNT intrinsic electrical conductivity that is sensitiveto CNT's chirality, diameter, length, etc. In the present study, the scntis an effective mean conductivity determined experimentally.

The contact or tunneling resistance at crossed CNT junctionsdepends on the distance between adjacent CNTs, their geometryand chirality, and the contact area among others. The current at thejunction can be described by the Landauer-Büttiker formula[23,38], since the contact region between CNTs is at the nanoscale.By solving the Landauer-Büttiker formula with a rectangular po-tential barrier, the contact resistance of an undeformed CNT junc-tion can be evaluated as [34,40],

Rc ¼ h2e2 MT

(4)

with

del derived from rigid CNT model.

T ¼

8>>>>>>><>>>>>>>:

exp� �ðd� DÞh ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

8meWcntp

�Dþ dvdw < d � Dþ 2dvdw

exp

0BB@ �ðd� D� 2dvdwÞh� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

8me

�Wcnt �Wpoly

�r þ �2dvdwh ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

8meWcntp

1CCA Dþ 2dvdw < d � dcutoff

(5)

S. Gong et al. / Polymer 55 (2014) 5488e5499 5491

where M is the total number of conduct channels in integer formdepending on CNT wall types, T denotes the transmission proba-bility for electrons, Wcnt and Wpoly are the work functions of CNTsand polymer matrix, respectively. Other parameters (h, e, me) arethe Planck constant, electron charge and mass, respectively.

2.2.2. Electrical resistance of deformed CNT percolating networkConsider a pair of deformed CNTs (j-th and k-th CNTs) at a

crossed CNT junction as shown in Figs. 1(b) and 2. Assume D0 is thenominal diameter of the deformed CNT and can be expressed as

D0 ¼ 12�d� d0vdw þ D

�(6)

T ¼

8>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>:

exp

0B@ �d0vdw min

h. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

8meW 0cnt_min

q1CA 0<d � �

2D0min � D

�þ d0vdw

exp

0B@ �d0vdw

h� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

8meW 0cnt

q1CA �

2D0min � D

�þ d0vdw < d � Dþ dvdw

exp� �ðd� DÞh ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

8meWcntp

�Dþ dvdw <d � Dþ 2dvdw

exp

0BB@ �ðd� D� 2dvdwÞh� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

8me

�Wcnt �Wpoly

�r þ �2dvdwh ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

8meWcntp

1CCA Dþ 2dvdw < d � dcutoff

(9)

Here, d0vdw is the Van der Waals distance between the walls of

adjacent deformed CNTs and D is the diameter of CNT beforedeformation. The experimental observations found that theintrinsic electrical conductivity of CNT is a function of wall defor-mation and can be expressed as [41],

scnt_d ¼ scnt exp�� h

�1� D0

D

�(7)

where h is the change rate of the intrinsic electrical conductivityof CNT under wall deformation and is determined experimen-tally. Eq. (7) shows that the wall deformation of CNTs will reducethe intrinsic electrical conductivity of CNT exponentially. Corre-spondingly, the intrinsic resistance of deformed/flattened CNTsegment between points 4 and 5 as shown in Fig. 1(c) is modifiedfrom Eq. (3) as,

Rint_d ¼Zl450

4

pscnt_dðD0Þ2dl ¼ Rint

Z10

�D0

D

��2

exp�h

�1� D0

D

�ds

(8)

where Rint is the intrinsic resistance of undeformed segment be-tween points 4 and 5, ds ¼ dl/l45 is the dimensionless length of CNTsegment. Obviously, the intrinsic resistance of a deformed CNTsegment is greater than its undeformed status.

Accordingly, the transmission probability for electrons to tunnelthrough deformed CNTs is revised, such that,

where D0min is the minimum diameter of deformed CNT, d0

vdw min isthe minimum distance between deformed CNTs, and W 0

cnt is thework function of deformed CNTs, which can be approximated bycurve fitting of the calculated values, such that [42,43],

W 0cnt ¼ Wcnt þ r

�1� D0

D

�(10)

where r is the change rate of work function under radial defor-mation. Then, the contact resistance at deformed CNT junctions(Rc_d) is calculated by Eq. (4).

The unknown parameter D0 or d0vdw can be determined by

minimizing a pseudo-potential energy at CNT junctions [33]. Itdepends on CNT's chirality, diameter, wall types (single or multi-ple). Once the CNTs are specified, the relationship between theinter-tube distance d and the parameter D0 or d0

vdw at CNT junctions

Fig. 3. Schematic of multiscale model for CNT agglomeration.

S. Gong et al. / Polymer 55 (2014) 5488e54995492

can be determined uniquely by the coarse grain atomistic simula-tion [33]. In the current study, this relationship is used as an inputfunction to the CNT percolating network model since only one typeof CNTs will be used in themodel each time. Any further decrease inthe inter-tube distance d, resulting from Eq. (2), will lead to thebending of CNTs rather than further flattening CNT walls todecrease the distance d0

vdw in order to reflect reality observed in theexperiment [35]. After the percolating network of deformed CNTs isbuilt with all variables associated with the deformation evaluated,the total resistivity of the CNT percolating network can be solvednumerically by the DulmageeMendelsohn decomposition and theKirchhoff's circuit law [44,45].

Fig. 4. (a) CNT agglomerates-conductive polymer RVC at the mesoscale and (b) CNT-polymer RVC at the nanoscale with CNT re-orientation model.

2.3. Multiscale CNT agglomeration model

The multiscale CNT agglomeration model considers the CNTagglomerates as second phase clusters embedded randomly inCNT-polymer composites in the macroscale as shown in Fig. 3.The total volume of CNTs (VCNT_total) can be divided into the twoparts: the total volume of agglomerated CNTs (VCNT_agg) insidethe second phases and the total volume of non-agglomeratedCNTs (VCNT_nonagg) in the matrix. All these second phases areassumed sphere-like [46] with the same shape, size and CNTloading in the current model. Inside the second phases and therest of non-agglomerated region, the CNTs are assumed to beuniformly distributed to form CNT percolating networks but withdifferent CNT loadings. Three parameters are introduced todescribe the state of CNT aggregation: the volume ratio ofagglomerated CNTs over the total CNTs (xagg ¼ VCNT_agg/VCNT_total),the nominal radius (ragg) of sphere-like second phases, and thevolume fraction of CNTs (Vf_CNT_agg) inside the second phases.The effective or nominal CNT loading of the CNT-polymer com-posites is obtained by the volume-weighted average of CNTloadings in the second phases and the CNT-polymer matrix. Theeffective electrical conductivity of the composites will be evalu-ated at three different scales as shown in Fig. 3. First, the CNTpercolating networks are generated in the second phases at thenanoscale. The averaged or effective electrical conductivity of thesecond phases at the mesoscale (micrometer scale) is determinedby evaluating the resistance of CNT percolating networks basedon the nanoscale intrinsic and contact resistances of CNTs by Eqs.

(8) and (9) and then averaging by Monte Carlo simulations. Thisprocess is shown in the upper part of Fig. 3. Second, the CNTpercolating networks are generated in the non-agglomeratedregion with the second phases embedded randomly. The sec-ond phases are treated as conductive fillers with the same elec-trical conductivity, shape and size. They are interconnected byCNTs in the polymer matrix, as shown in the lower right part ofFig. 3. Three types of conductance are involved: the tunnelingconductance at CNT junctions, the CNT itself, and the contactbetween CNTs and second phases. The same procedure used inevaluating the electrical conductivity of second phases is usedagain to find the conductive paths in the polymer matrix butwith the consideration of second phases and their associatedresistances. In the process, the CNTs are allowed to pass throughthe second phases and the contact between CNTs and secondphases is assumed perfect with zero contact resistance if the CNTtouches or passes through the surface of a second phase. Finally,the macroscale electrical conductivity of composites is averagedthrough Monte Carlo simulations.

S. Gong et al. / Polymer 55 (2014) 5488e5499 5493

2.4. Modelling of piezoresistivity of CNT-polymer composites

Consider a CNT-polymer composite experiences a uniaxial strainchange Dε in the x-axis direction as shown in Fig. 4. Generally, thesecond phases or CNT agglomerates in the polymer matrix areassumed as non-deformable spheres [47,48]. Only their positionsare changed in accordancewith the strain of thematrix as shown inFig. 4(a). The response of CNTs to the applied strain is modeled bythe widely accepted fiber re-orientation model [26,27] in theexisting CNT percolating network approaches, as shown in Fig. 4(b).In addition, the bending and wall deformation of CNTs [49] will beconsidered at the nanoscale after applying the strain.

The re-orientation model assumes that the midpoint ðxci ; yci ; zci Þof i-th CNT or the centroid of CNT agglomerate are moved trans-lationally in accordance with the magnitude of strain Dε and thePoisson's ratio of the composite n, such that,

�~xci ; ~y

ci ;~z

ci� ¼ �

xci þ xciDε; yci � nyciDε; z

ci � nzciDε

�(11)

Then, the starting and ending points of i-th CNT after strain canbe expressed as

�~x0i ; ~y

0i ;~z

0i

�¼ �

~xci ; ~yci ;~z

ci�� h�

~xci ; ~yci ;~z

ci�� �

x0i ; y0i ; z

0i

�i��li.~li�;�

~x1i ; ~y1i ;~z

1i

�¼ �

~xci ; ~yci ;~z

ci�þ h�

x1i ; y1i ; z

1i

�� �

~xci ; ~yci ;~z

ci�i� �

li.~li�

where

�x0i ;y

0i ;z

0i

�¼�x0i þ x0i Dε;y

0i � y0i nDε;z

0i � z0i nDε

��x1i ;y

1i ;z

1i

�¼�x1i þ x1i Dε;y

1i � y1i nDε;z

1i � z1i nDε

�~li ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�x1i � x0i

�2ð1þDεÞ2þ��

y1i � y0i�2þ�

z1i � z0i�2ð1� nDεÞ2

s

(12)

Obviously, the positions of crossed CNT junctions will bechanged due to the applied strain as shown in Fig. 5. Themovement can be decomposed into the normal and axial di-rections of a CNT. The normal component generally causes adistance change perpendicular to CNTs, while the axial compo-nent causes a junction moving along the CNT axis by a distance

Fig. 5. Schematic of CNT junction movement: (a) the normal component and (b) theaxial component. dm is the moving distance of junction along CNT before and afterstrain.

dm as shown in Fig. 5. In the first case, the contact and intrinsicresistance at the junction varies due to the variation of inter-tubedistance, as indicated by Eqs. (4), (8) and (9). Statistically, moreCNT junctions will move apart when subject to a tensile strain,although the Poisson's ratio effect will push some together in thetransverse directions. This will lead to a significant increase ofcontact resistance or even the breakdown of conducting CNTjunctions if the distance between CNTs increases beyond thecutoff distance. Therefore, the tensile strain will lead to an overallincrease in the total electrical resistance. This is in agreementwith the experimentally observed piezoresistive effect: a positivestrain causes a positive variation of resistance in the CNT-polymer composites. In contrast, the compressive strain gener-ally pushes CNTs together statistically, although the Poisson'sratio effect will pull some apart in the transverse directions. Thiswill create more conductive paths in the percolating networks byforming more CNT junctions and lead to an overall decrease inelectrical resistance, which is also in agreement with experi-mental observations - a negative strain causes a negative varia-tion of resistance. For the existing CNT conjunctions, the closerthey get, the contact resistance decreases but the intrinsicresistance in the vicinity of CNT junction will increase if the CNTdeformation occurs.

The variations of contact and intrinsic resistances with differentdistance between CNTs are shown in Fig. 6. For the given CNTs, itshows that the allowed minimum distance between deformedCNTs is around 5.5 nm. The existing models [23,26], assume con-stant contact and intrinsic resistances at CNT junctions ifd < D þ dvdw and there will be no piezoresistive response in thisregion, leading to the underestimation of overall piezoresistivity.Different from the existing models, the new model allows contin-uous variation of the contact and intrinsic resistances in this region,which will contribute to the overall piezoresistivity.

Finally, the strain gauge factor (K) of the CNT-polymer com-posites is calculated by comparing the electrical resistance changesbefore and after strain with respect to the applied strain,

K ¼ DR=R0Dε

(13)

where R0 is the electrical resistance before strain and DR is thestrain-induced variation of electrical resistance.

Fig. 6. Variations of contact and intrinsic resistances vs. inter-tube distance.

Table 1Physical properties of MWCNTs.

Diameter(nm)

Length(mm)

Aspect ratio(�)

Intrinsicconductivity(S m�1)

Type 1 MWCNTExperiment [31] 40e90 10e30 >100 1 � 104

Simulation 65 20 (a ¼ 21.9971,b ¼ 2.4)a

300 1 � 104

Type 2 MWCNTExperiment [31] <10 N/A 500e1500 1 � 104

Simulation 8 8 (a ¼ 9.0244,b ¼ 2.4)*

1000 1 � 104

a Parameters of the Weibull distribution for CNT length.

S. Gong et al. / Polymer 55 (2014) 5488e54995494

3. Results and discussion

3.1. Validation of newly developed deformable CNT percolatingnetwork model

The newly developed deformable CNT percolating networkmodel is validated against the existing experimental data inRef. [31]. Table 1 shows the properties of two types of multi-walledcarbon nanotubes (MWCNTs) used in the experimental investiga-tion. The actual diameters and lengths of the type 1 MWCNTsranged from 40 to 90 nm and from 10 to 30 mm, respectively.Accordingly, the averages of 65 nm and 20 mm were chosen as thenominal CNT diameter and length in the simulation, respectively.This results in the parameters of the Weibull distribution of CNTlength being a ¼ 21.9971 mm and b ¼ 2.4. The actual diameter oftype 2 MWCNTs was less than 10 nm with aspect ratios rangingfrom 500 to 1500. Thus, the nominal diameter and aspect ratio ofCNTswere set to 8 nm and 1000 in the simulation, respectively. Thisresults in the parameters of the Weibull distribution of CNT lengthbeing a ¼ 9.0244 mm and b ¼ 2.4. The experimental results inRef. [31] showed that these two types of CNTs resulted in differentmicrostructures in CNT-polymer composites. The type 1 MWCNTswere quite thick with relative low aspect ratios. It was relativelyeasy to disperse uniformly in polymers and no obvious agglomer-ation was observed in the experiment. Therefore, no effect of CNTagglomeration was considered in the simulation for the type 1MWCNT-polymer composites. Compared with the type 1 MWCNTs,the type 2MWCNTs were thinwith relative high aspect ratio. It wasmore difficult to disperse uniformly in polymers. They wereobserved bent, tangled and aggregated in polymers. Thus, the effectof CNT agglomeration cannot be ignored in the simulation. Basedon the experimental observation in Ref. [31], the state of CNTagglomeration is estimated as follows: (i) the shape of agglomer-ations is assumed as near spherical with a nominal radius (ragg)being 10 mm; (ii) the CNT loading in the agglomerationsVf_CNT_agg ¼ 15 vol.%; and (iii) the volume ratio of the agglomeratedCNTs over the total CNT in the CNT-polymer composite (xagg) is0.25. By changing the numbers of second phases in the composites,

Table 2Physical parameters used in the simulation.

Items Values Refs.

Van der Waals distance, dvdw 0.34 nm [50]Cutoff distance, dcutoff 1.4 nm [34]Conduction channels, M 460 [38]Change rate of scnt, h 5 [33]Poisson's ratio, v 0.3 [33]CNT, Wcnt 4.7 eV [33]Polymer, Wpoly 4.0 eV [51]

different nominal CNT loadings of CNT-polymer composites can beobtained by the volume-weighted average of CNT loadings ofagglomerated and non-agglomerated regions in the simulation.

In addition to these CNT type-specific parameters, the followingparameters (as shown in Table 2) were selected in the simulationbased on previous studies [33,34,38]. Considering the facts thatsome parameters in real samples might be changed during fabri-cation process, typical values within their possible ranges wereassumed in the simulation.

The comparisons of numerical predictions and experimentalmeasurements of electrical conductivity, piezoresistivity, and straingauge factor of type 1 MWCNT-polymer composites are shown inFig. 7. It shows that the predicted electrical conductivity agrees wellwith the experimental measurements in Fig. 7(a). The criticalpercolating threshold occurs near the weight fraction of 1 wt.%.Next, the real strain gauge factors were different in the tensile andcompressive strain states, where the gauge factor in tension ishigher than compression. Compared with the experimental data,the model predicts this characteristic satisfactorily as shownFig. 7(a) in the tensile and compressive strain states respectively.Very high sensitivity can be achieved, especially when the CNTloading is near the percolating threshold. The gauge factors in thetensile and compressive strain states decreases quickly as the CNTloading increases and seem to converge to 2, a value of conven-tional strain gages, when the CNT loading increases beyond 10wt.%.Finally, Fig. 7(b) and (c) show the measured and predicted piezor-esistivity of CNT-polymer composites vs. the applied strain atdifferent CNT loadings. The newly developedmodel agreeswith theexperimental measurement well at all CNT loadings. The CNT-polymer composites with lower CNT loadings show higher pie-zoresistivity. The piezoresistivity reaches the highest sensitivitynear the percolating threshold. The relationship between theapplied strain and electrical resistance change is highly nonlinearand asymmetric about zero strain in tension and compressionstates at low CNT loading, such as, 1 wt% [31]. The newly developedmodel predicts this characteristic quantitatively. This indicates thatthe CNT deformation, both CNT flattening and bending, is one of themajor mechanisms of the piezoresistivity. As the CNT loading in-creases beyond 10 wt.%, the piezoresistivity becomes linear andsymmetric about zero strain with a low sensitivity because theCNT-polymer composites become too conductive. It suggests thatthe highly conductive CNT-polymer composites are not favorablefor making high sensitive sensors.

The comparison between the predicted and measured data ofelectrical conductivity and piezoresistivity of type 2 MWCNT-polymer composites is shown in Fig. 8. Unlike type 1 MWCNT-polymer composites where no obvious CNT agglomeration wasobserved, there was obvious CNT agglomeration observed in thetype 2 MWCNT-polymer composites in experiment [31]. Accord-ingly, the multiscale CNT agglomeration model is used in thesimulation. The comparison has clearly shown that the numericalpredictions are in good agreement with the experimental data,confirming the ability of the newly developed multiscale CNTagglomeration model to predict the electrical conduction and pie-zoresistive properties of the CNT-polymer composites with CNTagglomeration. Compared Fig. 8 with Fig. 7, significant differencescan be seen in the electrical conductivity and piezoresistivity ofthese two types of composites. First, the electrical conductivity oftype 2 MWCNT-polymer composites is much lower than that oftype 1 MWCNT-polymer composites at the same level of CNTloadings by several orders of magnitude, see Figs. 7(a) and 8(a). Thisis attributed to the fact that much less conductive paths are formedin the polymer matrix of type 2 MWCNT-polymer composites dueto the CNT agglomeration. The CNT loading in the non-agglomerated region is much lower than the nominal CNT

Fig. 8. Comparison of experimental and predicted electrical conductivity and piezor-esistivity of type 2 MWCNT-polymer composites. (a) Electrical conductivities. (b)Experimental and (c) predicted resistance changes vs. applied strain.

Fig. 7. Comparisons of experimental and predicted electrical conductivity and pie-zoresistivity of type 1 MWCNT-polymer composites. (a) Electrical conductivities andGauge factors at various MWCNT loadings. (b) Experimental and (c) predicted resis-tance changes vs. applied strain.

S. Gong et al. / Polymer 55 (2014) 5488e5499 5495

loading of the composites due to the high concentration of CNTs inCNT agglomerates. Thus, there are less conductive paths in thematrix to bridge the CNTs in non-agglomerated regions with CNTagglomerates. Second, the CNT agglomeration makes the piezor-esistivity of CNT-polymer composites almost the same in the tensile

and compressive strain states with respect to the zero strain asshown in Fig. 8(b). This is quite interesting because the CNTagglomeration has the same effect in this aspect as the high CNTloadings on composites without CNT agglomeration, see Fig. 7(c). It

S. Gong et al. / Polymer 55 (2014) 5488e54995496

indicates that the CNT agglomeration is responsible for the reducednonlinearity and asymmetry of piezoresistivity about the zerostrain.

3.2. Parametric studies of electrical conductivity andpiezoresistivity

3.2.1. Impact of CNT agglomerationIn our simulation, the shape of agglomerations is assumed as

near spherical. Besides the shape of agglomerations, four parame-ters affect the state of agglomerations: the agglomeration extent(xagg), the agglomeration internal density (Vf_CNT_agg), the agglom-eration size (ragg) and the agglomeration numbers. Only the firstthree of the above parameters are independent variables and therest one can be calculated once the total CNT loading in the sampleis given. The influences of CNT agglomeration extent, internal

Fig. 9. Impact of CNT agglomeration extent (a, b), internal density (c, d) and size (e,

density and size on the electrical conductivity and piezoresistivityof CNT-polymer composites are investigated and the results areshown in Fig. 9. The rest of parameters are kept as the same as intype 2MWCNT-polymer composites for analysis. The CNT loading ischosen as 5 wt.% in all of the following piezoresistivity simulations.

As xagg increases from 0.05 to 0.35, the electrical conductivity ofCNT-polymer composites decreased by four orders of magnitude atthe low CNT loadings and two orders at the high CNT loadings(Fig. 9(a)). The results confirmed that the CNT agglomeration is themajor mechanism responsible for the reduced electrical conduc-tivity. As shown in Fig. 9(b), the main effect of CNT agglomerationon piezoresistivity is to weaken its nonlinearity and asymmetry atabout the zero strain. Fig. 9(c) and (d) shows the effect of CNTagglomeration internal density on the electrical conductivity andpiezoresistivity of CNT-polymer composites, respectively. As theinternal CNT loading of agglomerations raise, the electrical

f) on the electrical conductivity and piezoresistivity of CNT-polymer composites.

S. Gong et al. / Polymer 55 (2014) 5488e5499 5497

conductivity reduces monotonously while the piezoresistivity in-creases first and then decreases. As the size of agglomerates de-creases from 12.5 mm to 5 mm, the electrical conductivity of CNT-polymer composites increases by more than 15 times while thepiezoresistivity reduces slightly from 5 to 4 only. It shows clearlythat the electrical conductivity of the CNT-polymer composites ismore sensitive to the internal density and size of CNT agglomeratesthan the piezoresistivity does. Thus, any process that can decreasethe CNT agglomeration extent or reduce the internal density andsize of CNT agglomerates is beneficial in improving the electricalconductivity of CNT-polymer composites.

3.2.2. Impact of CNT aspect ratio and intrinsic conductivityAs discussed above, it is difficult to fully disperse carbon nano-

tubes uniformly in a polymer matrix especially with small tubediameters with high aspect ratio. With the existence of CNTagglomeration (xagg ¼ 0.25), the impact of CNT aspect ratio on theelectrical conductivity and piezoresistivity of CNT-polymer com-posites has been investigated by varying the aspect ratio from 800to 1200 for a given CNT length of 8 mm. Higher CNTaspect ratio for afixed tube length means a smaller CNT diameter. Fig. 10 shows alimited increase in electrical conductivity and decrease in piezor-esistivity as the aspect ratio decreases. It indicates that at a fixedstate of CNT agglomeration (xagg ¼ 0.25, Vf_CNT_agg ¼ 15 vol.%,ragg ¼ 10 mm), the electrical conductivity and piezoresistivity ofCNT-polymer composites are not sensitive to the aspect ratiowithin the given range. However, MWCNTs with smaller diameterwill increase the total specific surface area and thus the adsorp-tivity. This makes them easier to aggregate and may change thestate of agglomeration in the polymers, which in turn affect theelectrical property of CNT-polymer composites. This mechanismmay exist in real samples and need future study.

Fig. 10. Impact of CNT aspect ratio (a, b) and CNT intrinsic conductivity (c, d) o

Assume the existence of CNT agglomeration (xagg ¼ 0.25,Vf_CNT_agg ¼ 15 vol.%, ragg ¼ 10 mm), the impact of CNT intrinsicconductivity is simulated for the range from 103 to 106 S m�1 forMWCNTs [34] and the results are shown in Fig. 10(c,d). It can beseen that both the electrical conductivity and the piezoresistivityincreases significantly with the increase in the intrinsic conduc-tivity of CNTs, see Fig. 10(c) and (d). With the increase of theintrinsic conductivity from 103 to 106 S m�1, the piezoresistivityincreases remarkably from 2 to 14 in both cases of tensile andcompressive strain. This indicates that it is an effective way toimprove both the electrical conductivity and the piezoresistivity ofCNT-polymer composites by using CNTs with higher intrinsic con-ductivity, even in the case of CNT agglomeration.

3.2.3. Impact of Poisson's ratio of CNT-polymer compositesFinally, the influence of the Poisson's ratio of CNT-polymer

composites on the piezoresistivity is explored with the existenceof CNT agglomeration (xagg ¼ 0.25, Vf_CNT_agg ¼ 15 vol.%,ragg ¼ 10 mm). The simulation results are shown in Fig. 11 for thedifferent Poisson's ratio ranging from 0.1 to 0.5. It can be seen thatthe piezoresistivity increases in both tension and compressioncases almost symmetrically as the Poisson's ratio decreases. Undera tensile strain, higher Poisson's ratio results in higher compressivestrains in the transverse directions perpendicular to the tensilestrain. Consequently, the piezoresistivity decreases since the Pois-son's ratio induced-compressive strain will increase the electricalresistance as discussed before. In randomly dispersed CNT-polymercomposite, the Poisson's ratio of the composite is dependent on thePoisson's ratio of polymer matrix and the CNT loadings. Therefore,polymers with a small Poisson's ratio will be a good choice toachieve high piezoresistive CNT-polymer composites.

n electrical conductivity and piezoresistivity of CNT-polymer composites.

Fig. 11. Impact of Poisson's ratio of CNT-polyer composites on the piezoresistivity ofCNT-polymer composites.

S. Gong et al. / Polymer 55 (2014) 5488e54995498

4. Conclusions

In summary, a new multiscale 3D deformable CNT percolatingnetwork model has been developed to quantify the effect of CNTagglomeration on the electrical conductivity and piezoresistivity ofCNT-polymer composites. The numerical results show a goodagreement with existing experimental data and demonstrate thatthe new model is able to quantify the effect of CNT agglomerationon piezoresistivity properly. The study finds that the CNT agglom-eration is responsible for the reduced electrical conductivity andnonlinearity of piezoresistivity with respect to zero strain.Furthermore, the study shows that the higher intrinsic conductivityof CNTs, smaller Poisson's ratio of composites and better processesto avoid CNT agglomeration are beneficial to improve the electricalconductivity and piezoresistivity of CNT-polymer composites. Withcertain CNT agglomeration, reducing the internal density and sizeof CNT agglomeration will improve the electrical conductivity butweaken the piezoresistivity. All of these features can be exploitedand optimized quantitatively by the new model for potential ap-plications such as smart sensing and electronic devices made ofCNT-polymer composites.

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