A Monte Carlo model with equipotential approximation and tunneling resistance for the electrical conductivity of carbon nanotube polymer composites

Carbon 146 (2019) 125e138

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Carbon journal homepage: www.elsevier.com/locate/carbon

A Monte Carlo model with equipotential approximation and tunneling resistance for the electrical conductivity of carbon nanotube polymer composites Chao Fang a, d, Juanjuan Zhang b, c, d, Xiqu Chen a, George J. Weng d, * a

Department of Electrical and Electronic Engineering, Wuhan Polytechnic University, Wuhan, Hubei, 430023, PR China Key Laboratory of Mechanics on Environment and Disaster in Western China, The Ministry of Education of China, Lanzhou University, Lanzhou, Gansu, 730000, PR China c Department of Mechanics and Engineering Science, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou, Gansu, 730000, PR China d Department of Mechanical and Aerospace Engineering, Rutgers University, New Brunswick, NJ, 08903, USA b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 28 October 2018 Received in revised form 26 January 2019 Accepted 29 January 2019 Available online 31 January 2019

A Monte Carlo model with equipotential approximation and tunneling resistance is developed to predict the percolation threshold and electrical conductivity of carbon nanotube (CNT) polymer nanocomposites. We first establish a random CNT network, and then calculate their intrinsic and contact conductance. To provide a pathway for the current to flow from CNT to polymer, a thin coated surface (CS) is introduced. The CNTs, CS, and the two electrodes then constitute the three major components of the conduction process. To solve this problem, we develop the method of equipotential approximation to determine the electrical potentials of CNTs and CS, and further determine their coefficient matrix by the walk-onspheres method. In this way the electrical properties of CNT nanocomposites, both before and after percolation, are predicted. It is demonstrated that the developed theory compares well with three sets of experimental data for the electrical conductivity and several sets of data for the percolation threshold. The effects of barrier heights, polymer conductivity, aspect ratio, diameter (and chirality) of CNTs are also investigated. This equipotential approximation possesses the distinct features that it can break through the limits of ellipsoidal fillers and properly estimate the electrical conductivity with any shape, orientation and distribution of fillers. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Electrical properties Carbon nanotubes Nanocomposites Walk-on-spheres Equipotential approximation Tunneling

1. Introduction Polymer nanocomposites with carbon nanotube (CNT) fillers are of great interests due to their potential applications in nano electronic devices, actuators, electrostatic discharge protection [1], microwave absorption [2], electromagnetic wave shielding [3], piezoresistive sensors [4] and biosensors [5]. It has been widely reported that, by introducing a small amount of CNTs into polymer matrices, the electrical conductivity of CNT-based nanocomposites can be greatly enhanced. This is largely due to the high electrical conductivity and high aspect ratio of the CNTs, and the important contribution of electron tunneling. This phenomenon is also closely related to the issue of percolation threshold. As the filler concentration continues to increase, connectedness increases and

* Corresponding author. E-mail address: [email protected] (G.J. Weng). https://doi.org/10.1016/j.carbon.2019.01.098 0008-6223/© 2019 Elsevier Ltd. All rights reserved.

additional percolation paths continue to build up, thereby greatly enhancing the overall conductivity of the nanocomposite. The issues of percolation are complex. They depend strongly on the polymer type, fabrication parameters, aspect ratio of CNTs, disentanglement of CNT agglomerates, uniformity of spatial distribution of nano fillers, and degree of alignment [6]. Determination of the electrical conductivity of CNT nanocomposites has long been a challenging problem. With varying degrees of sophistication, it has been considered in the continuum context, and it has also been studied by Monte Carlo (MC) simulations. In the continuum context, a plausible way is to adopt the Mori-Tanaka (MT) approach ~ eda-Willis (PCW) approach [9]. Pan et al. [7,8] or the Ponte Castan [10] proposed a method utilizing the PCW approach and the Hashin-Shtrikman upper bound [11] to calculate the effective conductivity of a two-component system containing randomly oriented ellipsoidal inclusions. They pointed out that there exist two critical volume concentrations that respectively represent the

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C. Fang et al. / Carbon 146 (2019) 125e138

percolation threshold and the percolation saturation. Based on the MT method, Kim et al. [12] estimated the effective electrical conductivity considering multiple heterogeneous inclusions. It has also been developed to consider other parameters such as interphase thickness and tunneling distance in polymer nanocomposites [13]. In addition, the symmetric version of the effective medium theory originally developed by Bruggeman [14] - also called the coherent potential approximation - has also been recently applied to study the electrical conductivity of CNT nanocomposites [15], as well as the electrical conductivity, dielectric permittivity [16,17] and thermal conductivity [18] of graphene/polymer nanocomposites. Because of molecular interactions, the interphase is produced between polymer matrix and the filler, the property of which is different from both polymer and filler [19]. The most important property improvement in polymer nanocomposites can be attributed to the formation of interphase between polymer matrix and nanofillers [20,21]. Assuming the interphase to surround the CNT, Zare and Rhee [22e24] found that a better conductivity is obtained by thicker interphase. By the known Pukanszky model and the networking role of interphase, the percolation threshold of CNT polymer nanocomposites has also been studied [25,26]. MC simulations can provide the desirable feature of randomness in filler distributions that the continuum approach cannot fulfill. MC method, however, has not been widely used because of its prohibitive computational tasks. It appears that only very limited simulations have been carried out. In this regard, Hu et al. [27] have used a 3D resistor network to investigate the effects of CNT shape and aggregation. Bao et al. [28e30] have examined the effects of electron tunneling and CNT alignment on the electrical conductivity of CNT polymer nanocomposites. The method has also been used to study the influence of structural distortion of CNT walls at the crossed junctions [31,32]. In addition, Mora et al. [33] have developed a computational tool to quantify the amount of CNTs that actively participate in the conductive network. These simulations have provided significant insights into the complex nature of electrical conductance when the connected networks are random in the composites. Despite the complexities of MC simulations, an equipotential approximation with electron tunneling is presented here. It is based on the walk-on-spheres (WoS) method. In this model, the electrical conductivity of the CNT-polymer nanocomposites can be calculated for the full range of filler concentrations. Only four parameters are involved: (i) the electrical conductivity of polymer, s0, (ii) the diameter of CNTs, D, (iii) the aspect ratio of CNTs, a, and (iv) the barrier height, DE. The barrier heights can also describe the effect of interface characteristics. Factors such as the shape, orientation, and distribution of CNTs are implicit in the 3D network established in this MC. It will become evident that this model can be applied to calculate the conductance of a nanocomposite for fillers of any shapes, orientations, distributions, and concentrations in polymers. 2. The theory The development of the model takes five steps. The first one is the establishment of a CNT network with random distribution using MC simulations. The second one is the calculation of intrinsic conductance and contact conductance that considers the effect of electron tunneling in the network. In the third step the concept of equipotential approximation is introduced. Then with consideration of a thin layer of coated surface (CS) on each CNT, the electric potentials of CNTs are constructed. In this process, the tunneling current from one CNT to others, the direct current from one CNT to its own CS, and the tunneling current from one CNT to the two electrodes, are considered. In the fourth step, the WoS method is

applied to determine the electric potentials of CS in terms of the potentials of CNTs and electrodes. In the last step, the potentials of CNTs are obtained in terms of the potential of the electrodes, and the conductivity of the nanocomposite is established. 2.1. The CNT network CNTs are not ideally straight in the actual CNT network. To model a curved CNT, it is usually divided into several interconnected straight sections [27,33]. As curved CNTs tend to result in lower conductivity and higher percolation threshold, so such an approximation will have an effect on the predicted results. But when CNTs are randomly oriented, its effect is less significant. For instance, Hu et al. [27] found that when the maximum angle between segments increases from 0 to 60
, the percolation threshold increases only from 0.6% to 0.9%. In MC calculations, both the distance between each pair of CNTs and the distance from any point in the polymer to the CNT need to be calculated. If a CNT is to be preserved in a curved shape, it will greatly increase the amount of computation. For these reasons, CNTs are approximated by straight line elements here. Now consider a random distribution of CNTs generated by the MC simulation shown in Fig. 1(a), where the numbers 1, 2, 3 in the inset represent three types of contacts: end to end, lateral edge to lateral edge and end to lateral edge contact, respectively. Following [27,34], a schematic for these three types of CNT junctions is depicted in Fig. 1(b). Fig. 1(c) shows the geometric configuration of a CNT with the starting point ðx0i ; y0i ; z0i Þ and ending point ðx1i ; y1i ; z1i Þ in a representative polymer cuboid of dimensions Lx  Ly  Lz, with

x0i ¼ Lx  randðx ¼ x; y; zÞ:

(1)

Here, ‘rand’ denotes a uniform random number in the interval [0, 1]. The ending point can be expressed by



 x1i ; y1i ; z1i ¼ x0i ; y0i ; z0i þ li ðsin qi cos fi ; sin qi sin fi ; cos qi Þ; (2) where the subscript i is the index of the i-th CNT and (qi, 4i, li) are its azimuthal and polar angles, and length, respectively. In order to make random orientations of the CNTs, (qi, 4i) are uniformly generated as

qi ¼ cos1 ð2  rand  1Þ; fi ¼ 2p  rand:

(3)

To characterize the aspect ratio of CNTs, it is assumed that all CNTs have the same length li ¼ l and diameter D. If the ending point ðx1i ; y1i ; z1i Þ falls outside the representative cuboid volume, the protruded part should be truncated at the boundary z ¼ 0 and z ¼ Lz plane. We consider that the positive and negative voltages are applied to the two electrodes z ¼ Lz plane and z ¼ 0 plane, respectively. In the x and y directions, the protruded parts are relocated back into the representative cuboid volume by the periodic boundary conditions. In the nanoscale, CNTs are taken to be caped cylinders with diameter D and nominal length L, with L ¼ l þ D, as shown in Fig. 1(d). 2.2. Intrinsic conductance and contact conductance The conductance of carbon nanotubes is related not only to the diameter and length of the carbon nanotubes, but also to the type of

C. Fang et al. / Carbon 146 (2019) 125e138

(a)

(b)

(c)

(d)

127

Fig. 1. Schematics of (a) typical CNT network in a representative cuboid volume, where 1, 2, 3 represent three types of contacts, (b) the three types of contacts: end to end, lateral edge to lateral edge and end to lateral edge contact, (c) position and orientation of a CNT, and (d) a capped CNT. (A colour version of this figure can be viewed online.)

structure. They can be semiconducting or metallic depending on the chirality. For zigzag, armchair or other chiral carbon nanotubes, the conductance is the same as long as the diameter, length and type of structure are all the same. So we take zigzag carbon nanotubes to analyze the conductance of CNTs. A zigzag nanotube has a chirality of (m, 0). The bottom of the conduction band and the top of its valence band with respect to the Fermi level are [35,36]

Ev ¼ ±

  3ta0  2m ; v  3  D 

(4)

where a0 ¼ 0.142 nm is the length of carbonecarbon bonds, t ¼ 2.7 eV is the Hamiltonian matrix element between neighboring carbon atoms, D is the shell diameter, and v is an integer less than m. If m is a multiple of three, there is an integer v such that Ev ¼ 0, then the shell becomes metallic. The diameter of a zigzag CNT is related to its chirality by the following equation [36]

pffiffiffi 3a0 m D¼ :

(5)

p

At temperatures above 0 K, all subbands contribute to conduction. The total number of channels for each shell can be written as [37]

Nchan ¼

X

1 : expðjEv j=kB TÞ þ 1

Gintrinsic ¼

subbands

At room temperature T ¼ 300 K, kB is Boltzmann constant. Thus, the intrinsic conductance of a shell is

(7)

where G0 is the quantum conductance 2e2/h, e is the electron charge and h is Planck's constant. L0 is the mean free path. It has been suggested that the interwall conductance is negligible as compared to the intrawall ballistic conductance [38], and that the current flows through only one or a few of the outermost shells of a capped multi-walled CNTs (MWCNT) when the current is injected in the outer shell [39e41]. Therefore, we only consider the conductance of the outermost shell of CNT and ignore the conductance of the inner shells, so that the single-walled CNTs (SWCNT) and MWCNT are unified in our model. When the distance between two CNTs is at the nanoscale, electron tunneling between CNTs could occur. The tunneling effects cannot be ignored if the polymer thickness between two CNTs is smaller than the cutoff distance dcutoffz1.4 nm [28,42]. By solving €dinger equation with rectangular potential barrier, and the Schro assuming that the temperature effect is ignored at room temperature, the contact conductance can be evaluated as

Gcontact ¼ G0 Nchan t;

(8)

where t is the transmission probability for the electron to tunnel through the polymer obstacle between CNTs, which can be estimated by



(6)

G0 Nchan ; ð1 þ L=L0 Þ

t ¼ exp 

deff dtunnel

 ;

(9)

where the symbol dtunnel denotes the tunneling characteristic length

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C. Fang et al. / Carbon 146 (2019) 125e138

.pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dtunnel ¼ Z 8me DE;

(10)

where Z ¼ h/2p, me is the mass of electron, and DE is the barrier height between two CNTs. Considering an interface on the surface of CNTs, DE should be changed according to the different interface characteristics. In our model, CNTs are not allowed to penetrate into each other, so there is a distance between two CNTs. This distance, on the one hand, should be no less than the van der Waals separation distance, dvdW, due to Pauli's exclusion principle [43] and van der Waals repulsive force. On the other, it should also be no less than the actual separation distance, d, between two CNTs. So the effective distance between a pair of CNTs, denoted as deff, is determined by the larger of the two, as

deff ¼ maxðd; dvdW Þ;

(11)

A detailed approach to calculate d for all three types of contacts is provided by Hu et al. [27].

2.3. Equipotential approximation and construction of linear equations for the electrical potentials of CNTs Applying Eq. (9) and Eq. (11), we have



t  exp 

 dvdW : dtunnel

(12)

Usually dvdW ¼ 3e4 Å [28,44], and DE is about several eV [45]. It can be estimated that exp(-dvdW/dtunnel)z104~102, which means that the most transmission probability t is small and below 0.01. According to Eq. (7) and Eq. (8), the intrinsic resistance of the CNT is relatively negligible compared to the contact resistance between the CNTs with small L/L0. Buldum and Lu [46] pointed out that the contact resistance can vary several orders of magnitude with atomic scale manipulations. Moreover, it is found that the electrical conductivity of composites containing 0.10 wt% of Ag@CNTs was significantly improved [47], which demonstrated that the contact resistance of CNTs in composites plays a dominant role in contrast to the role of intrinsic resistance of CNTs. Similar conclusion was also drawn in Li et al. [48]. However, some experiments found that the CNT has a non-negligible intrinsic resistance [49] and shows a

(a)

resistance per unit length of 7 kU/mm when L [ L0 [50]. The mean free path L0 increase with nanotube diameter [51] and can be written as L0 ¼ 1000D [35]. If the aspect ratio of CNT a ¼ L/ D  2000, then Gcontact << Gintrinsic, that is, the contact resistance is substantially higher than the intrinsic resistance. In that case, there will be a sharp increase in the electric potential at the contact points, and an almost constant potential along each CNT. Fig. 2 shows such a variation of electrical potential along a percolation path. There are six CNTs with the lateral edge to lateral edge contact sketched on the percolation path in Fig. 2(a), as this type of contact is the most common among all three types of contact [27]. There is a jump of electrical potential from one CNT to another due to the contact resistance, and the potential of each CNT is basically constant as indicated in Fig. 2(b), expressed as Ui, i ¼ 1, 2, …, 6. Thus if CNTs are not long, they are approximately equipotential in each CNT and can be represented by an N  1 matrix UCNT, the elements of which are UiCNT , i ¼ 1, 2, …, N, the electrical potential of i-th CNT. This observation leads us to introduce the concept of equipotential approximation to address the conduction problem in CNT-based polymer nanocomposites. Using the equipotential approximation, the conductance on a path will increase by about 3% when a ¼ 2000, which has little effect on the result of the composite conductivity. It must be noted that, if a CNT is not on a percolation path, the variation of potential will not display the feature in Fig. 2(b), and current will flow from CNT to polymer. To account for the current from CNT to polymer, we introduce a thin coated surface (CS) on the periphery of each CNT, consisting of a cylindrical surface and two hemispheres at both ends with diameter D0 a little greater than D, as shown in Fig. 3. The average potential of CS is represented by a matrix UCS of N  1, the elements of which are UiCS , i ¼ 1, 2, …, N, the electrical potential of i-th CS associated with the i-th CNT. With the assembly of CNTs, their coated surfaces, and the two electrodes, the current flow from the i-th CNT can be illustrated in Fig. 4. It includes three parts: the red arrows indicate the tunneling currents from the i-th CNT to other CNTs, the black arrows indicate the direct current from the i-th CNT to i-th CS, and the blue arrows indicate the tunneling current from the i-th CNT to the two electrodes. Now we establish the inter-relations of the electrical conductance for these three mechanisms. According to Kirchhoff's current law, we have

(b)

Fig. 2. Equipotential approximation method when Gcontact << Gintrinsic. (a) A percolation path, (b) electrical potential on a percolation path. (A colour version of this figure can be viewed online.)

C. Fang et al. / Carbon 146 (2019) 125e138

N X

129

N
 X
 UiCNT  UjCNT þ UiCNT  UjCS GCNTCNT GCNTCS ij ij

j¼1

þ

j¼1 2 X




UiCNT  UjElectrode GCNTElectrode ij



j¼1

¼ 0; i ¼ 1; 2; :::; N;

(13)

where the first term is the sum of tunneling current from i-th CNT to other CNTs, the second term is direct current from i-th CNT to its own coated surface, and the third term is the sum of tunneling current from i-th CNT to two electrodes, represented by UjElectrode. In this expression GCNTCNT is the contact conductance between i-th ij CNT and j-th CNT, GCNTCS is the direct conductance between i-th ij CNT and j-th CS, and GCNTElectrode is the contact conductance beij tween the i-th CNT and j-th electrode. According to Eqs. (8-11), GCNTCNT can be evaluated as ij

( GCNTCNT ij

¼

. i h
G0 Nchan exp  max dCNT dtunnel ij ; dvdW 0

isj i¼j (14)

where dCNT is the distance between i-th CNT and j-th CNT, and is ij equal to the distance between the line segment ðx0i ; y0i ; z0i Þ  ðx1i ; y1i ; z1i Þ and ðx0j ; y0j ; z0j Þ  ðx1j ; y1j ; z1j Þ minus D. GCNTCS is the direct ij conductance between i-th CNT and j-th CS. As D0 is close to D, GCNTCS can be calculated by ij

 GCNTCS ¼ ij

Fig. 3. A thin coated surface. (A colour version of this figure can be viewed online.)

2pDLs0 =ðD0  DÞ 0;

i¼j isj

(15)

where s0 is the electrical conductivity of polymer. The calculation of GCNTCS in Eq. (15) cannot consider the case that the i-th CS coij incides with other CNTs, so the distance between i-th CS and i-th CNT needs to be small enough to avoid such coincidence. For this reason, we take D' ¼ D þ dvdW. Next, the contact conductance between CNTs and the electrodes needs to be considered. The electrical potential of electrodes is expressed by a 2  1 matrix, UElectrode, the elements of which are U1Electrode and U2Electrode . It is assumed that there is also a minimum distance dvdW between the CNT and the electrode as that between can be written as CNTs. Then following Eqs. (8-11), GCNTElectrode ij

. 
dtunnel ; GCNTElectrode ¼ G0 Nchan exp  dCNTElectrode ij ij

(16)

where dCNTElectrode is the distance between the i-th CNT and the jij th electrode, so

dCNTElectrode ij

h
 i max min z0i ; z1i  D=2; dvdW j ¼ 1; h
 i ¼ : max min Lz  z0 ; Lz  z1  D=2; d j ¼ 2: vdW i i 8 <

(17) With these three sets of conductance, Eq. (13) can be simplified to a set of linear equations for the electrical potentials of the CNTs, as




 I  Q CNTCNT U CNT ¼ Q CNTCS U CS þ Q CNTElectrode U Electrode ; (18)

Fig. 4. Schematic diagram of current through i-th CNT. (A colour version of this figure can be viewed online.)

where I is an N  N identity matrix, and

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C. Fang et al. / Carbon 146 (2019) 125e138

3 U1CNT 6 U CNT 7 7 ¼6 4 2« 5 2

U

CNT

3 U1CS 6 U CS 7 7 ¼6 4 «2 5; 2

; U

CS

UNCNT

" U

Electrode

¼

UNCS

U1Electrode U2Electrode

#

(19) 2

Q CNTCNT

Q CNTCNT 6 11 CNTCNT 6 Q21 ¼6 4 « CNTCNT QN1 2 0 6 CNTCNT 6 Q21 ¼6 4 « CNTCNT QN1 2

Q CNTCS

Q CNTCS 6 11 6 CNTCS ¼ 6 Q21 4 « CNTCS QN1 2 Q CNTCS 6 11 6 0 ¼6 4 « 0

CNTCNT Q12 CNTCNT Q22 « CNTCNT QN2

/ / 1 /

CNTCNT Q12 0 « CNTCNT QN2

/ / 1 /

CNTCS Q12 CNTCS Q22 « CNTCS QN2

/ / 1 /

0

/ / 1 /

CNTCS Q22

« 0

2

Q CNTElectrode

Q CNTElectrode 6 11 6 CNTElectrode ¼ 6 Q21 4 « CNTElectrode QN1

3 CNTCNT Q1N CNTCNT 7 7 Q2N 7 5 « CNTCNT QNN 3 CNTCNT Q1N CNTCNT 7 7 Q2N 7 5 « 0

3 CNTCS Q1N CNTCS 7 7 Q2N 7 5 « CNTCS QNN 3 0 7 7 0 7 5 « CNTCS QNN

3 CNTElectrode Q12 CNTElectrode 7 7 Q22 7 5 « CNTElectrode QN2

Fig. 5. WoS sampling of random walk in 3D. (A colour version of this figure can be viewed online.)

( UðrÞ ¼

(20)

and

QijCNTCNT ¼

N
P j¼1

QijCNTCS ¼

N
P j¼1

GCNTCNT ij

GCNTCNT ij

QijCNTElectrode ¼

N
P j¼1

GCNTCNT ij ;  P 2 CNTElectrode þ þ GCNTCS G ij ij j¼1

GCNTCS ij ;  P 2 CNTCS þ þ Gij GCNTElectrode ij

GCNTCNT ij

j¼1

2.4. WoS method for the construction of linear equations for the electrical potentials of CS Considering the matrix material to be a uniform conducting medium, its electrical potential distribution satisfies the Laplace equation

r2G; r2vG;

r2Ci ; i ¼ 1; 2; :::; N; r2Zj ; j ¼ 1; 2:

(23)

where Ci is the space occupied by the i-th CNT, which can be regarded as a set of points, the distance from which to the line segment ðx0i ; y0i ; z0i Þ  ðx1i ; y1i ; z1i Þ is less than or equal to D/2. Z1 and Z2 represent the plane z¼0 and z¼Lz respectively. The solutions to (22) can be generated numerically via WoS MC simulations of the path that is stopped at a distance ε or smaller from vG. The red dashed arrow in Fig. 5 shows a path r(0)-r(1)-r(2)r(3)-r(4) from the r point in the polymer to C1 through the random walk on sphere. r(0) is the start point of random walk. Drawing a sphere as large as possible with r(0) as the center without intersecting any CNT and electrodes, r(1) is a random point on the surface of the sphere. By repeating this process, we can get r(2), r(3) …. In Fig. 5, If the distance between r(4) and the k-th CNT is small enough, then the end of the random walk is the k-th CNT. There is also a certain probability of reaching other CNTs or electrodes from r. Then, by the strong law of large numbers, we have

j¼1

GCNTElectrode ij :  P 2 CNTCS CNTElectrode þ þ Gij Gij (21)

V2 uðrÞ ¼ 0; uðrÞ ¼ UðrÞ;

UiCNT ; Electrode Uj ;

M 1 X UðvGk Þ; M/∞ M

uðrÞ ¼ lim

where vGk indicates the end of the k-th random walk [52,53]. Eq. (24) could be simplified to

2 3 M N 2 X

X

1 X CNT Electrode 4 uðrÞ ¼ lim Uj P Cj jr þ Uj P Zj jr 5; M/∞ M j¼1 j¼1 k¼1

(25) where P{Cj|r} and P{Zj|r} represent the number of events arriving at Cj and Zj from the point r, respectively. Let r be a random point on i-th CS, then we get the formula for calculating UiCS

(22)

where G is a polymer domain in 3D with smooth boundary, and vG is the boundary of G. Then U(r) is the voltage on vG, which can be expressed by

(24)

k¼1

UiCS ¼

N X j¼1

where

QijCSCNT UjCNT þ

2 X j¼1

QijCSElectrode UjElectrode ;

(26)

C. Fang et al. / Carbon 146 (2019) 125e138

QijCSCNT

M

1 X ¼ lim P Cj jCSi ; M/∞ M k¼1

M

1 X P Zj jCSi : M/∞ M

QijCSElectrode ¼ lim

(27)

k¼1

Thus Eq. (26) can be simplified to

U CS ¼ Q CSCNT U CNT þ Q CSElectrode U Electrode ;

(28)

where

2

Q CSCNT

Q CSCNT 6 11 6 CSCNT ¼ 6 Q21 4 « CSCNT QN1 2

Q CSElectrode

CSCNT Q12 CSCNT Q22 « CSCNT QN2

Q CSElectrode 6 11 CSElectrode 6 Q21 ¼6 4 « CSElectrode QN1

/ / 1 /

3 CSCNT Q1N CSCNT 7 7 Q2N 7 5 « CSCNT QNN (29)

3 CSElectrode Q12 CSElectrode 7 7 Q22 7: 5 « CSElectrode QN2

2.5. Key steps of WoS algorithm for the calculation of the i-th row elements of QCSCNT and QCSElectrode associated with the i-th CNT Step 1: Set k ¼ 1, P{Cj|CSi} ¼ 0, P{Zj|CSi} ¼ 0. Set the initial value of k, P{Cj|CSi}, and P{Zj|CSi}. k is the order number of random walks, and k ¼ 1 indicates the first random walk. At the beginning, the initial value of P{Cj|CSi} and P{Zj|CSi} are set to zero. Step 2: Initialize r(0) ¼ r, and r is a random point on i-th CS. Here r is a position, the potential of which we want to calculate. Let r be a random point on i-th CS. r(0) is the starting point of k-th random walk. So r(0) ¼ r, as shown in Fig. 5. Step 3: If d(r(n), vG)>ε, then 3.1 Set r(n þ1) ¼ r(n)þVnd(r(n), vG), where Vn is a random unit vector in 3D space. 3.2 If r(n þ1) falls outside the representative cuboid volume Lx  Ly  Lz, then it must be relocated back into the representative cuboid volume by the periodic shift. 3.3 Repeat step 3. If d(r(n), vG)ε, and further if d(r(n), vG) ¼ d(r(n), 



  P Cj CSi )P Cj CSi þ 1; ns0  



, P Cj CSi )P Cj CSi þ 0:5 ;n ¼ 0 PfCi jCSi g)PfCi jCSi g þ 0:5

Cj),

then

if d(r(n), vG) ¼ d(r(n), Zj), then PfZj jCSi g)PfZj jCSi g þ 1 Here n represents the order number of steps, and r(n) represents the position of the n-th step. ε is a small positive number. d(r(n), vG) is the distance from r(n) to vG, so

 h

i
d r ðnÞ ; vG ¼ min d r ðnÞ ; Cm ; d r ðnÞ ; Zn ; m ¼ 1; 2; :::; N; v m;v

¼ 1; 2; where d(r(n), Cm) is the minimum distance from r(n) to all CNTs

131

obtained by periodic extension of m-th CNTs, and d(r(n), Zv) is the distance from r(n) to the electrode v. If d(r(n), vG)>ε, which means the k-th random walk does not end, then draw a sphere with r(n) as the center of the sphere and d(r(n), vG) as the radius, and take a random point on the surface of the sphere as r(n þ 1) and continue to random walk. If d(r(n), vG)ε, which means the k-th random walk ends, then if (n) d(r , vG) ¼ d(r(n), Cj), then the end of the k-th random walk falls on the j-th CNT, P{Cj|CSi} plus 1; If d(r(n), vG) ¼ d(r(n), Zj), then the end of the k-th random walk falls on the electrode j, P{Zj|CSi} plus 1. In Fig. 5, we can see that when d(r(4), vG)ε, it will take four steps to reach 1-th CNT. It is noted that the randomly distributed CNTs might penetrate each other, and in fact, the contacting CNTs at junctions are separated by a distance dvdW. If the starting point of the random walk is in the coincident space of i-th CNT and j-th CNT, then the probability of walking to i-th CNT and j-th CNT is 50% each. So a special case here to deal with is to determine whether n is equal to 0. If n ¼ 0, which indicates that the initial position r is in the coincident space of i-th CNT and j-th CNT, then add 0.5 to P{Cj|CSi} and P{|CiCSi} respectively. Step 4: Let k ¼ kþ1, and return to step 2 if k < M. Here M is a larger integer, the total number of random walks from i-th CS. If k does not reach M, then return to step 2 and start the next random walk from i-th CS. If k reaches M, go to step 5. Step 5: According to Eq. (27), we get

QijCSCNT ¼

M

1 X P Cj jCSi ; j ¼ 1; 2; :::; N; M k¼1

QijCSElectrode ¼

M

1 X P Zj jCSi ; j ¼ 1; 2: M k¼1

Through these 5 steps, we can obtain the i-th row elements of QCS-CNT and QCS-Electrode that are associated with the i-th CNT. Let i ¼ 1, 2, 3, …, N and repeat the above 5 steps, the values of all the elements of the matrices QCS-CNT and QCS-Electrode can be calculated.

2.6. The conductivity of the nanocomposite Substitute UCS from Eq. (28) into Eq. (18), we obtain UCNT in terms of UElectrode as




 I  Q CNTCNT  Q CNTCS Q CSCNT U CNT 
¼ Q CNTCS Q CSElectrode þ Q CNTElectrode U Electrode :

(30)

The electrical potential of each CNT UCNT can be calculated by solving the linear equation (30). Before we compute the current flowing through the nanocomposite, the average potential Uz¼ dvdW of the plane z ¼ dvdW is also calculated using the WoS method. The plane z ¼ dvdW does not cut any CNT because of the assumed minimum distance dvdW between the CNT and the electrode, then the current through the plane z ¼ dvdW is

I¼


 Lx Ly s0 U z¼dvdW  U1Electrode dvdW   U1Electrode ;

þ

N X


UiCNT GCNTElectrode i1

i¼1

(31)

where the first term is the direct current and the second term is the tunneling current from the plane z ¼ dvdW to the electrod 1. So at the end we arrive at the conductivity of the nanocomposite

132

C. Fang et al. / Carbon 146 (2019) 125e138

s¼


ILz



U2Electrode  U1Electrode Lx Ly

:

(32)

3. Results and discussion In order to place the developed Monte Carlo model in perspective, we now apply it to determine the electrical conductivity of three CNT polymer composites and compare the calculated results with the experimental data. We also use it to examine the effects of several key parameters on the overall conductivity of the nanocomposite. In all of these, the van der Waals separation distance between the walls of CNTs is taken as dvdW ¼ 0.39 nm, as given in Ref. [44]. 3.1. Comparison with experiments To compare our results with the test data, the following parameters were selected. The conductivity of most polymers s0 is on the orders of 1018 to 106 S/m. The barrier height varies from 1 to 7 eV [28,42]. The CNT diameter is taken D ¼ 1 nme50 nm, according to Eq. (5), and the corresponding chirality m ¼ 12e640. In Fig. 6, our theoretical predictions show very good agreement with the three sets of data [54e56] considered. Since the conductivity of the CNTs is much larger than the conductivity of the polymer, the electrical conductivity of the composite increases with the addition of CNTs. When CNTs concentration is small, the distance between the CNTs is so large that the contact resistance may exceed the resistance of the polymer, so the electrical conductivity of the composite is close to the conductivity of the polymer. As the CNTs concentration increases, the decrease of the distance between the CNTs leads to the decrease of contact resistance and the increase of overall conductivity. Once CNTs constitute the percolation paths, which means that the distance between two adjacent CNTs on the path is less than the cut-off distance, the conductance of the composite results from the combination of a number of contact resistors in series and parallel. The electrical conductivity of the composite sharply increases due to the dramatic reduction of contact resistance. When more and more percolation paths are formed, the increase of electrical conductivity tends to a more stable state. Our theoretical predictions indicate that the developed MC model is able to predict these main characteristics, both for a low and a high CNT volume fraction. It can also be seen that the model is suitable for various types of CNT, such as CNT with high aspect ratio a ¼ 2000 in Fig. 6 (a), low aspect ratio a ¼ 21 in Fig. 6 (b), SWCNTs in Fig. 6 (a) and MWNTs in Fig. 6 (b) and (c), metallic outermost shell CNTs in Fig. 6 (b) and semiconductor outermost shell CNTs in Fig. 6 (a) and (c). Our theory involves four material constants: (i) the electrical conductivity of polymer, s0, (ii) the diameter of CNTs, D, (iii) the aspect ratio of CNTs, a, and (iv) the barrier height, DE. The first three could all be found from the original experimental papers; only the barrier height needs to be assumed. The chirality, m, is related to D through Eq. (5), which for instance leads to m ¼ 640, in Fig. 6 (c). There are two main reasons for a non-smooth curve before percolation threshold: (i) The CNT geometry model established by MC method entails certain degree of randomness, especially when the CNT volume fraction is small, and (ii) The number of random walks is not sufficiently large which could affect the precision when D'/D approaches one. 3.2. Effects of barrier height, DE Since the tunneling effect occurs between two CNTs with a small

distance, the tunneling conductance is closely related to the medium surrounding the CNT. Barrier height DE can be considered as the difference of the work functions between the CNT and the interface. The work function of the interface should be between that of CNT and the polymer. If the work function of the interface is close to the work function of CNT, then DE is small; conversely, DE is large. Therefore, DE is an important parameter that can describe the characteristics of interface. Fig. 7 shows its effects on the percolation threshold and overall conductivity of the nanocomposite with different DE. It is seen that, as the barrier height DE decreases from 7 eV to 1 eV, the percolation threshold decreases and the conductivity increases. It can also be seen from Eqs. (10) and (9) that, as the barrier height DE increases, the tunneling probability t decreases. The decrease of the contact conductance causes the number of percolation paths to drop or even disappear, and the percolation threshold is increased. After percolation threshold, the percolation paths build up and the electrical conductivity of the nanocomposite is proportional to the contact conductance. According to Eqs. (8-10), the interval between two adjacent curves is

log10 sjDE¼1eV  log10 sjDE¼3eV log10 sjDE¼3eV  log10 sjDE¼5eV pffiffiffi pffiffiffi pffiffiffi ¼ 31 5 3 log10 sjDE¼5eV  log10 sjDE¼7eV pffiffiffi pffiffiffi ; ¼ 7 5 (33) which is consistent with the results of our simulation in Fig. 7. To illustrate the effect of interface - or the barrier height - in our model, we compare our calculations with the predictions of Zare and Rhee [22], which was based on the approach of interphase thickness, on the electrical conductivity of the nanocomposite. Both types of calculated results are compared with the experimental data for epoxy/SWCNT (D ¼ 2 nm, l ¼ 2 mm) [57] in Fig. 8(a), and for ultrahigh molecular weight polyethylene/MWCNT (D ¼ 16 nm, l ¼ 8 mm) [58] in Fig. 8(b). In our calculations, we chose s0 ¼ 108 S/ m, DE ¼ 6.3 eV, m ¼ 25 (D ¼ 1.957 nm), a ¼ 1000 for epoxy/SWCNT and s0 ¼ 5.5  1010 S/m, DE ¼ 5 eV, m ¼ 204 (D ¼ 15.971 nm), a ¼ 500 for ultrahigh molecular weight polyethylene/MWCNT. Good agreement between our MC calculation considering the work function of interface and the continuum calculation of Zare and Rhee considering interphase thickness is found, and both are also in good agreement with the test data. This shows that both approaches can be used to represent the tunneling mechanism at the interface in the CNT polymer composites. This is probably the first time that these two distinctly different approaches are brought together. 3.3. Influence of electrical conductivity of polymer matrix, s0 The influence of electrical conductivity of polymer is illustrated in Fig. 9. It can be seen that the percolation threshold is independence of the electrical conductivity of polymer. As s0 decreases, the electrical conductivity of nanocomposites decreases synchronously before percolation threshold and remains unchanged after that. It is easy to distinguish from Fig. 9 that the exact percolation threshold value is 0.3%e0.4%. Before percolation threshold, the CNTs do not constitute percolation paths. For any given i-th CNT, if its distance from the electrode z ¼ 0 is so large that GCNTElectrode z0, the tunneling current is zero; or if its distance i1 from the electrode z ¼ 0 is small enough to cause GCNTElectrode > > 0, then the solution of Eq. (30) is UiCNT zU1Electrode i1 because the i-th CNT does not constitute a percolation path to the electrod 2. Therefore, GCNTElectrode z0 or UiCNT  U1Electrode z0 , i1 which results in an about zero tunneling current in equation (31),

C. Fang et al. / Carbon 146 (2019) 125e138

(a)

133

(b)

(c) Fig. 6. Comparison of predicted electrical conductivity with existing experimental data. (a) with Ounaies et al. data [54], (b) with Ngabonziza et al. data [55], and (c) with Hu et al. data [56].

and the total current is determined by the direct current. The role of CNT is only to shorten the distance between the positive and negative electrodes in the polymer, which is expressed as to increase the potential U z¼ddvW . So the electrical conductivity of the nanocomposite is slightly larger than that of polymer. As the CNT concentration increases beyond the percolation threshold, the percolation paths is formed. Both GCNTElectrode and UiCNT  U1Electrode i1 are non-zero for the closest i-th CNT to the negative electrode on a path, leading to a non-zero tunneling current. Taking m ¼ 12, DE ¼ 3 eV, dCNTElectrode ¼ dvdW , s0 ¼ 10-10 S/M, Lx ¼ Ly ¼ 10 mm for i1 an example, according to Eq. (16), GCNTElectrode z107 S , while the i1 conductance of the polymer between z ¼ 0 and z ¼ dvdW plane is LxLys0/dvdWz10e11 S, which is four orders of magnitude smaller than the tunneling conductance. Therefore, once the percolation

paths are built up, the conductivity of the composite increases sharply. The tunneling current is independent of s0, so the electrical conductivity of nanocomposites after percolation threshold is not affected by s0. 3.4. Effects of aspect ratio of CNTs, a We present the electrical conductivity of the nanocomposites as a function of the volume fraction for different aspect ratios, a ¼ 2000, 1000, 500, 200 in Fig. 10(a) and a ¼ 100, 50, 20, 10, 5 in Fig. 10(b). By comparison, we observe that a higher CNT aspect ratio causes a lower percolation threshold. The percolation threshold is related to the formation of the percolation path, and a high CNT aspect ratio

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C. Fang et al. / Carbon 146 (2019) 125e138

Fig. 7. Tunneling effects of different barrier heights on electrical conductivity of nanocomposites.

Fig. 9. Effects of electrical conductivity of polymer on the electrical conductivity of the nanocomposites.

can build up percolation paths at a low CNT concentration according to the established random CNT network. As CNT concentration increases, the electrical conductivity of the nanocomposites increases and faster increases in the percolation threshold region. Before percolation threshold, larger aspect ratio causes greater conductivity because it can more effectively shorten the distance between the positive and negative electrodes in the polymer. After percolation threshold, as the aspect ratio increases, the electrical conductivity increases because of the reduced number of CNTs and junctions on a percolation path. If the effective distance deff ¼ dvdW at each junctions, the conductivity is inversely proportional to the number of junctions on a percolation path, and proportional to the

length of CNTs. Therefore, when the volume fraction is fixed and far exceeds percolation threshold, the conductivity of the nanocomposites is proportional to the aspect ratio, as shown in Fig. 10(a) and (b). 3.5. Influence of CNT diameter, D (and chirality, m) The influence of CNT diameter (and chirality) on the electrical conductivity of the nanocomposites is very significant. Fig. 11 (a) and (b) show the conductivity of the nanocomposites versus volume fraction with various chirality or diameters for metallic outermost shells and semiconductor outermost shells, respectively.

Fig. 8. Electrical conductivity of nanocomposites vs volume fraction of CNT, compared with Zare and Rhee’ s predictions [22], (a) experimental measurement of Gojny et al. for epoxy/SWCNT [57], and (b) experimental measurement of Lisunova et al. for ultrahigh molecular weight polyethylene/MWCNT [58]. (A colour version of this figure can be viewed online.)

C. Fang et al. / Carbon 146 (2019) 125e138

(a)

135

(b)

Fig. 10. Effects of aspect ratio of CNTs on the electrical conductivity of the nanocomposites, (a) a ¼ 2000, 1000, 500, 200, (b) a ¼ 100, 50, 20, 10, 5.

(a)

(b)

Fig. 11. Effects of the chirality on the electrical conductivity of the nanocomposites. (a) Outermost shell is metallic, (b) outermost shell is semiconductor.

By comparison, it is found that the electrical conductivity of the nanocomposites with metallic shell CNT and semiconductor shell CNT shows the same percolation threshold for the similar diameter. As the diameter increases from 1 nm to 30 nm, the percolation threshold is only increased from 0.4% to 0.7%. Results in Fig. 11 suggest that, the conductivity of the nanocomposites below the percolation threshold is not affected by m. While volume fraction is above percolation threshold, the conductivity of the nanocomposites is a result of the competition between two factors. (i) As m increases, the number of CNTs per unit volume decreases if the

volume fraction is fixed, resulting in an increase in the average distance between CNTs, and thus the contact conductance decreases. (ii) As m increases, the number of channels for the outermost shell increases and the contact conductance increases. For metallic shells, the number of channels remains 2 at D < 10 nm and increases to 4 as D increases to 30 nm according to Eq. (6). So, the conductivity of the nanocomposites decreases as m increases, and this decrease is slower at D > 10 nm. And for semiconductor shells, the number of channels increases sharply from 2  106 to 1 as D increases from 1 nm to 10 nm, which leads to the increase of the

136

C. Fang et al. / Carbon 146 (2019) 125e138

conductivity of the nanocomposites as m increases at D < 10 nm. With the increase of D from 10 nm to 30 nm, the number of channels increases from 1 to 3.6, then the influence of these two factors cancel each other out, so that the conductivity of the nanocomposites is basically unchanged with the increase of m at D > 10 nm. The number of channels in metallic shells and semiconductor shells is very close at D > 30 nm, which causes nearly the same conductivity of the nanocomposites with m ¼ 383 and m ¼ 384. 3.6. Dependence of percolation threshold on a, the aspect ratio of CNTs Finally, we study the dependence of percolation threshold on the aspect ratio of CNTs. The calculated results are plotted against the aspect ratio L/D in Fig. 12, where the line is theoretically calculated [27,59,60], and the experimentally reported percolation data for various composite materials are also marked by different symbols [48,54e56,61e71]. Based on the effective medium theory, Wang et al. [60] developed a continuum model to determine the electrical conductivity of CNT-based nano composites. Using an excluded volume percolation theory based model, the theoretical critical volume percolation threshold has an analytical expression [59]. Based on 3D resistor network modeling, Hu et al. [27] numerically simulated the electrical conductivity of nanocomposites with the tube-tube contacts of zero resistance, and the percolation threshold versus aspect ratio is very close to the analytical expression of Pfeifer. Considering tunneling effect, Bao et al. [28] get a lower percolation threshold about 0.005 in nanocomposites containing MWCNTs with the average diameter 50 nm and length 5 mm. The simulated percolation threshold in this paper is generally 10%e20% lower than that of Pfeifer and Hu et al. at L/ D ¼ 5e2000 because of the introduction of the tunneling effect. Our calculated results are generally consistent with those of other theories while noting that the experimental data have some fluctuations at large aspect ratio. The scattering of the experimental data is likely to be related to the interface conditions. Different

processing techniques could lead to different interface characteristics and affect the barrier heights. In Fig. 12, the percolation threshold in our theory was calculated at a constant barrier height, 3 eV, while different processing conditions may have resulted in different barrier heights. Recently Zare and Rhee [72] indicated that the formation of interphase - which reflects the barrier height in this theory - could shift the percolation threshold to a smaller value. As already demonstrated in Fig. 7, smaller barrier height would lead to a smaller percolation threshold and higher electrical conductivity. Indeed the electrical properties of the nanocomposites are intimately related to the interface effects. In spite of the notable scattering in experimental data, our theory is seen to be able to capture the major trend of the percolation threshold. 4. Concluding remarks In this paper we have developed a MC approach with equipotential approximation and electron tunneling to determine the electrical conductivity of CNT polymer nanocomposites. In this process we have clarified the fact that the contact conductance between every pair of crossing CNTs plays a more dominant role than the intrinsic conductance for the conductivity of nanocomposites with dispersed CNTs. Using the equipotential approximation, we have built two sets of linear equations by the introduction of a thinly coated surface. The potential of each CNT has been obtained by solving the linear equations, whose coefficient matrices are calculated by the WoS method. This model unifies SWCNT and MWCNTS, and is capable of calculating the percolation threshold and electrical conductivity for both metallic outermost shell and semiconductor outermost shell. It is also confirmed that the electrical conductivity both before and after the percolation threshold can be successfully simulated by the proposed equipotential approximation. The developed MC approach has been highlighted with direct comparison to three sets of experimental data. It has also been used to demonstrate the effects of barrier height, conductivity of polymer matrix, aspect ratio, diameter (and chirality) of CNT fillers on the overall conductivity of the nanocomposites. The predicted dependence of percolation threshold on the inclusion aspect ratio is also verified with existing theories and test data. In addition to ellipsoidal inclusions, this method is suitable for fillers of any shape, orientation and distribution. It is believed that this approach could have wide applicability in the study of electrical properties of carbon-based nanocomposites. Acknowledgements Chao Fang thanks the support of Education Department of Hubei under the Grant No. D20181802 and the support provided by the China Scholarship Council (CSC) during a one-year visit to Rutgers University. Juanjuan Zhang thanks the support of National Natural Science Foundation of China under the Grant No. 11702120 and the support provided by the China Scholarship Council (CSC) during a one-year visit to Rutgers University. George J. Weng thanks the support of NSF CMMI 1162431. References

Fig. 12. Dependence of the percolation threshold as a function of inclusion aspect ratio. (A colour version of this figure can be viewed online.)

[1] Y.M. Huang, X.Q. Lee, F.C. Macazo, M. Grattieri, R. Cai, S.D. Minteer, Fast and efficient removal of chromium (VI) anionic species by a reusable chitosanmodified multi-walled carbon nanotube composite, Chem. Eng. J. 339 (2018) 259e267. [2] V.D. Phadtare, V.G. Parale, G.K. Kulkarni, H.-H. Park, V.R. Puri, Enhanced microwave absorption of screen-printed multiwalled carbon nanotube/ Ca1xBaxBi2Nb2O9 (0x1) multilayered thick film composites, J. Alloy. Comp. 765 (2018) 878e887. [3] C. Wang, V. Murugadoss, J. Kong, Z.F. He, X.M. Mai, Q. Shao, Y.J. Chen, L. Guo,

C. Fang et al. / Carbon 146 (2019) 125e138

[4]

[5]

[6] [7]

[8] [9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26]

[27]

[28]

[29]

[30]

C.T. Liu, S. Angaiah, Z.H. Guo, Overview of carbon nanostructures and nanocomposites for electromagnetic wave shielding, Carbon 140 (2018) 696e733. H. Souri, J. Yu, H. Jeon, J.W. Kim, C.-M. Yang, N.-H. You, B.J. Yang, A theoretical study on the piezoresistive response of carbon nanotubes embedded in polymer nanocomposites in an elastic region, Carbon 120 (2017) 427e437. Z.S. Aghamiri, M. Mohsennia, H.-A. Rafiee-Pour, Fabrication and characterization of cytochrome c-immobilized polyaniline/multi-walled carbon nanotube composite thin film layers for biosensor applications, Thin Solid Films 660 (2018) 484e492. C.W. Nan, Y. Shen, J. Ma, Physical properties of composites near percolation, Annu. Rev. Mater. Res. 40 (2010) 131e151. G.J. Weng, Some elastic properties of reinforced solids, with special reference to isotropic ones containing spherical inclusions, Int. J. Eng. Sci. 22 (1984) 845e856. G.J. Weng, The theoretical connection between MorieTanaka’s theory and the HashineShtrikmaneWalpole bounds, Int. J. Eng. Sci. 28 (1990) 1111e1120. ~ eda, J.R. Willis, The effect of spatial distribution on the effective P.P. Castan behavior of composite materials and cracked media, J. Mech. Phys. Solid. 43 (1995) 1919e1951. Y. Pan, G.J. Weng, S.A. Meguid, W.S. Bao, Z.-H. Zhu, A.M.S. Hamouda, Percolation threshold and electrical conductivity of a two-phase composite containing randomly oriented ellipsoidal inclusions, J. Appl. Phys. 110 (12) (2011) 123715. Z. Hashin, S. Shtrikman, A variational approach to the theory of the effective magnetic permeability of multiphase materials, J. Appl. Phys. 33 (1962) 3125e3131. S.Y. Kim, Y.J. Noh, J. Yu, Prediction and experimental validation of electrical percolation by applying a modified micromechanics model considering multiple heterogeneous inclusions, Compos. Sci. Technol. 106 (2015) 156e162. Y. Zare, K.Y. Rhee, A simple model for electrical conductivity of polymer carbon nanotubes nanocomposites assuming the filler properties, interphase dimension, network level, interfacial tension and tunneling distance, Compos. Sci. Technol. 155 (2018) 252e260. D.A.G. Bruggeman, Calculation of various physics constants in heterogenous substances I Dielectricity constants and conductivity of mixed bodies from isotropic substances, Ann. Phys. 24 (1935) 636e664. Y. Wang, G.J. Weng, S.A. Meguid, A.M. Hamouda, A continuum model with a percolation threshold and tunneling-assisted interfacial conductivity for carbon nanotube-based nanocomposites, J. Appl. Phys. 115 (2014) 193706. R. Hashemi, G.J. Weng, A theoretical treatment of graphene nanocomposites with percolation threshold, tunneling-assisted conductivity and microcapacitor effect in AC and DC electrical settings, Carbon 96 (2016) 474e490. X. Xia, Y. Wang, Z. Zhong, G.J. Weng, A frequency-dependent theory of electrical conductivity and dielectric permittivity for graphene-polymer nanocomposites, Carbon 111 (2017) 221e230. Y. Su, J.J. Li, G.J. Weng, Theory of thermal conductivity of graphene-polymer nanocomposites with interfacial Kapitza resistance and graphene-graphene contact resistance, Carbon 137 (2018) 222e233. Y. Zare, K.Y. Rhee, S.-J. Park, Predictions of micromechanics models for interfacial/interphase parameters in polymer/metal nanocomposites, Int. J. Adhesion Adhes. 79 (2017) 111e116. Y. Zare, K.Y. Rhee, Dependence of Z Parameter for tensile strength of multilayered interphase in polymer nanocomposites to material and interphase properties, Nanoscale Res. Lett. 12 (1) (2017) 42. Y. Zare, K.Y. Rhee, Development of Hashin-Shtrikman model to determine the roles and properties of interphases in clay/CaCO3/PP ternary nanocomposite, Appl. Clay Sci. 137 (2017) 176e182. Y. Zare, K.Y. Rhee, A simple methodology to predict the tunneling conductivity of polymer/CNT nanocomposites by the roles of tunneling distance, interphase and CNT waviness, RSC Adv. 7 (2017) 34912e34921. Y. Zare, K.Y. Rhee, Development of a model for electrical conductivity of polymer/graphene nanocomposites assuming interphase and tunneling regions in conductive networks, Ind. Eng. Chem. Res. 56 (32) (2017) 9107e9115. J.-M. Zhua, Y. Zare, K.Y. Rhee, Analysis of the roles of interphase, waviness and agglomeration of CNT in the electrical conductivity and tensile modulus of polymer/CNT nanocomposites by theoretical approaches, Colloid. Surface. A 539 (2018) 29e36. R. Razavia, Y. Zare, K.Y. Rhee, A model for tensile strength of polymer/carbon nanotubes nanocomposites assuming the percolation of interphase regions, Colloid. Surface. A 538 (2018) 148e154. H.-X. Li, Y. Zare, K.Y. Rhee, The percolation threshold for tensile strength of polymer/CNT nanocomposites assuming filler network and interphase regions, Mater. Chem. Phys. 207 (2018) 76e83. N. Hu, Z. Masuda, C. Yan, G. Yamamoto, H. Fukunaga, T. Hashid, The electrical properties of polymer nanocomposites with carbon nanotube fillers, Nanotechnology 19 (21) (2008) 215701. W.S. Bao, S.A. Meguid, Z.H. Zhu, G.J. Weng, Tunneling resistance and its effect on the electrical conductivity of carbon nanotube nanocomposites, J. Appl. Phys. 111 (9) (2012), 093726. W.S. Bao, S.A. Meguid, Z.H. Zhu, M.J. Meguid, Modeling electrical conductivities of nanocomposites with aligned carbon nanotubes, Nanotechnology 22 (48) (2011) 485704. W.S. Bao, S.A. Meguid, Z.H. Zhu, Y. Pan, G.J. Weng, A novel approach to predict the electrical conductivity of multifunctional nanocomposites, Mech. Mater. 46 (2012) 129e138.

137

[31] S. Gong, Z.H. Zhu, S.A. Meguid, Anisotropic electrical conductivity of polymer composites with aligned carbon nanotubes, Polymer 56 (2015) 498e506. [32] S. Gong, Z.H. Zhu, J. Li, S.A. Meguid, Modeling and characterization of carbon nanotube agglomeration effect on electrical conductivity of carbon nanotube polymer composites, J. Appl. Phys. 116 (19) (2014) 194306. [33] A. Mora, F. Han, G. Lubineau, Estimating and understanding the efficiency of nanoparticles in enhancing the conductivity of carbon nanotube/polymer composites, Res. Phys. 10 (2018) 81e90. [34] O. Maxian, D. Pedrazzoli, I. Manas-Zloczower, Conductive polymer foams with carbon nanofillers - modeling percolation behavior, Express Polym. Lett. 11 (5) (2017) 406e418. [35] A. Naeemi, J.D. Meindl, Compact physical models for multiwall carbonnanotube interconnects, IEEE Electron. Device Lett. 27 (5) (2006) 338e340. [36] Z. Kordrostami, M.H. Sheikhi, R.M. Zadegan, Modeling electronic properties of multiwall carbon nanotubes, Fullerenes, Nanotub. Carbon Nanostruct. 16 (2008) 66e77. [37] H.J. Li, W.G. Lu, J.J. Li, X.D. Bai, C.Z. Gu, Multichannel ballistic transport in multiwall carbon nanotubes, Phys. Rev. Lett. 95 (8) (2005) 086601. [38] Y.-G. Yoon, P. Delaney, S.G. Louie, Quantum conductance of multiwall carbon nanotubes, Phys. Rev. B 66 (7) (2002), 073407. [39] B. Bourlon, C. Miko, L. Forro, D.C. Glattli, A. Bachtold, Determination of the intershell conductance in multiwalled carbon nanotubes, Phys. Rev. Lett. 93 (17) (2004) 172103. [40] S. Frank, P. Poncharal, Z.L. Wang, W.A. de Heer, Carbon nanotube quantum resistors, Science 280 (5370) (1998) 1744e1746. , T. Nussbaumer, [41] A. Bachtold, C. Strunk, J.-P. Salvetat, J.-M. Bonard, L. Forro €nenberger, AharonoveBohm oscillations in carbon nanotubes, Nature C. Scho 397 (1999) 673. [42] S. Gong, Z.H. Zhu, E.I. Haddad, Modeling electrical conductivity of nanocomposites by considering carbon nanotube deformation at nanotube junctions, J. Appl. Phys. 114 (7) (2013), 074303. [43] T. Hertel, R.E. Walkup, P. Avouris, Deformation of carbon nanotubes by surface van der Waals forces, Phys. Rev. B 58 (20) (1998) 13870e13873. [44] B. De Vivo, P. Lamberti, G. Spinelli, V. Tucci, L. Vertuccio, V. Vittoria, Simulation and experimental characterization of polymer/carbon nanotubes composites for strain sensor applications, J. Appl. Phys. 116 (5) (2014), 054307. [45] M. Shiraishi, M. Ata, Work function of carbon nanotubes, Carbon 39 (12) (2001) 1913e1917. [46] A. Buldum, J.P. Lu, Contact resistance between carbon nanotubes, Phys. Rev. B 63 (16) (2001), 161403 (R). [47] P.C. Ma, B.Z. Tang, J.-K. Kim, Effect of CNT decoration with silver nanoparticles on electrical conductivity of CNT-polymer composites, Carbon 46 (2008) 1497e1505. [48] C. Li, E.T. Thostenson, T.-W. Chou, Dominant role of tunneling resistance in the electrical conductivity of carbon nanotubeebased composites, Appl. Phys. Lett. 91 (22) (2007) 223114. [49] H.C. Chin, A. Bhattacharyya, V.K. Arora, Extraction of nanoelectronic parameters from quantum conductance in a carbon nanotube, Carbon 76 (2014) 451e454. [50] S.D. Li, Z. Yu, C. Rutherglen, P.J. Burke, Electrical properties of 0.4 cm long single-walled carbon nanotubes, Nano Lett. 4 (10) (2004) 2003e2007. [51] C.T. White, T.N. Todorov, Carbon nanotubes as long ballistic conductors, Nature 393 (1998) 240e242. [52] J.F. Reynolds, A proof of the random-walk method for solving laplace's equation in 2-D, Math. Gaz. 49 (1965) 416e420. [53] A.E. Kyprianou, A. Osojnik, T. Shardlow, Unbiased ‘walk-on-spheres’ Monte Carlo methods for the fractional Laplacian, IMA J. Numer. Anal. 38 (3) (2018) 1550e1578. [54] Z. Ounaiesa, C. Park, K.E. Wise, E.J. Siochi, J.S. Harrison, Electrical properties of single wall carbon nanotube reinforced polyimide composites, Compos. Sci. Technol. 63 (11) (2003) 1637e1646. [55] Y. Ngabonziza, J. Li, C.F. Barry, Electrical conductivity and mechanical properties of multiwalled carbon nanotube-reinforced polypropylene nanocomposites, Acta Mech. 220 (1e4) (2011) 289e298. [56] N. Hu, Z. Masuda, G. Yamamoto, H. Fukunaga, T. Hashida, J.H. Qiu, Effect of fabrication process on electrical properties of polymer/multi-wall carbon nanotube nanocomposites, Compos. Part A-Appl. S. 39 (2008) 893e903. [57] F.H. Gojny, M.H. Wichmann, B. Fiedler, I.A. Kinloch, W. Bauhofer, A.H. Windle, K. Schulte, Evaluation and identication of electrical and thermal conduction mechanisms in carbon nanotube/epoxy composites, Polymer 47 (6) (2006) 2036e2045. [58] M.O. Lisunova, Ye.P. Mamunya, N.I. Lebovka, A.V. Melezhyk, Percolation behaviour of ultrahigh molecular weight polyethylene/multi-walled carbon nanotubes composites, Eur. Polym. J. 43 (3) (2007) 949e958. [59] S. Pfeifer, S.-H. Park, P.R. Bandaru, Analysis of electrical percolation thresholds in carbon nanotube networks using the Weibull probability distribution, J. Appl. Phys. 108 (2) (2010), 024305. [60] Y. Wang, G.J. Weng, S.A. Meguid, A.M. Hamouda, A continuum model with a percolation threshold and tunneling-assisted interfacial conductivity for carbon nanotube-based nanocomposites, J. Appl. Phys. 115 (19) (2014) 193706. [61] J. Sandler, M.S.P. Shaffer, T. Prasse, W. Bauhofer, K. Schulte, A.H. Windle, Development of a dispersion process for carbon nanotubes in an epoxy matrix and the resulting electrical properties, Polymer 40 (1999) 5967e5971. [62] D.S. Mclachlan, C. Chiteme, C. Park, K.E. Wise, S.E. Lowther, P.T. Lillehei, E.J. Siochi, J.S. Harrison, AC and DC percolative conductivity of single wall

138

[63]

[64]

[65]

[66]

[67]

C. Fang et al. / Carbon 146 (2019) 125e138 carbon nanotube polymer composites, J. Polym. Sci. Part B 43 (22) (2005) 3273e3287. N. Grossiord, J. Loos, L.V. Laake, M. Maugey, C. Zakri, C.E. Koning, A.J. Hart, High-conductivity polymer nanocomposites obtained by tailoring the characteristics of carbon nanotube fillers, Adv. Funct. Mater. 18 (20) (2008) 3226e3234. J. Vilcakova, P. Saha, O. Quadrat, Electrical conductivity of carbon fibres/ polyester resin composites in the percolation threshold region, Eur. Polym. J. 38 (12) (2002) 2343e2347. J.Z. Kovacs, R.E. Mandjarov, T. Blisnjuk, K. Prehn, M. Sussiek, J. Muller, K. Schulte, W. Bauhofer, On the influence of nanotube properties, processing conditions and shear forces on the electrical conductivity of carbon nanotube epoxy composites, Nanotechnology 20 (15) (2009) 155703. S. Kara, E. Arda, F. Dolastir, O. Pekcan, Electrical and optical percolations of polystyrene latex-multiwalled carbon nanotube composites, J. Colloid Interphase Sci. 344 (2) (2010) 395e401. F. Dalmas, R. Dendievel, L. Chazeau, J. Cavaille, C. Gauthier, Numerical simulation of electrical conductivity in three-dimensional entangled fibrous networks, Acta Mater. 54 (11) (2006) 2923e2931.

[68] Y.R. Hernandez, A. Gryson, F.M. Blighe, M. Cadek, V. Nicolosi, W.J. Blau, Y.K. Gun’ko, J.N. Coleman, Comparison of carbon nanotubes and nanodisks as percolative fillers in electrically conductive composites, Scripta Mater. 58 (1) (2008) 69e72. [69] Y.J. Kim, T.S. Shin, H.D. Choi, J.H. Kwon, Y.-C. Chung, H.G. Yoon, Electrical conductivity of chemically modified multiwalled carbon nanotube/epoxy composites, Carbon 43 (1) (2005) 23e30. [70] J.L. Luo, C.W. Zhang, Z.D. Duan, B.L. Wang, Q.Y. Li, K.L. Chung, J.G. Zhang, S.C. Chen, Influences of multi-walled carbon nanotube (MCNT) fraction, moisture, stress/strain level on the electrical properties of MCNT cementbased composites, Sens. Actuators, A 280 (2018) 413e421. [71] M.A. Mazo, A. Tamayo, A.C. Caballero, J. Rubio, Enhanced electrical and thermal conductivities of silicon oxycarbide nanocomposites containing carbon nanofibers, Carbon 138 (2018) 42e51. [72] Y. Zare, K.Y. Rhee, Development and modification of conventional Ouali model for tensile modulus of polymer/carbon nanotubes nanocomposites assuming the roles of dispersed and networked nanoparticles and surrounding interphases, J. Colloid Interface Sci. 506 (2017) 283e290.