Development of a Model for Electrical Conductivity of Polymer

Jul 25, 2017 - Department of Mechanical Engineering, College of Engineering, Kyung Hee ... affects the conductive networks and conductivity.9 However,...
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Development of a Model for Electrical Conductivity of Polymer/ Graphene Nanocomposites Assuming Interphase and Tunneling Regions in Conductive Networks Yasser Zare† and Kyong Yop Rhee*,‡ †

Young Researchers and Elites Club, Science and Research Branch, Islamic Azad University, Tehran, Iran Department of Mechanical Engineering, College of Engineering, Kyung Hee University, Yongin 446-701, Republic of Korea

‡

ABSTRACT: In this study, a conventional model for the resistivity of conductive composites is developed for polymer/ graphene nanocomposites taking into account the interphase and tunneling spaces in the conductive networks. The developed model considers the effects of the graphene dimensions, volume fraction of graphene in the conductive networks, contact diameter, number of contacts between nanosheets, orientation angle, interphase thickness, and tunneling distance on the conductivity of the nanocomposites. The experimental results for conductivity and the reasonable effects of different parameters on the conductivity confirm the developed model. Thin and large nanosheets, small tunneling distances, thick interphases, large contact diameters, low orientation angles, and high fractions of percolated nanosheets in the networks can improve the conductivity of a nanocomposite. Moreover, the thickness of the nanosheets and the tunneling distance cause the largest variations in the conductivities of nanocomposites.

1. INTRODUCTION Graphene can be considered as the best nanofiller for polymer nanocomposites in which polymer is the main phase and graphene is the added phase, because it shows significant extents of electrical conductivity and mechanical properties.1−8 Studies on polymer/graphene nanocomposites have attracted much interest in recent years. The extraordinary aspect ratio (diameter per unit thickness) of graphene nanosheets reduces the percolation threshold in nanocomposites, which positively affects the conductive networks and conductivity.9 However, the large surface energy and strong interaction of graphene nanosheets limit their uniform dispersion in polymer matrixes.10,11 Researchers have attempted to solve this problem by many methods including in situ polymerization in the presence of nanoparticles, solution mixing, ultrasonication, and melt processing.12,13 Electrons can be transferred through the contact regions between adjacent nanosheets in conductive networks.14,15 In fact, a short distance between nanosheets can also improve the conductivity of nanocomposites, through the tunneling mechanism. The tunneling conductivity as a function of filler size, polymer matrix, contact distance, and contact area has been studied to a limited extent in previous works.16−18 Moreover, the interphase regions around nanoparticles (between polymer matrix and nanofiller) usually influence the properties of polymer nanocomposites.19−21 The effects of interphase thickness and stiffness on the mechanical performance of nanocomposites have been studied extensively.22−24 It was reported that the interphase forms a high volume of © XXXX American Chemical Society

nanocomposites and plays a reinforcing role. Similarly, the interphase areas can shift the percolation threshold to small volume fractions, which can positivity change the conductivity of nanocomposites.25−27 The effects of the interphase on the percolation threshold and mechanical properties of polymer nanocomposites were described in previous articles.28−32 However, there is no simple model for the conductivity of nanocomposites assuming tunneling effects and interphase regions. Clearly, these terms contribute to smaller percolation thresholds and superior conductivities for graphene nanocomposites as compared to conventional composites. In other words, the tunneling effect and interphase spaces are the main factors that improve the electrical conductivity in polymer nanocomposites in comparison to microcomposites because the percolating level of particles is reduced by these factors, which upgrades the conductive networks and nanocomposite conductivity. Modeling studies on the conductivity of polymer/graphene nanocomposites are limited. Most earlier works linked the percolation threshold to the filler aspect ratio and assessed the conductivity in terms of the conventional power-law model.33−35 In fact, previous studies did not consider the roles of interphase and tunneling regions in the percolation level and conductivity. The main novelty and advantage of this work involve the assumption of these terms in Received: Revised: Accepted: Published: A

April 2, 2017 July 12, 2017 July 25, 2017 July 25, 2017 DOI: 10.1021/acs.iecr.7b01348 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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replaced by D. Moreover, the conduction of graphene is expressed as σf = 1/ρf. Therefore, eq 3 can be rewritten for polymer/graphene nanocomposites as

the percolation threshold and conductivity of graphene nanocomposites. Weber and Kamal36 suggested a model for the volume resistivity of conductive composites in terms of the roles of the filler volume fraction, filler dimensions, filler conduction, contact number, contact diameter and angle between nanoparticles, and direction of current. However, this model cannot address the novel aspects of polymer nanocomposites such as interphase and tunneling regions. In this work, the model suggested by Weber and Kamal36 is developed for polymer nanocomposites containing graphene nanosheets. The developed model considers the effects of the volume fraction of graphene in the conductive networks, the contact diameter, the orientation angle, and the tunneling distance on the conductivity of nanocomposites. In addition, the volume fraction of graphene in the conductive networks is expressed in terms of the percolation threshold and effective filler fraction depending on the graphene dimensions, interphase thickness, and tunneling distance. Therefore, the developed equations properly show the roles of interphase and tunneling regions in the percolation threshold, effective filler concentration, and conductivity. Many experimental results are applied to evaluate the developed equations. In addition, the effects of different model parameters on the conductivity are plotted to justify the developed model and to determine the efficiencies of the parameters. The developed equations can facilitate the estimation of the percolation level and conductivity in graphene nanocomposites in terms of the effects of the interphase and tunneling regions.

σ=

For a three-dimensional (3D) random distribution of nanoparticles in nanocomposites,38 it is assumed that 1 cos2 θ = (6) 3 Moreover, ϕN is expressed as ϕN = fϕf

1 0.59 + 0.15m

(1)

f=

(2)

(8)

(9)

where ti is the interphase thickness. The effective volume fraction of graphene in the nanocomposite can be expressed in terms of the total concentrations of nanoparticles and interphase regions as

ϕNdcl cos θ (3)

This model can be upgraded for polymer/graphene nanocomposites assuming the structure and properties of graphene nanosheets in addition to the dissimilar aspects of nanocomposites compared to composites. The cross-sectional area of graphene nanosheets can be presented as A f = tD

1 − ϕp1/3

⎛ 2t ⎞ ϕi = ϕf ⎜ i ⎟ ⎝ t ⎠

2

A f ρf X

ϕf 1/3 − ϕp1/3

where ϕp is the volume fraction of the percolation threshold. This equation can be applied for polymer/graphene nanocomposites. However, the interphase and contact regions affect the effective filler concentration and percolation threshold. The volume fraction of the interphase areas in a nanocomposite containing graphene39 is defined as

where the maximum m value was reported as 15. The conductivity of fiber composites can be obtained as the inverse of ρ, namely σ=

(7)

where f denotes the fraction of nanoparticles in the networks and ϕf is the total volume fraction of nanoparticles. The f parameter for polymer−carbon nanotube (CNT) nanocomposites14 is given by

where Af is the fiber cross-sectional area, ρf is the fiber resistivity, ϕN is the volume fraction of networked fibers, dc is the diameter of the contact area, l is the fiber length, and θ is the angle between the fibers and the direction of the current. X is a function of the number of contacts (m), given by X=

(5)

Figure 1. Schematic of contact regions between overlapped graphene nanosheets in nanocomposites.

A f ρf X ϕNdcl cos2 θ

tX

Morphological images illustrate overlapping between nanosheets in nanocomposites instead of tip-to-tip contacts.37 Accordingly, the conduction mostly occurs through the overlapped nanosheets, as shown in Figure 1.

2. THEORETICAL APPROACHES Weber and Kamal36 proposed that the longitudinal resistivity of polymer/fiber composites assuming fiber−fiber contacts (body contacts rather than end to end) can be expressed as ρ=

ϕNdcσf cos2 θ

⎛ 2t ⎞ ϕeff = ϕf + ϕi = ϕf ⎜1 + i ⎟ ⎝ t ⎠

(10)

Furthermore, the percolation threshold in nanocomposites containing randomly oriented graphite nanosheets was suggested40 to be given by

(4)

ϕp =

where D and t are the diameter and thickness, respectively, of the graphene nanosheets. Also, l, as the length of the fibers, is B

27πD2t 4(D + λ)3

(11) DOI: 10.1021/acs.iecr.7b01348 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Figure 2. Experimental results and calculations of conductivity according to the developed model for (a) PVA/graphene,42 (b) PI/graphene,43 (c) PET/graphene,44 and (d) ABS/graphene45 nanocomposites.

where λ is the tunneling distance between neighboring nanoparticles. However, D ≫ λ removes the effect of the tunneling distance, giving ϕp =

27πt 4D

ϕN =

27πt 4D + 2(Dt i + Dλ)

1 − ϕp1/3

(12)

(13)

ϕeff 1/3 − ϕp1/3

expressing that the percolation threshold depends inversely on the graphene diameter, interphase thickness, and tunneling distance. This equation can be utilized to predict the interphase thickness and tunneling distance in terms of the percolation level. When the roles of interphase and tunneling regions are assumed in the effective filler concentration and percolation threshold, f in eq 8 can be rewritten as f=

σ=

1 − ϕp1/3

(15)

ϕeff dcσf cos2 θ 3

()

tX

λ ζ

(16)

where ζ is a tunneling parameter equal to 0.1 nm. Equation 16 presents a simple model for the conductivity of graphene nanocomposites taking into account the graphene dimensions, volume fraction of graphene in the conductive networks, contact diameter, number of contacts between nanosheets, orientation angle, interphase thickness, and tunneling distance.

ϕeff 1/3 − ϕp1/3 1 − ϕp1/3

ϕeff

Additionally, the tunneling distance considerably affects the conductivity of nanocomposites, because it controls the level of electron transfer at contact regions between adjacent nanosheets (see Figure 1). Many studies have related the tunneling distance (λ) to ϕf−1/3.14,41 Because the developed model expresses a linear relationship between conductivity and ϕf, it can be assumed that the conductivity is a function of λ−3. Assuming the latter equation and substituting λ−3 into eq 5 establishes the developed model for the conductivity of polymer/graphene nanocomposites as

The interphase regions around graphene nanosheets and the contact spaces mainly shift the percolation threshold to smaller filler concentrations. The roles of these parameters in the percolation threshold can be regarded as ϕp =

ϕeff 1/3 − ϕp1/3

(14)

3. RESULTS AND DISCUSSION 3.1. Evaluation by Experimental Measurements. Many experimental results have been applied to confirm the

Thus, the volume fraction of conductive networks in the nanocomposite is expressed as C

DOI: 10.1021/acs.iecr.7b01348 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Figure 3. (a) 3D and (b) contour plots of conductivity as a function of t and λ parameters at average levels of the other factors according to the developed model.

nanocomposites. The highest values of (m, dc) were found in the PET/graphene nanocomposite. As observed in Figure 2, the highest values of conductivity were obtained in the PET/ graphene sample, which confirms the calculations of (m, dc). On the other hand, the lowest values of (m, dc) were calculated for the ABS/graphene sample. The poor conductivity of this sample at high filler concentrations (Figure 2d) confirms the obtained results. Therefore, the model and equations developed in the present study can properly express the percolation threshold, interphase thickness, tunneling distance, number of contacts between nanosheets, and contact diameter in terms of experimental measurements of the electrical conductivity. 3.2. Parametric Analyses. The developed model can determine the effects of different parameters on the conductivities of nanocomposites. Parametric analyses can justify the developed model and show the efficiency of each parameter in terms of its effects on the conductivity of graphene nanocomposites. Three-dimensional (3D) and contour plots are applied to display the effects of two parameters on the conductivity at average levels of other parameters. The following average and constant levels of the parameters were assumed in all calculations: t = 2 nm, ϕf = 0.01, D = 1 μm, ti = 5 nm, λ = 5 nm, σf = 105 S/m, m = 8, dc = 50 nm, and cos2(θ) = 1/3. Figure 3 demonstrates the effects of the parameters t and λ on the conductivity of nanocomposites at average levels of the other parameters using the developed model. The highest conductivity of 14 S/m is achieved at t = 1 nm and λ = 2 nm, whereas values of t > 2 nm and λ > 4 nm mainly decrease the conductivity to about 0. Actually, the most desirable conductivity is obtained at the lowest values of t and λ, but the higher levels of these parameters produce an insulated nanocomposite. Accordingly, very thin graphene nanosheets and short tunneling distances can improve the conductivity of nanocomposites. On the other hand, thick nanosheets and large distances between nanosheets significantly weaken the conductivity. These findings demonstrate that these parameters mainly change the conductivity of nanocomposites from 0 to 14 S/m, so they should be controlled to achieve the desired conductivity. Thin nanosheets enlarge the interphase regions and reduce the percolation threshold in nanocomposites, because they increase the specific surface areas of the nanoparticles and the number of contacts between nanosheets. Because the conductive networks in nanocomposites are promoted by

developed model. Four samples including poly(vinyl alcohol) (PVA)/graphene (t = 2 nm, D ≈ 2 μm),42 polyimide (PI)/ graphene (t = 3 nm, D ≈ 5 μm),43 poly(ethylene terephthalate) (PET)/graphene (t = 2 nm, D ≈ 2 μm),44 and acrylonitrile butadiene styrene (ABS)/graphene (t = 1 nm, D ≈ 4 μm)45 are considered. The experimental results for electrical conductivity give the percolation thresholds (ϕp) as 0.0035, 0.0015, 0.005, and 0.0013, respectively, for these samples. These levels are the critical filler volume fraction of graphene in the samples at which the conductivity significantly improves. Using eq 13 for the percolation threshold, the values of (ti, λ) as the interphase thickness and tunneling distance, respectively, were calculated as (5, 5), (7, 9), (3, 4), and (3, 3) nm for PVA/graphene, PI/ graphene, PET/graphene, and ABS/graphene nanocomposites, respectively. The thickest interphase and the largest tunneling distance were obtained for the PI/graphene sample, whereas the lowest values of (ti, λ) were observed for the PET/graphene and ABS/graphene nanocomposites. These calculations demonstrate that the interphase thickness and tunneling distance mainly affect the percolation threshold of graphene nanosheets in nanocomposites and the proper assumptions of these terms are essential for calculating the correct percolation threshold. In other words, the absence of these terms overpredicts the percolation threshold in polymer/graphene nanocomposites. The developed model can estimate the conductivities of the samples by applying the graphene dimensions, interphase thickness, and tunneling distance. The conduction of graphene (σf) and cos2(θ) are considered to be 105 S/m and 1/3, respectively. Figure 2 illustrates the experimental results for conductivity and the calculations according to the developed model for the reported samples. All calculations at different levels of graphene concentrations agree with the experimental data. Thes results demonstrate that the developed model can properly estimate the electrical conductivity in the reported samples. Therefore, the developed model can be applied for the conductivities of polymer/graphene nanocomposites assuming the interphase and contact regions. The values of (m, dc) as the number of contacts and contact diameter (nm) were also calculated using the experimental conductivities and the developed model. The calculated values of (m, dc) are (3, 5), (10, 50), (13, 50), and (1, 1) for the PVA/ graphene, PI/graphene, PET/graphene, and ABS/graphene samples, respectively. A better conductivity is achieved with higher values of these parameters, because a high number of contacts and a large contact area improve the conductivity of D

DOI: 10.1021/acs.iecr.7b01348 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Figure 4. Effects of ti and dc on the conductivity of the nanocomposite based on the developed model at average levels of the other parameters: (a) 3D and (b) contour plots.

Figure 5. Dependences of the conductivity on D and θ at average values of the other parameters: (a) 3D and (b) contour plots.

According to eq 10, a thick interphase increases the effective concentration of graphene in the nanocomposite . Moreover, a smaller percolation threshold is obtained with a thicker interphase, because the interphase regions can form network structures in the nanocomposite. In fact, a thick interphase around nanosheets grows the extent of networks in the nanocomposite, because the nanosheets participate in the conductive networks. As a result, the observation of a higher conductivity is expected for a thicker interphase. On the other hand, a large contact diameter increases the contact area between overlapping nanosheets in the nanocomposite (see Figure 1). Under these conditions, the surface area of nanosheets in the contact regions increases, which strengthens the tunneling conductivity. In other words, a large contact area decreases the contact resistance, because of the existence of large conductive nanosheets that improve the electron transport through the contact regions. Accordingly, a higher conductivity is obtained for a larger contact area between two adjacent nanosheets. An inverse relationship between contact resistance and contact diameter was reported in a previous article.14 Conclusively, the developed model shows the correct correlation between conductivity and contact diameter. Figure 5 shows the conductivity as a function of the parameters D and θ at average values of the other parameters. The conductivity improves at high D and low θ, but the lowest conductivity is observed at the highest values of θ. A conductivity of 0.6 S/m was calculated at D > 2 μm and θ = 0°, whereas an insulating surface was observed at θ > 70°.

larger interphase regions, larger surface areas of nanoparticles, and numerous contacts between nanosheets, a high conductivity is obtained under these conditions. In other words, thin nanosheets can raise the conductivity of nanocomposites, because they positively affect the conductive networks for electron transfer. The tunneling distance also inversely governs the conductivity of nanocomposites. The contacts between nanosheets commonly contain the polymer layer covered by conductive nanosheets. When the tunneling distance increases, the insulated polymer layer controls the tunneling resistance rather than the graphene nanosheets. As a result, a high tunneling distance results in a poor conductivity, because of the increased level of tunneling resistance and poor electron transfer. Many reports have also confirmed the inverse trend between conductivity and tunneling distance.41,46 Therefore, it is logical that a better conductivity can be obtained with a smaller tunneling distance, as expressed by the present model. The effects of ti and dc on the conductivity of nanocomposites are illustrated in Figure 4. Both ti and dc directly influence the conductivity of nanocomposites. The highest conductivity of 0.9 S/m is obtained with ti = 10 nm and dc = 100 nm, whereas an insulated nanocomposite is observed at ti < 4 nm and dc < 40 nm. Thus, a thick interphase and large contact diameter improve the conductivity of nanocomposites, but a nanocomposite with a thin interphase and small contact diameter shows very poor conductivity. E

DOI: 10.1021/acs.iecr.7b01348 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Figure 6. Conductivities of nanocomposites calculated according to the developed model at different values of ϕeff and ϕp: (a) 3D and (b) contour plots.

Figure 7. (a) 3D and (b) contour plots for the correlations of conductivity with the parameters f and m based on the developed model.

current can effectively transfer the electrons, whereas an orientation of θ = 90° produces poor electron current. Actually, large and dense conductive networks at θ = 0° can efficiently transfer the electrons, producing a desirable conductivity. In contrast, the inappropriate orientation of nanosheets decreases the effectiveness of electron transport, which weakens the conductivity. According to Figure 5, θ > 70° mainly decreases the conductivity at different levels of D. As a result, it is important to prevent the unsuitable orientation of the nanosheets in nanocomposites. However, common techniques for the preparation of nanocomposites such as melt mixing, solution mixing, and in situ polymerization randomly distribute the nanoparticles in the nanocomposite. In fact, controlling and characterizing the orientation angle in nanocomposites produced by common techniques are very difficult. Therefore, a random distribution of nanoparticles is frequently assumed in the nanocomposites. Some special techniques can be developed for the production of polymer nanocomposites to obtain a desirable orientation angle. The variations in nanocomposite conductivity at different values of ϕeff and ϕp based on the developed model are plotted in Figure 6. The highest conductivity of 0.45 S/m is observed at ϕeff = 0.1 and ϕp = 0.001, whereas a poor conductivity of 0.03 S/m is obtained at ϕeff < 0.025 and ϕp > 0.004. It is evident that a more desirable conductivity is obtained at higher ϕeff and lower ϕp. Therefore, the effective concentration of nanoparticles and the percolation threshold should be controlled to obtain a good conductivity in the nanocomposite.

Accordingly, a high nanosheet diameter and a low orientation angle desirably affect the conductivity of nanocomposites. In other words, the diameter of the nanosheets and the orientation angle directly and inversely control the conductivity of nanocomposites, respectively. Larger nanosheets shift the percolation threshold to smaller filler fractions, because such nanosheets have a higher potential for networking. Undoubtedly, the distance between large nanosheets decreases, which increases the possibility of networking. Moreover, according to eq 14, large nanosheets increase the efficiency of conductive networks in nanocomposites. As a result, larger nanosheets positively change the conductivity of nanocomposites. Previous articles also showed the desirable effects of large nanosheets on the geometric percolation threshold.47,48 These works suggested an inverse relationship between the percolation threshold and the aspect ratio of the nanosheets in which a high aspect ratio of large nanosheets decreases the percolation level. Accordingly, large nanosheets decrease the percolation threshold, which expands the conductive networks and produces high conductivity. The orientation angle of nanosheets can largely govern the conductivity of nanocomposites. The orientation of nanosheets parallel to the direction of the current (θ = 0°) increases the conductivity of nanocomposites, but the orientation of nanosheets perpendicular to the electron current (θ = 90°) leads to a very poor conductivity. These effects are logical, because the orientation of nanosheets in the direction of the F

DOI: 10.1021/acs.iecr.7b01348 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research ϕeff directly affects the conductivity of nanocomposites, because it represents the efficiency of nanoparticles in the nanocomposite. In fact, a high effective filler concentration increases the level of percolated nanosheets in the conductive networks ( f in eq 14), which directly influences the conductivity. Accordingly, the direct relationship between ϕeff and conductivity is attributed to the formation of large conductive networks in the nanocomposite. However, a high ϕeff value is obtained with a thick interphase and thin nanosheets, whereas the poor interfacial interaction/adhesion between the polymer matrix and graphene nanosheets and the aggregation/agglomeration of nanoparticles during processing produce a thin interphase and thick nanosheets,49−51 which considerably diminish ϕeff. Thus, it is important to regulate these parameters to increase the efficiency of nanosheets in the nanocomposite. The percolation threshold shows the minimum content of nanosheets for the establishment of conductive networks in the nanocomposites. Additionally, according to eq 15, a high percolation threshold negatively affects the number of nanosheets belonging to conductive networks. As a result, it is logical to observe an inverse relationship between the conductivity and percolation threshold, because a high percolation level deteriorates the performance of conductive networks. Experimental and theoretical studies on the conductivity of nanocomposites also reported an inverse relationship between conductivity and percolation level.45,52 Therefore, the developed model correctly expresses the negative effect of a high percolation threshold on the conductivity. Figure 7 depicts the dependences of conductivity on f and m predicted by the developed model. The maximum conductivity of 0.9 S/m is observed at f = 0.7 and m = 19, whereas the conductivity decreases to 0.07 S/m at f < 0.15 and m < 7. As a result, these parameters representing the fraction of percolated nanosheets in the networks and the number of contacts, respectively, directly govern the conductivity of graphene nanocomposites. These findings are reasonable considering the formation and extent of conductive networks in such nanocomposites. A high f value shows the contribution of a large number of nanosheets to the conductive networks. Thus, a higher f value indicates the formation of larger and denser networks in the nanocomposite, which produces a better conductivity. It was mentioned that f depends on the effective concentration of nanosheets in the nanocomposite and the percolation threshold. Generally, thin and large nanosheets, thick interphases, and small tunneling distances can present suitable levels for effective filler concentrations and percolation thresholds, which result in high f values. Moreover, a high number of contacts between nanosheets can result in conductive networks in the nanocomposite. Accordingly, a high m value desirably affects the conductivity of nanocomposites, because it diminishes the distance and resistance between adjacent nanosheets. Thus, the developed model reasonably describes the effects of f and m on the conductivity of nanocomposite.

analyses. The experimental results for percolation threshold and electrical conductivity and the parametric examinations verify the prediction abilities of the developed equations. The highest conductivity of 14 S/m was calculated at t = 1 nm and λ = 2 nm, whereas values of t > 2 nm and λ > 4 nm chiefly decrease the conductivity to about 0. Therefore, thin nanosheets and small tunneling distances obtain the most desirable conductivity. These parameters cause the highest variations in the conductivity of nanocomposites. In addition, a thick interphase and large contact diameter increase the conductivity, but a nanocomposite with a thin interphase and small contact diameter shows very poor conductivity. A conductivity of 0.6 S/m was found at D > 2 μm and θ = 0°, whereas an insulating behavior was observed at θ > 70°. Thus, a large nanosheet diameter and low orientation angle improve the conductivity of nanocomposites. Furthermore, a more desirable conductivity is obtained with a higher effective filler concentration and a lower percolation threshold. The maximum conductivity of 0.9 S/m was observed at f = 0.7 and m = 19, whereas the conductivity decreased to 0.07 S/m at f < 0.15 and m < 7. As a result, the fraction of percolated nanosheets in the networks and the number of contacts directly affect the conductivity of nanocomposites.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +82 31 201 2565. Fax: +82 31 202 6693. E-mail: [email protected]. ORCID

Yasser Zare: 0000-0002-1293-8878 Notes

The authors declare no competing financial interest.

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REFERENCES

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4. CONCLUSIONS The conventional model proposed by Weber and Kamal36 for the resistivity of conductive composites was developed for polymer/graphene nanocomposites by including interphase thickness and tunneling distance. The develop model was supported by many experimental results and parameter G

DOI: 10.1021/acs.iecr.7b01348 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

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DOI: 10.1021/acs.iecr.7b01348 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.iecr.7b01348 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX