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Technical Brief

# Effect of Temperature on Electrical Resistivity of Carbon Nanotubes and Graphene Nanoplatelets NanocompositesOPEN ACCESS

[+] Author and Article Information
Amirhossein Biabangard Oskouyi

Department of Mechanical Engineering,
University of Alberta,
e-mail: biabanga@ualberta.ca

Uttandaraman Sundararaj

Department of Chemical and Petroleum Engineering,
University of Calgary,
e-mail: ut@ucalgary.ca

Pierre Mertiny

Department of Mechanical Engineering,
University of Alberta,
e-mail: pmertiny@ualberta.ca

1Corresponding author.

Manuscript received June 30, 2014; final manuscript received February 27, 2015; published online April 2, 2015. Assoc. Editor: Debjyoti Banerjee.

J. Nanotechnol. Eng. Med 5(4), 044501 (Nov 01, 2014) (6 pages) Paper No: NANO-14-1044; doi: 10.1115/1.4030018 History: Received June 30, 2014; Revised February 27, 2015; Online April 02, 2015

## Abstract

The effect of the temperature on the electrical resistivity of polymer nanocomposites with carbon nanotube (CNT) and graphene nanoplatelets (GNP) fillers was investigated. A three-dimensional (3D) continuum Monte Carlo (MC) model was developed to first form percolation networks. A 3D resistor network was subsequently created to evaluate the nanocomposite electrical properties. The effect of temperature on the electrical resistivity of nanocomposites was thus investigated. Other aspects such as polymer tunneling and filler resistivities were considered as well. The presented comprehensive modeling approach is aimed at providing a better understanding of the electrical resistivity behavior of polymer nanocomposites in conjunction with experimental works.

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## Introduction

The electrical resistivity of polymer nanocomposites may be affected by temperature. This characteristic makes them promising materials for temperature sensors for a wide range of applications. High sensitivity, linearity, stability, a wide operating range, and low cost are desirable characteristics for an ideal temperature sensor [1]. Thermocouples exhibit linear behavior, but their comparatively high production cost restricts their applications. Transistor-based temperature sensors have a linear behavior over a wide operating range, but they are associated with drawbacks such as low sensitivity [1]. Alternatively, thermoresistive nanocomposite polymers can be employed for temperature sensing applications. Low production cost and the ease of tailoring their geometry can be listed as the main advantages for this group of sensors. The electrical properties of polymers filled with CNT and GNP have been the subject of extensive research in recent years [2,3,4,5,6,7,8,9,10,11-2,3,4,5,6,7,8,9,10,11]. Several experimental and numerical studies have been devoted to the investigation of the percolation threshold (i.e., the transition between electrical insulator and conductor), electrical resistivity properties, and piezoresistivity effects. However, only a few studies have been conducted to investigate the effect of the temperature on the electrical characteristics of polymeric nanocomposites [1,12,12-14,14-15]. Karimov et al. [16] investigated the effect of temperature on the resistivity and the Seebeck coefficient of a CNT-polymer nanocomposite. Yang et al. [17] studied the effect of temperature on the resistivity of aligned CNT-hydrogel nanocomposites. They showed that the resistance of the composite decreases linearly with increasing temperature. Neitzert et al. [1] demonstrated that the resistivity-temperature behavior of CNT-epoxy nanocomposites can be described by the exponential description developed by Sheng [18] for tunneling conduction in disordered materials. Matzeu et al. [19] developed a temperature sensor based on a CNT/styrene composite. They reported a negative temperature coefficient for the developed sensor. Further, they showed that the sensitivity of the sensor increases as the filler loading is decreasing. Sibinski et al. [20] examined the effect of temperature on the resistivity of CNT filled polymers. Their study indicated that the nanocomposite resistivity-temperature curve exhibits quasi-linear characteristics. To the best of authors' knowledge, only scarce information on numerical or analytical studies devoted to the resistivity-temperature behavior of nanocomposites is available in the technical literature. Consequently, a need exists for suitable modeling approaches, in particular since suitable models will allow for the efficient design and optimization of polymeric nanocomposites for specific temperature sensing applications. These aspects motivated the present authors to conduct an investigation of the resistivity-temperature characteristics of CNT and GNP filled polymers. A 3D continuum MC model was developed, in which randomly dispersed fillers were included in a representative volume element (RVE). In subsequent modeling steps, finite element modeling was employed to evaluate the nanocomposite electrical characteristics and temperature effects.

## Modeling Procedure

###### Initial Model Generation Using MC Approach.

MC simulation is a statistical technique that has widely been used to evaluate the electrical behavior of conductive composites, and several such models have been employed to predict the percolation threshold of polymers filled with conductive inclusions, see, e.g., Refs. [6-11]. The 3D continuum MC model developed in the present work constitutes the basis for the successive evaluation of temperature effects in electrically conductive nanocomposites. The model provides for a random and uniform CNT and GNP distribution inside a cubic RVE with edge length L.

Undertaking an approach similar to that provided in Ref. [21], a MC model was developed to evaluate the percolation behavior of a system of CNT particles dispersed in 3D space. Three random numbers (xi,yi,zi) were generated in the interval (0, L), indicating the coordinate of one end of a rod-shaped CNT. The other end of the CNT was determined by choosing a random point on the surface of a sphere with radius lf, where the center of the sphere is located at the initial end of the CNT and lf is the length of the CNT. The algorithm introduced by Marsaglia [22] was herein employed to choose a random point on the surface of the sphere, which guarantees the random orientation of the generated CNT. The Mersenne Twister [23] algorithm was employed in the present study for random number generation due to its long period and low computational cost. Rod-shaped CNT elements were thus randomly created and added to the RVE. In general, a large RVE size is desirable to guarantee the randomness of the MC simulation. However, the computational cost of the simulation also increases considerably with RVE size. Hence, periodic boundary conditions were applied that decrease the computational cost by limiting the size of the RVE while preserving the randomness of the MC model. Any CNT with an end point located outside the RVE was cut, and the segment located outside the RVE was transferred to the opposite face of the RVE [24]. In this manner, the obtained RVE becomes a quasi unit cell allowing for the formation of bigger RVE structures. The existence of a percolation network and electrical conduction was herein evaluated for a composite RVE composed of several smaller ones. The concept of the periodic boundary conditions is illustrated by Fig. 1.

An approach similar to the above was undertaken to model disk-shape GNP distributed in a 3D RVE. A random point was chosen inside the RVE indicating the center of the circular disk representing a platelet. A unit vector perpendicular to the disk plane was generated employing the procedure described above in the context of CNT. Circular GNP with a certain disk radius were added to the RVE only when geometrically feasible, that is, when disks did not intersect with each other. The procedure for avoiding intersecting disks was described in earlier works [25,26], where additional details on the MC modeling approach for GNP nanocomposites are given as well.

###### Modeling Electrical Connection Between Filler Particles.

Considering quantum mechanical effects, the polymer occupying the space between the conductive filler particles was represented in this study by virtual tunneling resistors through which electrons can “pass”, making the nanocomposite conductive. The tunneling resistivity, ρ, for a quantum mechanical rectangular barrier and a low voltage level with respect to the thickness of the insulator, d, is described by [27]Display Formula

(1)$ρ=h2e22mλexp(4πdh2mλ)$

where h, m, and e are, respectively, the Planck constant, the mass of an electron, and the quantum of electrical charge; λ is known as the barrier height in connection with quantum mechanical tunneling effects. Numerically evaluating Eq. (1) for a set of appropriate parameters reveals that tunneling conductivity decreases sharply with increasing insulator thickness, see Fig. 2. A cut-off distance was therefore assumed at which a tunneling resistor no longer contributes appreciably to the overall resistivity of the nanocomposite. In the present modeling approach, the shortest mutual distance between filler particles was determined for all inclusions inside the RVE, and any pair of filler particles with a smallest mutual distance less than the electron tunneling cut-off distance was grouped as a cluster, indicating that electrical current can pass through the tunneling resistor formed between these adjacent particles. Hu et al. [21] set the electron tunneling cut-off distance to be 1 nm for CNT based polymeric composites. This distance corresponds to a conductivity of 102s m−1, which is substantially greater than the CNT conductivity [15]. Bao et al. [28] considered an electron tunneling cut-off distance of 1.4 nm. At this value, the transmission probability is less than 10−4 for most polymers with a quantum mechanical barrier height ranging from 1 to 5 eV. In this study, an electron tunneling cut-off distance of 1.4 nm was assumed for CNT nanocomposites as well. For polymers filled with circular conductive nanoplatelets the present authors showed in a previous work [25] that tunneling resistors with length greater than 2 nm have no appreciable contribution toward the overall nanocomposite resistivity.

Increasing numbers of electrical connections between filler particles and their clusters ultimately leads to a transition of the nanocomposite from electrical insulator to conductor. The filler volume fraction at which this transition occurs is known as the percolation threshold. At this filler volume, a percolation network is formed that allows electrical current to pass through the material. The schematic shown in Fig. 3 illustrates the mechanism which renders a polymer nanocomposite electrically conductive as the result of dispersed conductive inclusions.

In order to determine the shortest distance between two CNT, the following technique was employed. A line segment in 3D space was described as follows:Display Formula

(2)$x-x1ix2i-x1i=y-y1iy2i-y1i=z-z1iz2i-z1i=t 0≤ti≤1$

In Eq. (2), $(x1i,y1i,z1i)$ and $(x2i,y2i,z2i)$ are the coordinates of the end points of the ith nanotube inside the RVE. The shortest distance d between the ith and jth nanotube can be calculated by minimizing the following function:Display Formula

(3)$d=(xi-xj)2 + (yi-yj)2 + (zi-zj)2$

where considering Eq. (2), d is a function of ti and tj which are within the interval [0,1]. In order to determine the shortest distance between two circular nanodisks representing GNP, the algorithm developed by Almohamad and Selim [29] was employed. Details on this approach can be found in Ref. [26].

As explained earlier, a nanocomposite is rendered conductive by an electrical current I passing through an insulator matrix due to quantum tunneling effects. Referring to [30] the effect of the temperature T on the tunneling current density J for intermediate voltages, where eVλ, can be described byDisplay Formula

(4)$J(V,T)=J(V,T0){1+[3×10-9×d2T2/(λ-V/2)]}$

where d is expressed in Angstrom, T in degrees Kelvin, and λ in Volts; T0 represents absolute zero Kelvin (−273.15 °C).

###### Model Completion Using Finite Element Method.

Using a finite element modeling approach, the nanocomposite resistivity was evaluated forming 3D resistor networks that consisted of rod-shaped CNT or disk-shape GNP elements, and tunneling resistors. Note that besides the tunneling resistivity also the intrinsic resistivity of CNT was taken into account when predicting the electrical behavior of CNT nanocomposites. Using the procedures described above, the shortest mutual distance between filler particles and the points on them corresponding to the shortest distance were determined. Particles having a shortest mutual distance of less than the tunneling cut-off distance were connected by a tunneling resistor with a resistivity given by Eqs. (1) and (4). If the shortest distance between two CNT was less than a certain minimum value, a constant distance of dmin was assumed for the resistivity calculation. According to Bao et al. [28] the minimum separation between CNT should not be less than the van der Waals separation distance, dvdw, which, for example, is 3.4 Å (0.34 nm) for graphene sheets. Considering the influence of polymer chains, Yu et al. [24] showed that the minimum separation distance should be on the order of 1 nm for the case of good filler dispersion. Based on these considerations, the distance between CNT was determined as follows:Display Formula

(5)$dR={dmin0≤d≤D+dmind-DD+dmin≤d$

where D, d, and dR are the CNT diameter, the shortest mutual distance between CNT axes, and the length of the tunneling resistor, respectively. The schematic in Figs. 4 and 5 illustrate the developed concept for the formation of a tunneling resistor and a resistor network.

In the finite element code developed to evaluate the resistivity of the resistor network, the governing equation for the kth resistor with resistance Rk connecting the ith and jth nodes is given as follows:Display Formula

(6)$[IiIj]=[1/1RkRk -1/1RkRk-1/1RkRk 1/1RkRk]{ViVj}$

The conductivity matrix for the resistor network, [Keq], was formed by assembling the individual conductivity matrices from Eq. (6). It was assumed that an electrode is formed by the parallel faces of the RVE, and a given electrical direct current (DC) is passed through the nanocomposite. So, the boundary condition was set to be −Iinp and Iinp at the electrode nodes. The voltage at the electrodes was obtained usingDisplay Formula

(7)$[V]=inv[Keq]{I}$

In order to avoid singularity effects for the conductivity matrix, resistors without any contribution to the overall resistivity of the nanocomposite were not considered in composing the conductivity matrix. In other words, only the resistors that are involved in forming the electrical network were taken into account for computing the electrical properties.

## Results and Discussion

###### CNT Nanocomposites.

The developed MC model with dispersed CNT comprises a cubic RVE with edge length of L = 15 μm. Multiwall CNT (MWCNT) were herein considered with length of lf = 5 μm and diameter of D = 50 nm, corresponding to an aspect ratio of 100. The given RVE dimension was chosen based on findings by Chen et al. [31], who evaluated the percolation threshold for a system of CNT fillers with length and diameter of 200 nm and 2 nm, respectively. The percolation thresholds predicted for RVEs with a side length of 600 nm, 800 nm, and 1000 nm yielded values of 0.551%, 0.553%, 0.548%, respectively (with a standard deviation of 0.0025). The repeatability of these results and the small value of the standard deviation indicate that when L/lf is set to three or higher the simulation results show no appreciable sensitivity to the RVE size. Results obtained from the current MC simulation exhibited a filler volume fraction of 0.61% for the formation of a percolation network, which is in good agreement with 0.6165 vol. % reported in Ref. [21] as the percolation threshold. In a second step, the electrical behavior of the conductive nanocomposite was evaluated employing the developed finite element model. The electrical resistivity of nanocomposites was predicted for different electrical properties of the polymer matrix, i.e., λ = 0.5, 1, 1.5 eV. MWCNT were assumed as filler material with resistivities ranging from 10−4 to 10−7 Ωm [32]. As an example, Fig. 6 illustrates the electrical behavior of a nanocomposite evaluated at 300 K, for a matrix with λ = 0.5 eV filled with CNT having a resistivity of 0.5 × 10−6 Ωm.

It is well known that beyond the percolation threshold the conductivity of nanocomposites can be described by the power law relation given, see, e.g., Ref. [33]Display Formula

(8)$σ=σ0(Vf-Vfc)t$

where Vf and Vfc are the filler volume fraction and percolation threshold, respectively, and σ0 and t are constants.

The nanocomposite electrical behavior was evaluated at different temperatures ranging from 200 to 550 K. Considering the resistivity-temperature relationship of Eq. (4), a nonlinear behavior of the resistivity versus temperature curve is to be expected. However, besides temperature also the separation of particles and the electrical field govern the electrical behavior of the nanocomposite. These nonlinear governing parameters appear to work destructively leading to a quasi-linear behavior. The resistivity of the nanocomposite at a temperature of absolute zero, ρ0, was predicted through linear extrapolation; this value was subsequently employed to normalize nanocomposite resistivity data. The computed numerical results were compared with existing experimental works [1,34] as shown in Fig. 7. Simulation predictions presented in this figure are qualitatively in good agreement with experimental values. It should be noted that the reported experimental results were obtained for nanocomposites containing CNT with an average diameter of 10 nm and a length varying from 0.1 to 10 μm. Comparing the experimental and numerical results indicates an insignificant difference in gradient G, where G = (Δρ/ρ0)/T. Evaluating the electrical resistivity for composites with different polymers reveals that the quantum mechanical barrier height of the polymer plays a major role for the resistivity-temperature characteristic of the nanocomposite. As shown in Fig. 8, the electrical resistivity is more sensitive to a change in temperature for polymers with a lower barrier height.

Next, the effect of filler resistivity on the CNT nanocomposite resistivity-temperature behavior was investigated. Figure 9 shows the variation of electrical resistivity with respect to temperature for CNT nanocomposites for three different CNT resistivities and three different filler loadings (0.65%, 1%, and 1.5%). It can be observed that for diminishing CNT resistivities, the nanocomposite resistivity is increasingly dependent on temperature. No significant effect of filler loading on the resistivity-temperature behavior can be ascertained for the low and high resistivities, that is, the effect of filler concentration appears diminished. However, for the intermediate CNT resistivity of 10−5 Ωm the filler concentration affects the electrical resistivity by approximately 5%. It is conjectured that the observed behavior for the different CNT resistivities is rooted in the interrelation between CNT resistivity and tunneling resistivity. As described by Eq. (4), the effect of temperature on the tunneling resistivity is governed by the particle separation and electrical potential difference, which in turn are ruled by the filler loading. To fully appreciate the underlying effects, further studies should be conducted on the contribution of filler volume fraction and filler resistivity on nanocomposite resistivity.

###### GNP Nanocomposites.

Modeling of GNP nanocomposites at different temperatures was conducted for nanodisks with a diameter of 100 nm and thickness of 0.34 nm. Nanocomposite resistivity data were normalized with respect to a 300 K condition. Simulation results showing the effect of temperature on nanocomposite resistivity are depicted in Fig. 10. As illustrated in this figure, the resistivity of GNP nanocomposites decreases monotonically with increasing temperature. Simulation results further indicate that the sensitivity of the nanocomposite resistivity to temperature decreases as filler loading is increasing. Comparing the CNT nanocomposite resistivity-temperature behavior with that of GNP nanocomposites shows that the resistivity of GNP nanocomposite is affected to a greater extent by temperature, suggesting a higher suitability of GNP nanocomposites for temperature sensing application.

As shown above, the resistivity of CNT and GNP nanocomposites decreases as temperature is increasing which is a characteristic behavior of semiconductors. Mott's “variable range hopping” (VRH) model [15,35] was employed to describe the observed nanocomposite behavior, i.e.,Display Formula

(9)$ρ=ρ0exp[(T0T)1/D+1]$

where $ρ0$ and T0 are constants and D is the dimensionality of the conduction process. The resistivity of GNP and CNT nanocomposites in terms of the absolute temperature were described using Eq. (9) and the results are presented in Figs. 11 and 12. The results indicate that Mott's VRH model is an effective tool to describe the resistivity-temperature behavior of GNP nanocomposites, whereas the proposed model is less suitable for delineating the resistivity-temperature behavior of CNT nanocomposites.

## Conclusions

A comprehensive numerical study was conducted to evaluate the effect of temperature on the electrical behavior of CNT and GNP filled polymers. The study was carried out to investigate the resistivity-temperature behavior of nanocomposites involving a range of parameters, including filler electrical resistivity, quantum tunneling barrier height of the polymer matrix, and filler loading. Comparing the results obtained from the modeling work with existing experimental data confirmed good qualitative agreement. The simulations showed that a significant temperature effect can be realized for nanocomposites with favorable material properties, especially for GNP nanocomposites. This dependence of electrical resistivity on temperature motivates future research to explore the feasibility of using CNT and GNP nanocomposites for temperature sensing applications.

## Acknowledgements

This research was supported by the following organizations: Alberta Innovates-Technology Futures, ROSEN Swiss AG, and Syncrude Canada Ltd.

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## References

Neitzert, H. C., Vertuccio, L., and Sorrentino, A., 2011, “Epoxy/MWCNT Composite as Temperature Sensor and Electrical Heating Element,” IEEE Trans. Nanotechnol., 10(4), pp. 688–693.
Heo, C., and Chang, J. H., 2013, “Polyimide Nanocomposites Based on Functionalized Graphene Sheets: Morphologies, Thermal Properties, and Electrical and Thermal Conductivities,” Solid State Sci., 24(Oct.), pp. 6–14.
Kim, J. S., Yun, J. H., Kim, I., and Shim, S. E., 2011, “Electrical Properties of Graphene/SBR Nanocomposite Prepared by Latex Heterocoagulation Process at Room Temperature,” J. Ind. Eng. Chem., 17(2), pp. 325–330.
Manivel, P., Kanagaraj, S., Balamurugan, A., Ponpandian, N., Mangalaraj, D., and Viswanathan, C., 2014, “Rheological Behavior and Electrical Properties of Polypyrrole/Thermally Reduced Graphene Oxide Nanocomposite,” Colloids Surf. A, 441(Jan.), pp. 614–622.
Patole, A. S., Patole, S. P., Kang, H., Yoo, J. B., Kim, T. H., and Ahn, J. H., 2010, “A Facile Approach to the Fabrication of Graphene/Polystyrene Nanocomposite by In Situ Microemulsion Polymerization,” J. Colloid Interface Sci., 350(2), pp. 530–537. [PubMed]
Natsuki, T., Endo, M., and Takahashi, T., 2005, “Percolation Study of Orientated Short-Fiber Composites by a Continuum Model,” Physica A, 352(2–4), pp. 498–508.
Ma, H. M., and Gao, X. L., 2008, “A Three-Dimensional Monte Carlo Model for Electrically Conductive Polymer Matrix Composites Filled With Curved Fibers,” Polymer, 49(19), pp. 4230–4238.
Dalmas, F., Dendievel, R., Chazeau, L., Cavaille, J. Y., and Gauthier, C., 2006, “Carbon Nanotube-Filled Polymer of Electrical Conductivity in Composites. Numerical Simulation Three-Dimensional Entangled Fibrous Networks,” Acta Mater., 54(11), pp. 2923–2931.
Li, C., Thostenson, E. T., and Chou, T. W., 2008, “Effect of Nanotube Waviness on the Electrical Conductivity of Carbon Nanotube-Based Composites,” Compos. Sci. Technol., 68(6), pp. 1445–1452.
Li, C. Y., and Chou, T. W., 2007, “Continuum Percolation of Nanocomposites With Fillers of Arbitrary Shapes,” Appl. Phys. Lett., 90, p. 174108.
Li, C. Y., Thostenson, E. T., and Chou, T. W., 2007, “Dominant Role of Tunneling Resistance in the Electrical Conductivity of Carbon Nanotube-Based Composites,” Appl. Phys. Lett., 91, p. 223114.
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## Figures

Fig. 1

Schematic of periodic boundary conditions used for MC modeling of CNT nanocomposites (elements in dashed lines indicate CNT crossing the RVE boundary)

Fig. 2

Tunneling conductivity versus insulator thickness for λ = 1.5 eV

Fig. 3

Schematic for the formation of a percolation network at the percolation threshold

Fig. 4

Schematic of the electron tunneling mechanism in CNT and GNP based conductive nanocomposites

Fig. 5

Schematic of the conductivity mechanism in conjunction with a percolation network

Fig. 6

Electrical conductivity of a CNT nanocomposite with σCNT = 0.5 × 10−6 Ωm, λ = 0.5 eV

Fig. 7

Qualitative comparison of experimental data [1,34] with simulation results for a CNT nanocomposite with σCNT = 0.5 × 10−6 Ωm, λ = 0.5 eV, Vf = 1%

Fig. 8

Effect of polymer electrical properties on the resistivity-temperature behavior for a CNT nanocomposite with Vf = 1%

Fig. 9

Effect of CNT resistivity on the resistivity-temperature behavior of CNT nanocomposites with λ = 0.5 eV

Fig. 10

Normalized resistivity as a function of the temperature, from simulations with 100 nm nanodisks and λ = 0.5 eV

Fig. 11

Fitting of GNP nanocomposite resistivity data according to the VRH model proposed by Mott [15,35]

Fig. 12

Fitting of CNT nanocomposite resistivity data according to the VRH model proposed by Mott [15,35] (λ = 0.5 eV, Vf = 0.65%)

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