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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

In this study, a three-dimensional continuum percolation model was developed based on a Monte Carlo simulation approach to investigate the percolation behavior of an electrically insulating matrix reinforced with conductive nano-platelet fillers. The conductivity behavior of composites rendered conductive by randomly dispersed conductive platelets was modeled by developing a three-dimensional finite element resistor network. Parameters related to the percolation threshold and a power-low describing the conductivity behavior were determined. The piezoresistivity behavior of conductive composites was studied employing a reoriented resistor network emulating a conductive composite subjected to mechanical strain. The effects of the governing parameters,

The development of nano-scale electrically conductive fillers and associated low cost fabrication methods has stimulated considerable interest in employing such particles to render otherwise insulating polymers conductive. Conductive nano-fillers, such as graphene platelets and exfoliated graphite, were found to impart significant conductivity and piezoresistivity behavior to polymers, which makes such material systems an excellent choice for electrical applications, such as strain sensors. Nano-platelet based conductive polymers have been the subject of a number of experimental studies [

Yi and Tawerghi [

The limited quantity of published works related to the modeling of platelet based conductive polymers, and limitations associated with the employed techniques motivated the present authors to conceive an alternative approach for modeling the electrical behavior of polymers filled with conductive nano-platelet fillers. Published research on the electrical properties of nano-platelet based nanocomposites has further been limited to studying the electrical conductivity of polymers with exfoliated sheets with in-plane dimensions varying from a few to several hundred micrometer and thicknesses ranging from a few to several nanometers. To achieve further enhancements in composite properties the synthesis of submicron size single layer graphene sheets may be an attractive proposition, and information on the percolation and electrical behavior of such reinforced polymers is therefore desirable. The latter aspects are also addressed in the present contribution.

A 3D Monte Carlo model was developed to study the percolation, conductivity and piezoresistive behavior of composites filled with randomly dispersed impenetrable conductive nano-disks. In the present study a Monte Carlo model was first developed to form a representative volume element filled with randomly dispersed nano-platelet conductive inclusions. In a second stage a 3D finite element based resistors network model was used to analyze the conductivity behavior of nano-platelet based conductive polymers. Previous studies have shown that conductivity of such polymers for volume fractions greater than the percolation threshold can be described by a power-low expression [

where _{p} is the percolation threshold,

The developed 3D continuum Monte Carlo model is based on conductive inclusions that are randomly dispersed inside a cubic representative volume element (RVE) with side length _{c},_{c},_{c}) were generated between (0,_{1}_{2}

where _{i}_{i}_{i}

Electrical connection between two conductive inclusions arises from two different mechanisms,

where

The volume fraction of the conductive particles at the onset of the percolation network is called percolation threshold. This concept is illustrated by the schematic shown in

Subjecting a polymer with conductive inclusions to mechanical strain, as shown in

The computational code for the Monte Carlo simulation was developed in the FORTRAN language to numerically investigate the 3D continuum percolation problem of conductive nano-disks. Considering the high computational cost of the Monte Carlo simulation, multiple processor computation was employed to allow for rapid code execution on a Linux cluster.

The developed model was evaluated numerically considering a polymer matrix such as epoxy containing conductive nano-disk filler comprised of graphene platelets. Hence a constant nano-disk thickness of 0.34 nm was assumed, which is equal to the thickness of a single graphene layer [_{t}_{t}

As discussed earlier, a cut-off distance can be assumed at which the effect of tunneling resistors with length greater than this distance is negligible. The conductivity of nanocomposites was evaluated for different values of electron tunneling distance. Considering the results shown in

It was stated previously that the polymer matrix chiefly affects the electrical resistivity of a nano-composite, especially for lower filler volume fractions. As such, the contribution of matrix electrical properties,

Balberg _{p}. The present study confirmed that a single expression with specific values of _{p} is insufficient to represent the entire nanocomposite conductivity and percolation behavior. This is illustrated in _{u} _{p}, 1.4_{p}). _{u}. Employing the power-law equation to describe the composite behavior for higher volume fractions, on the other hand, yields a mean value of 5.56 for the critical exponent, _{u}. Comparing _{u} is expected to be restricted to a small region near the percolation threshold, and non-universal percolation behavior becomes dominant as the filler volume fraction is increasing. Based on the above considerations, results obtained in the present study for conductive circular nano-disks indicate a critical exponent that is filler content dependent and converges toward the universal value with filler loading approaching the percolation threshold. This conclusion is congruent with findings from Johner

Ambrosetti ^{2}^{2}(

In another work, Lee and Torquato [^{2} to exclude the area of the penetrable shell that is equivalent to the tunneling distance. As illustrated by

Employing the methods described in Section 2.3, the piezoresistivity behavior of nanocomposites with nano-platelet inclusions was investigated. Simulation results shown in

In

As mentioned above, piezoresistivity in conductive composites arises from two major mechanisms, that is, changing particle proximity and orientation. Increasing the tensile strain increases the average interparticular distance, so an increase in resistivity of the conductive nanocomposite is expected. On the other hand, tensile strain raises the propensity for the formation of clusters, leading to a decrease in electrical resistivity. This conjecture is supported by the results shown in

The observed “switching behavior” and the existence of a critical strain have been reported in several other studies. Kalanadhabhatla [

It is interesting to note that for some intermediate filler volume fractions,

Using a combined Monte Carlo and finite element modeling framework a theoretical study was carried out to investigate the electrical conductivity and piezoresistivity behavior of nano-composites filled with conductive nano-platelets such as graphene. The power law relationship describing the conductivity of such polymers for filler volume fractions greater than the percolation threshold was investigated and discussed. The critical exponent was approximately constant for all of the nano-disk sizes considered herein, which led to the conclusion that the exponent is controlled by filler geometry rather than size. In addition, the critical exponent in the power law description depends on the filler volume fraction. Near the percolation threshold, the exponent was found to match the often-cited value of 2, whereas the exponent deviates from this value for higher filler loadings. Modeling results further indicate a reduction in percolation threshold for decreasing nano-disks sizes and a strong effect of matrix electrical properties,

The authors would like to acknowledge the support by the following organizations: Alberta Innovates—Technology Futures, ROSEN Swiss AG, and Syncrude Canada Ltd.

All authors contributed to the development of this paper. Amirhossein Biabangard Oskouyi is a doctoral graduate student. He chiefly conducted the modeling work under the supervision and guidance of professors Uttandaraman Sundararaj and Pierre Mertiny.

The authors declare no conflict of interest.

Tunneling conductivity

Schematic of representative volume element with individual nano-platelets (gray), and platelets forming clusters (white) and a percolation network (black).

Schematic of three-dimensional percolation network modeled by tunneling resistors (A).

Schematic of nano-platelet composite subjected to tensile strain.

Effect of tunneling cut-off distance (1 nm, 2 nm and 3 nm) on the resistivity of nano-platelet composites for an insulator barrier height of λ = 0.5 eV and a platelet diameter of

Effect of insulator barrier height λ on the resistivity of conductive nano-platelet composites plotted

Graph defining the percolation region depicted as nano-platelet composite resistivity

Deviation of nano-platelet composite resistivity from a universal power-law description with

Nano-platelet composite resistivity _{p}, 1.4_{p}).

Nanocomposite resistivity

Values for the critical exponent,

Graph depicting filler volume fraction at the percolation threshold _{p}

Percolation threshold of oblates of revolution as a function of aspect ratio and ratio of soft shell thickness to the major axis. The dashed line depicts the percolation threshold trend when

Percolation threshold of oblates of revolution as a function of aspect ratio. The soft shell volume was deducted in order to make the data analogous to a tunneling percolation problem with platelet based nanocomposites. (adopted from [

Percolation threshold for 2D disks dispersed in a 2D domain. (adopted from [

Resistivity

Graphs depicting the effect of filler volume fraction on piezoresistivity with tensile strain ε applied to unconstrained nano-platelet composites with particle size of _{f} = 7.5%; (_{f} = 8.8%; (_{f} = 11%.

Piezoresistivity behavior and critical strain for a CNT-graphene nano-platelet hybrid composite. (adopted from [