Predictive Model of Graphene Based Polymer Nanocomposites: Electrical Performance

1.1 Electrical Simulation Models

Despite the high financial and time cost of material preparation and experimental work, there are a few computational models predicting the electrical response of graphene based polymer nanocomposites. In the literature, the electrical simulation models are divided into two main categories- the percolation threshold models and the full electrical response. As far as the percolation models are concerned, there is a wide variety of proposed methodologies taking into account the filler shape, aspect ratio, tunnelling distance, overlapping, and the formation of agglomerations. Oskouyi et al. [16] apply the Monte Carlo method to model the percolation threshold for disk-shaped fillers, simulating the conductive network formed by inclusions like graphene nanoplatelets (GNP). Later, Ambrosetti et al. [17] conducted a numerical study to investigate a system’s percolative properties consisting of hard oblate ellipsoids of revolution surrounded with soft penetrable shells. Since the previously mentioned computational works were focusing only on material systems with constant filler material and geometrical properties, Otten et al. [18] developed an analytical approach to investigate the percolation behaviour of polydisperse nanofillers of platelet-based composites. However, their modelling approach was subject to certain limitations, that is, the platelet thickness and the tunnelling decay length have to be of the same order of magnitude, and the diameter of the disk-like fillers needs to be much larger than the disk thickness. In terms of exploring the effect of filler overlapping and agglomerations, Xia et al. [19] proposed a computational methodology under which it was possible to predict the percolation threshold for identical ellipses with the overlapping effect for a 2D structure, while Vovchenko et al. [20] predicted the percolation threshold of composites filled with intersecting circular discs in a 3D structure. Besides, the conventional Monte Carlo approaches to predict the percolation threshold for materials reinforced with 2D particles, Mathew et al. [21] conducted a Monte Carlo study on the percolation of hard platelets in a 3D continuum system considering the rate of order in the microstructure, proving its effect on the percolation threshold with the employment of isotropic-nematic (IN) transition.

Although there is a satisfactory number of numerical methods predicting the percolation threshold of nanocomposites doped with fillers of various shapes and sizes, there are a few simulation tools on the prediction of the full electrical response of these nanocomposites. At first, Hicks et al. [22] developed a tunnelling-percolation model to investigate electrical transport in graphene based nanocomposites, covering the need of a suitable model to predict the full electrical response of semiconducting 2D element reinforced materials. This model, however, is applicable only to rectangle shaped nanoparticles forming 2D networks, while in common graphene reinforced nanocomposites, fillers exhibit a wide range of shapes, mainly circular-ellipsoid ones, and the conductive network formed is considered to be a 3D one. In a later work, Ambrosetti et al. [23] studied the electrical conductivity of an insulating matrix reinforced with conductive ellipsoids by assuming that an expected curve of electrical conductivity variation would be applied and finally being reduced in a geometrical form taking into account the inter-particle distance and the tunnelling distance. Oskouyi et al. [24] developed a 3D Monte Carlo model to study the percolation, conductivity, and piezoresistive behaviour of composites filled with randomly dispersed impenetrable conductive nano-disks. In this study, a Monte Carlo model was first developed to form a representative volume element (RVE) filled with randomly dispersed nano-platelet conductive inclusions. Then, a 3D finite element based resistors network model was used to analyse the conductivity behaviour of nano-platelet based conductive polymers. Finally, they developed a continuum model to determine the effective AC and DC electrical properties of graphene nanocomposites [25]. The proposed theory consisted of three major components, embodying the most fundamental characteristics of the graphene nanocomposites, i.e. percolation threshold, interface effects and additional contribution of electron hopping and micro capacitor structures to interfacial properties.

Knowing that, such numerical models are not easily applicable on industrial and engineering case study, as their employment is complicated and their results are limited to certain case. In this paper, a multi-scale multi-physics finite element model (FEM) is presented which would be easily applied to most cases, extended to more sophisticated material architecture in respect of being scientifically structured and proven. The current approach on multi scale modelling consists of the creation of a unit cell and a micro-scale nanocomposite model (Representative Volume Element RVE) on a commercial finite element modelling environment. The unit cell consists of a single rectangular or elliptical graphene plate surrounded by a thin layer of polymer. This polymer layer represents the inter-plate volume between successive graphene reinforcements, in which conduction phenomena (tunnelling effect, electron hopping) take place. The unit cell is loaded with a constant electric potential to compute the resistance matrix representing this system. The RVE is a rectangular block, on whose resistance matrix elements previously obtained through the unit cell is randomly distributed. This distribution represents the random position of graphene on the bulk volume of polymer nanocomposites. The orientation of graphene is simulated by the 3D random orientation of the corresponding element local coordinate system.

2.1 Unit Cell

Similarly to the approach of hard core-penetrable shell has been assumed on several studies predicting the percolation threshold [17, 22], the unit cell consists of a single-layer graphene plate surrounded by a thin layer of polymer as shown in Fig. 1 (rectangular shape) and Fig. 2 (ellipsoid shape).

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Equation (1) describes the current flow between two conductive electrodes when they are separated by an insulating film, under the condition that the top of the energy gap of the insulator is above the Fermi level of the electrodes. The electronic current can flow through the insulating region between the two electrodes if: (i) the electrons in the electrodes have enough thermal energy to surmount the potential barrier; and (ii) the flow in the conduction band or the barrier is thin enough to permit its penetration by the electric tunnel effect. Knowing that the electrons’ thermal energy could additionally contribute to the electric current, this effect was neglected requesting low-temperature conditions.

where d₁ and d₂ are the polymer thicknesses of two different unit cells named UC₁ and UC₂, R_tunnel(d₁ + d₂) is the tunnelling resistance exhibited between the conductive particles UC₁ and UC₂ with total tunnelling distance d₁ + d₂, and R_tunnel(d₁) and R_tunnel(d₂) are the tunnelling resistances exhibited in the polymer layer in each one of the unit cells UC₁ and UC₂, while m is the electron mass, h is the Plank’s constant and λ is the height of barrier.

The constant C would be given by equation (4), depicted in Fig. 3 and its equivalent value would be the result of the equation (5). The equivalent value of C for each d₂ is presented in Table 2, where it is possible to conclude that assuming C being equal to 0.3 would be reasonable to represent each case.

$$ {R}_{tunnel}\left({d}_1+{d}_2\right)=C\left({R}_{tunnel}\left({d}_1\right) \exp \left(\frac{4\pi {d}_1}{h}\sqrt{2m\lambda}\right)+{R}_{tunnel}\left({d}_2\right) \exp \left(\frac{4\pi {d}_2}{h}\sqrt{2m\lambda}\right)\right) $$

(4)

$$ {C}_{equivalent}\left({d}_{\max }-{d}_{\min}\right)={\displaystyle \underset{d_{\min }}{\overset{d_{\max }}{\int }}C\left(\delta \right)d\delta } $$

(5)

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Table 2

C_equivalent in function of d₂

d2 (nm)	Cequivalent
0.2	0.37
0.4	0.37
0.6	0.38
0.8	0.34
1.0	0.20

This model was built using the commercial FEA program ANSYS 16.2. 3D 20-Node Hex couple-field solid element was used with its piezoresistive behaviour activated through the available key-option 101, while both graphene and polymer isotropic electrical behaviour was assumed. It is important to mention that the piezoresistive coupling induced by this key-option was neglected by not defining the piezoresistive matrix.

The values of the parameters used on the analysis of the unit cell are presented on the Tables 3 and 4, while the unit cell geometrical parameters are schematically presented in Fig. 4.

Table 3

Graphene geometrical parameters

Graphene shape		Rectangular	Elliptical
Graphene plate thickness	t (nm)	0.45	0.45
Graphene minor side dimension	a (nm)	10	2∙5.64
Aspect ratio	AR (−)	1,5,10,50	1,2,5,7,10
Graphene major side dimension	b (nm)	AR∙a	AR∙a

Table 4

Unit cell material properties

Material	Electrical behaviour
Graphene	Electrical conductivity	S/m	107
Polymer	Tunnelling resistivity	Ωm	Equation (3)
	Cequivalent	-	0.3
	Height of barrier (λ)	eV	0.5–2.5
	½ Tunnelling distance (d)	nm	0.2–1.0

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As far as the single-layer graphene thickness is concerned, reports from literature show a distinct variation of the measured thickness of single layer graphene, which could be attributed to the measurement method and the graphene purity [27, 28, 29, 30, 31, 32, 33, 34]. It has been stated that the thickness of a single-layer could range between 0.3 nm and 1.6 nm. However, in accordance with the thickness measured by the imaging mechanism of tapping mechanism AFM, it was estimated around 0.4 nm [35]. It has been shown from SEM images that graphene particles could have several shapes, which are related to the manufacturing of the graphene and the breakage of the sheets during the homogenisation/dispersion of graphene/polymer materials. Generally, taking into account that the most common shapes able to describe a 2D structure are the rectangle and the ellipse as well as that these shapes have been considered for the prediction of percolation threshold. These two shapes are chosen for exploring their effect on the conduction of graphene/polymer nanocomposites.

In this analysis, the height of barrier (i.e. energy state of the insulating material to be reached by the electrons so as to conduction occur) is set as a parameter its effect needs to be explored, and not as a constant variable. The height of barrier usually takes values between 0.5 and 2.5 eV [28, 29] and is dependent of the polymer structure (crystalline, amorphous structure, crosslinking grade), and the chemical composition in the inter-particle region. In the literature, there is one major technique on height of barrier experimental calculation reported [28, 29, 30, 31]. A specimen, consisting of two conductive electrodes separated by a thin layer of the insulating material of interest, is loaded under direct current. The current-voltage (I-V) response is plotted and fitted with the theoretical equation describing the junction.

2.2 Representative Volume Element (RVE)

The nanocomposite materials is modelled by a square block of side equal to multiplied b (k∙b) and thickness of multiplied a (k∙a), as shown in Fig. 5. The number of graphene seeds on the block is defined by equations (6, 7) as a function of nanocomposites volume fraction and graphene’s geometry. The number of elements representing each graphene layer (γ) would equal to the number of division in the direction of loading (eix) divided by the factor k. As a consequence the number of elements representing graphene reinforcement total in the volume (n_g) is defined by equation (8). The number of elements perpendicular to the loading direction (eiz) is given by equations (9, 10), while the number of elements through the block’s thickness (eiy) is \( \raisebox{1ex}{$eiz$}\!\left/ \!\raisebox{-1ex}{$AR$}\right. \).

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3.1 Unit Cell Analysis

After the analysis of the unit cell in respect of all the parameters mentioned on Tables 3 and 4, it is possible to analyse the effect of these features on the unit cell resistance, which represents the local resistance of the nanocomposites. The rise of the aspect ratio leads to significant decrease in the resistance. In longitudinal direction (x-direction), the resistance saturates to a constant value in function of the ½ tunnelling distance for filler’s aspect ratio being greater than 50, as it was fully studied in the [36]. Moreover, polymer and all the features controlling its response under any excitation (degree of polymerization, voids, humidity, temperature etc.) have serious effect on the electrical conductivity of a nanocomposite through the energy quantity of height of barrier appearing on the tunnelling resistivity. The effect of height barrier on the unit cell resistance in respect of tunnelling distance and for the case of rectangular graphene with aspect ratio of 1 and 5 is depicted in Fig. 6. An increase in the height of barrier results in the rise of the local resistance even for orders of magnitude. This increasing trend is common for every case of unit cell studied, while becomes more evident with the increase of tunnelling distance.

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Finally, the effect of the shape of the graphene layer on the unit cell resistance is considered in respect of the direction and the tunnelling distance is depicted for all examined aspect ratios and height of barrier of 0.5 eV [36]. It could be stated that ellipse shaped graphene sheets exhibit higher local resistance in the main axis directions than the rectangle shaped ones. However, for the case of the through thickness resistance, for the common aspect ratios ellipse and rectangle shaped graphene plates show the same electrical resistance. It is important to mention that for the purposes of comparison, the area of the rectangle shaped graphene layers is equal to that of the elliptical shaped graphene layers for each aspect ratio.

3.2 RVE Analysis

Since it is not practical to model infinite material and to eliminate the edge effect, it is important to conduct a convergence study on the size of the square block. The ratio of the sample’s average electrical conductivity to its standard deviation has been considered to be the convergence parameter. As it could be noticed in Fig. 7, for k = 5 the sample characteristics have converged to a steady state, indicating this size as appropriate for our study.

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The percolation probability, in accordance to the procedure suggested by the [21], is fitted by the function (14) and the inflection point of this function is considered to be the percolation threshold. In Fig. 8, the percolation probability, in respect of the volume fraction, is depicted for the case of rectangle graphene sheets and a height of barrier λ = 0.5 eV. The proposed fitting function is suitable to describe this data set achieving 99.999 % accuracy on fitting process.

$$ P\left({V}_f\right)=\frac{1}{2}\left(1+ \tanh \left(A\left({V}_f-B\right)\right)\right) $$

(14)

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As far as the effect of aspect ratio on the electrical conductivity in respect of the graphene volume fraction is concerned, the rise in the aspect ratio leads to a decrease on the volume fraction on which the onset of conductance occurs. In addition to this, for a constant value of volume fraction, the electrical conductivity increases with raising the aspect ratio. These results suggest that fillers with high aspect ratio are able to form more stable and efficient conductive network in the volume of nanocomposite at lower volume fraction. The above conclusion is supported also by the fact that with the rise of volume fraction the deviation on the statistical sample is significantly reduced leading to more uniform sample close to uniquely defined electrical conductivity for a nanocomposite of a specific volume fraction [36].

In Fig. 9, the effect of the height of barrier on the electrical conductivity is depicted for the case of rectangle shaped graphene with aspect ratio equal to 1 and 5. Although, the percolation threshold seems not to be affected by the height of barrier, since the onset of percolation is mainly geometrically oriented as it has been proven by [25]. For the case of constant volume fraction, the rise of the height of barrier causes rapid decrease on the electrical conductivity even by a decrease of an order of magnitude in the case of increasing the height of barrier by 0.5 eV.

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Finally, the effect of the graphene shape is calculated in respect of the volume fraction for the case of height of barrier being 0.5 eV [36]. The graphene shape does not affect the trend of the nanocomposite’s electrical response in function of the volume fraction (V_f). Although for graphene particles of the same aspect ratio and volume, the nanocomposite reinforced with rectangle shaped particles exhibits higher electrical conductivity than the one of the ellipse shaped, while the rectangle shaped graphene particles show comparable percolation threshold to that of the ellipse graphene. Yi et al. [37] suggested that this observation could be explained as the corner angles of squares and rectangles make it easier for the plates to touch each other, therefore enhancing the current passing from one particle to another.

3.3 Percolation Model

For the description of a nanocomposite’s electrical behaviour, there are two main theories being suggested, the percolation law -which describes the variation of the electrical conductivity of the nanocomposite in function of the volume fraction after the percolation threshold having being obtained- and the excluded volume theory-which predicts the percolation threshold in respect of the geometrical features of the reinforcement (dimensions, shape, 2D-3D conductive networks).

Taking into consideration the excluded volume theory and assuming that each particle is considered equivalent to a circular one, the excluded volume theory for the circular disk [38] could be modified to define the upper and lower boundaries of the volume fraction in which the percolation threshold should be expected (Eq. 15). In Fig. 10, it is obvious that the percolation threshold is reduced with increasing aspect ratio, the values were successfully found to lie between the assumed percolation bounds. Moreover, the shape of the graphene does not show any effect on the percolation threshold, since the corresponding values for each case are close enough to consider any intense effect of the filler’s shape.

$$ \left.\begin{array}{c}\hfill 1- \exp \left(-1.4\frac{V}{V_{ex}}\right)\le {V}_c\le 1- \exp \left(-2.8\frac{V}{V_{ex}}\right)\hfill \\ {}\hfill V=\pi {\left(\frac{d}{2}\right)}^2t\hfill \\ {}\hfill {V}_{ex}={\pi}^2{\left(\frac{d}{2}\right)}^3\hfill \end{array}\right\}\overset{\frac{d}{2}-a\sqrt{\frac{AR}{\pi }}}{\to }1- \exp \left(-1.4t\frac{\sqrt{\pi }}{a\sqrt{AR}}\right)\le {V}_c\le 1- \exp \left(-2.8t\frac{\sqrt{\pi }}{a\sqrt{AR}}\right) $$

(15)

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The percolation law is a power equation (16) introduced in [39], which is able to describe to electrical behaviour of an insulating material reinforced with conductive particles after the percolation threshold. In equation (16), σ _c is the electrical conductivity of the composite, V _f is the volume fraction, V _p is the percolation threshold, σ _o is a factor usually calculated to be close to the electrical conductivity of the conductive phase and t is the critical exponent. For a three dimensional conductive network, t usually takes value between 1.65 and 2.0, which is accepted as a universal value [39]. Higher values of t indicate the presence of tunnelling phenomena and the filler having extreme geometries (i.e. high aspect ratio) [39]. In Fig. 11, the fitting of the finite element model results in accordance to the percolation law is presented, showing a good correlation. Figure 12 depicted the effect of the geometry of the filler, and the height of barrier on the percolation law variables. σ_o increases with rising aspect ratio converging to the filler electrical conductivity, while it decreases with increasing height of barrier. The increase of the aspect ratio leads to more stable electrical networks which are mainly governed by the fillers conductance and not the tunnelling resistance formed in the interparticle region, leading σ_o to converge to the electrical conductivity of the filler. On the other hand, the rise of the height of barrier which induces a rise on the tunnelling resistance in the interparticle volume has as a consequence the decrease of the σ_o for the case of low aspect ratios. One of the most significant findings related to the percolation law is the agreement of the observation to the general law regarding the critical exponent. An increase of the aspect ratio (defining an extreme geometry) or even any rise of the height of barrier (defining a more intense tunnelling phenomenon) has a result an important increase of the value of the critical exponent far above the prediction for a common three dimensional conductive network.

$$ {\sigma}_c={\sigma}_o{\left({V}_f-{V}_p\right)}^t $$

(16)

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From Fig. 12, some important notices could be extracted regarding the development of an effective design tool for preparing appropriate material structures to serve the specified needs of a certain application. First of all, it could be concluded that Fig. 12 could be characterised as constitution curves, since the effect of the each variable (height of barrier, geometry and filler type) on the electrical response of the material could be depicted in a simple and easily understood manner, in respect of the percolation law parameters. These easily produced graphs could be used as a design tool for effective manufacturing nano-reinforced materials with desired electrical properties. Knowing the effect of several geometrical, material and manufacturing parameters on the final electrical performance of the nanocomposite in terms of the variation of percolation law variables, the number of “trial & error” procedures are reduced to the necessary ones to control the material system characteristics so as to produce a conductive nanocomposite with electrical response comparable to the one expected from the numerical analysis.

3.4 Preliminary Model Validation on the Prediction of the Percolation Threshold

The aforementioned methodology is employed on experimental material results found in literature for the prediction of the percolation threshold of these materials and their possibility of conductance under given circumstances was calculated. At first, the methodology is employed for the material structure of Poly(vinylidene fluoride)/Graphene proposed in the experimental work of Yu et al. [40]. In this work, PVDF/Graphene nanocomposite was prepared by adopting a solution method. The graphene nanosheets exhibited an average thickness around 1 nm, while the particle size range from 0.1-7 μm with mean diameter around 2 μm as it could be calculated in the Fig. 1c quoted in the work [40]. Two cases of reinforcement with constant thickness of 1 nm – minimum diameter of 0.1 μm and average diameter of 2 μm- and random distribution of the filler in the polymer bulk volume are modelled. The percolation threshold is calculated for each case as well as the percolation probability. It could be easily noted by Fig. 13, that the experimental value of percolation threshold (4.5 wt%) is well predicted by assuming the average value of the filler size distribution. To predict the percolation threshold of a graphene based nanocomposites, it is not necessary to simulate the size distribution of the filler, but just to use the average value of this.

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Since, the percolation threshold is not only highly dependent on the particle size but on the preparation process which affect the filler distribution on the bulk volume of the polymer, the effect of the manufacturing process should be taken into account. When a nanocomposite is manufactured with a solution method, it could be sufficiently assumed that the distribution of the filler is random. On the other hand, if the material was prepared with solution casting of films which were then stacked layer by layer during a hot-press process, it could be assumed that the in-plane filler distribution could be random due to the solution casting of the films, whereas a through-the-thickness Gaussian distribution could be assumed due to the hot-pressing of the films. These assumptions are used for the prediction of the percolation threshold of the Poly(vinylidene fluoride)/Exfoliated Graphite nanoplatelets suggested in the experimental work of He et al. [41], while they were supported by the SEM image of the fractured cross-section of PVDF/xGnP nanocomposite with V_f = 0.76 vol% showing nanoplatelets being concentrated in the middle of the specimen. Since, in the experimental work there was not any experimental study on the particle distribution into the polymer volume, there was a parametrical study on the effect of the standard deviation assuming the average was constant and set in the middle of the specimen thickness. In accordance to the experimental work, the graphite platelets had a diameter of 25 μm and thickness of 20 nm and these dimensions were introduced in the model. For reasons of comparison the same study is conducted with the randomly distributed filler, while the obtained percolation probability is compared to the one with the filler being normally distributed with an average μ = 62,5 μm and standard deviation σ = 20.8 μm depicted in Fig. 14. The percolation threshold is 3.83 vol% for randomly distributed filler and 2.34 vol% when the filler followed the Gaussian distribution proving a profound difference between these two modelling assumptions.

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Authors stated that the percolation threshold of the proposed material structure was estimated to be 1.09 vol%. As a consequence, the parametrical study on the effect of the standard deviation was made for a constant volume fraction of the experimentally obtained percolation threshold, so as to explore the effect of the standard deviation on the percolation probability. The results of this study are depicted in Fig. 15. For V_f = 1.09 %, the sample exhibits probability of conductance for standard deviation σ lower than 12 μm, while the studied V_f would be considered as percolation threshold of the nanocomposite (in correspondence to the definition of percolation threshold, it is the V_f where the 50 % of the sample is conductive, or the percolation probability is 50 %)- if the σ is 7.35 μm. Taking into consideration the above conclusions on the study of the percolation threshold of PVDF/Exfoliated Graphite Nanoplates prepared by He et al. [41], the importance of studying statistical samples and calculating the percolation probability of each sample during experimental works is recognized. This feature should be included to electrical conductivity characterization quantities, so as to be fully possible to compare the effectiveness of different material manufacturing techniques, the electrical performance of similar material structures prepared by different research groups as well as to assist the validation process of suggested simulation methods. Moreover, the parametrical study on the effect of standard deviation on the percolation probability for a given value of V_f with experimentally known percolation probability could be an additional tool on a nanocomposite’s filler distribution characterization in conjunction with imaging techniques (SEM images mapping for instance).

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