Electrical percolation in graphene–polymer composites

Electrically conductive composites comprising polymers and graphene are extremely versatile and have a wide range of potential applications. The conductivity of these composites depends on the choice of polymer matrix, the type of graphene filler, the processing methodology, and any post-production treatments. In this review, we discuss the progress in graphene–polymer composites for electrical applications. Graphene filler types are reviewed, the progress in modelling these composites is outlined, the current optimal composites are presented, and the example of strain sensors is used to demonstrate their application.


Introduction
A composite is a material that consists of two or more materials or phases, whose final properties differ from those of the original materials. Polymers are extremely versatile materials that make good matrices (or hosts) in composites due to their extensive range of properties and processability. However, in almost all cases pure polymers are insulating, and this limits their application in some sectors. Adding conductive fillers to the polymer matrix yields composites that are also conductive, and these composites have many applications including conductive adhesives [1], lightning strike protection [2], electromagnetic shielding [3], anti-static components [4], and strain sensors [5].
Traditional carbon-based fillers (such as carbon nanotubes (CNTs) [6] and carbon black (CB) [7]) make excellent candidates for conductive composites because of their high electrical conductivity combined with good mechanical properties. In the case of CNTs, they also posses high aspect ratios, which leads to electrically conductive composites at low CNT loadings [8]. However, there are limitations. While low percolation thresholds have been observed, achieving a high ultimate conductivity requires a greater loading of filler, and this often makes the composite brittle and unsuitable for many applications [9][10][11]. Therefore, alternative fillers are being sought.
Graphene has attracted significant scientific attention as a potential conductive filler. Graphene is a twodimensional sheet of carbon atoms, sp 2 -bonded into a hexagonal arrangement [12]. It has an exciting combination of properties [13], amongst which is its superb electrical conductivity (6 × 10 5 S m −1 [14]) when in a suitable isolated environment. There is now a family of graphene-related materials (GRMs), which differ from pure graphene in that they may contain multiple stacked layers (such as graphene nanoplatelets), or different chemical structures (such as graphene oxide). As composites have become a focus for industrial applications of graphene, these GRMs have drawn the most attention as fillers in graphene composites. Crucially, the structural differences between GRMs and pristine graphene affect the performance of the resulting composite.
There are also many other factors that impact a composite's electrical performance. There are different processing methods including graphene production methods and post-production routines. A careful understanding of each of these is required to achieve the ultimate performance improvements.
Herein we review the recent progress in the understanding of charge percolation in graphenebased poly mer composites and how this science has led to performance improvements in these materials. Firstly, we discuss the intrinsic conductive properties of GRMs produced through different routes, high-lighting how the production route strongly influences the resulting electrical properties. Next, the theoretical aspects of electrical percolation are presented. We then review recent results on the electrical properties of graphene-based composites, summarising the reported composite conductivities as a function of fabrication method and graphene type (these results are generally post-2010, with results before then covered in previous reviews [15,16]). Finally, as an example of how these developments can be applied, the effect of strain on conducting composites is explored.

Electrical conductivity of different types of graphene
These yield rGO with less oxygen functionalities, a more restored sp 2 network, and an increased electrical conductivity. For example, a bulk powder of CRGO formed after exposure to hydrazine recovered a conductivity of 200 S m −1 [29]. Similar values were also obtained from individual monolayers  S m −1 ) [36]. Another chemical treatment has been reported that involved immersion in FeI 2 at 95 °C, which recovered a conductivity of 6 × 10 4 S m −1 [37]. Electrochemical reduction through a voltageinduced approach has also been proposed. This has been used to reduce GO (in single sheets and thin films) by applying a potential (pulses of 6 V) between the tip of a conductive atomic force microscope (c-AFM) and a counter-electrode on the substrate surface [38] or between two interdigitated microelectrodes [39] in ambient condition. The reduction results from an electrochemical process whereby hydrogen ions are generated through the oxidation of water present as a thin adsorbed layer at the GO surface. This approach recovered a conductivity of ∼1 S m −1 , making it possible to engineer rGO micro-devices and micro-circuits in a few minutes.
There is a strong relationship between the structure of rGO and its electrical conductivity. Mohan et al [28] reported the synthesis and subsequent reduction of GO in order to maximize the electrical conductivity. They reported an electrical conductivity as high as 10 3 S m −1 for an rGO film reduced with hydroiodic acid (HI). Furthermore, Shin et al [32] demonstrated that the conductivity of graphite oxide films could be modulated using reducing agents (such as sodium borohydride, NaBH 4 ) and they elucidated the underlying reduction mechanism. This involved the formation of boron oxides that were subsequently removed at higher concentrations. They found that increasing the concentration of NaBH 4 led to higher C/O ratios and subsequently to higher electrical conductivities, as shown in figure 1(b). Using NaBH 4 instead of hydrazine to reduce GO is advantageous because, during hydrazine reduction, nitrogen-containing functional groups can form, whose presence then leads to an increase in sheet resistance [32].
Thermal treatments have also been applied, and generally perform better at recovering conductivity compared to chemical treatments. Thin films of rGO (∼10 nm thick) displayed increasing conductivities from 3 S m −1 to 10 × 10 4 S m −1 by tuning the annealing temperature (T ann ) from 200 °C to 940 °C [40]. Becerril et al obtained conductivities of ∼5 × 10 4 S m −1 after being heated to 1100 ° C whilst still maintaining 80% transmittance [41]. As a comparison, the same study found a hydrazine treatment followed by annealing at 400 °C gave ∼5 × 10 3 S m −1 with a similar transmittance. More recently, thin films of rGO have reached 9 × 10 4 S m −1 , after annealing at 1000 °C in a Ar/H 2 atmosphere [42]. By annealing at even higher temperatures (>2000 °C) using arc discharge it is possible to recover conductivities of 2 × 10 5 S m −1 [43], with the downside here of scalability.
Despite some success, these reduction methods have not been able to recover the pristine conductivity of graphene. Even the complete removal of oxygen groups does not fully recover the properties of pristine graphene as defects in the lattice remain. Therefore, only isolated conductive regions of graphene are recovered, and electrons are required to hop between (g) Inkjet printing liquid-exfoliated graphene offers good conductivities without aggressive treatments (adapted from [33] with permission of The Royal Society of Chemistry). (h) The measured conductivities of the available graphene sources. Exfoliated material possesses the highest conductivities but production challenges limit its application; materials produced by each of the other methods are more readily produced. these neighbouring regions, reducing the conductivity [36]. For this reason, covalently functionalised graphene may not be a suitable material for high-end electrical applications, and flakes that remain unmodified are likely to be more advantageous [44].

Liquid exfoliation
It is possible to exfoliate graphite in liquid without heavily functionalising the graphene sheets. Paton et al produced exfoliated graphene in a liquid using surfactants to stabilise the graphene sheets [45]. Flakes typically 1-10 layers in thickness and with lateral dimensions of 500-1000 nm were produced [5,46]. This route yielded large volumes of highquality graphene that was solution-processable and with few defects. The presence of surfactant, however, can decrease the conductivity of the films [47]. More recently, inkjet-printed thin-films of liquid exfoliated graphene were reported. The graphene was printed repeatedly to form films of different thickness, as shown in the optical microscope images in figures 1(c) and (d). SEM images (figures 1(e) and (f)) reveal that the flakes were randomly orientated within the film. Repeated printings of the graphene flakes increased the film's conductivity, and the maximum recorded conductivity was 3000 S m −1 [33]. This value is relatively high compared to other solution-processable techniques, as shown in figure 1(g), and also avoids the use of harsh chemicals and high temperatures.

Graphene nanoplatelets
Another promising candidate for composites is graphene (or graphite) nanoplatelets (GNPs), which are typically much cheaper than other forms of graphene, and are commercially available on the tonne scale. Production methods vary, but most exfoliate bulk graphite using thermal expansion, or mechanical agitation, or both. The dimensions, primarily the thickness, of GNPs dictate their intrinsic conductivity, with thinner GNPs having the highest values: GNPs with thickness of 50 nm, 5 nm and 3 nm have conductivities of 7 × 10 4 S m −1 , 1 × 10 5 S m −1 and 1.5 × 10 5 S m −1 respectively [48]. The reason for this thickness effect is thought to be the poor c-plane conductivity of graphite. Thus producing thin flakes (<10 layers) remains a priority for the preparation of conductive graphene-based composites.
In summary, the conductivity of graphene materials produced using different synthetic routes spans almost 8 orders of magnitude: from the poor conductivity of heavily functionalised graphene oxide (10 −2 S m −1 ) to the exceptional conductivity of graphene produced by mechanical exfoliation (10 6 S m −1 ). These values are summarised in figure 1(h). However, for practical applications such as polymer nanocomposites, there is a balance of quality against the ability to produce the material on the required scale. So while mechanically exfoliated and CVD graphene have the advantage of high conductivities, they are unlikely to have the production capacity to meet the demands of composites. GO can overcome this disadvantage because it has a solution-processable, scalable production route. However, the oxygen containing functionalities decrease the conductivity of the graphene. This has been overcome to a certain extent with reduction methods, but pristine graphene is yet to be recovered. Liquid exfoliated graphene and GNPs remain mostly unfunctionalised and so do not suffer the same quality degradation, and also have the advantage of scalable production routes. For these reasons, they show the most promise for conductive polymer composites.

Percolation theory
The versatility of polymers make them attractive for many applications but they are generally unsuitable for conducting applications because most polymers are insulating. Conductive composites can be achieved by adding a conductive filler to the polymer matrix. For a random distribution of this filler, a conducting network will form at a specific loading, known as the percolation threshold (p c ). When the filler loading reaches p c , the conductivity of the composite rises suddenly and the graph of conductivity versus loading takes the characteristic S-shape, demonstrating the three characteristic regimes: insulating, percolating and conductive (shown in figure 2(a)).
The conductive behaviour in these three regimes can be understood from the microstructure of the composite. When there is no conductive path through the filler, no charge can flow, and the composite remains insulating. On the other hand, if the filler forms a directly connected network, electrons can move through this network, and the composite is conductive. Between these two extremes, conduction takes place when the fillers are not in direct contact but are connected via electrons tunnelling through an interface formed between the filler and the matrix; the conductivity in this case is lower than that where a direct network has been formed. This tunnelling conduction is a quantum phenomenon where the contribution between the nearest neighbours is the most dominant [49][50][51][52]. Tunnelling transport has been directly observed in a carbon-nanofiber polymer composite using conductive-tip atomic force microscopy [53].
The percolation properties can also be affected when the filler forms aggregated structures that are then interconnected by individual filler particles. Often this agglomeration can promote percolation. In a similar vein, the fillers can form a phase-separated, co-continuous morphology comprising of graphenerich and -poor phases within the volume of the composite. This is known as selective localization [54,55] and often leads to a conductive composite at lower loadings.
Both simulations and experiments have shown that fillers with larger aspect ratios lead to reduced percolation thresholds (the modelling of this effect is discussed further in section 3.2) [58,59]. Thus, graphene is one of the most efficient fillers available due to its large aspect ratio. For example, Zhang et al filled PET with graphene and graphite and measured the percolation threshold for each filler [56]. The threshold for graphene (0.47 vol%) was significantly lower than that measured for graphite (∼4 vol%), as shown in figure 2(b). These experimental results were supported by simulations performed by Ambrosetti et al, who found that conductive, platelike objects with an aspect ratio of 100 yielded a composite with a percolation threshold 26 times lower than those formed with objects of aspect ratio 1 [58]. This ideal behaviour caused by high-aspect-ratio fillers can, however, be strongly affected by the preparation method. For example, Du et al [57] prepared high density polyethylene nanocomposites with graphene nanosheets (GNSs) and carbon nanotubes. They found that the samples filled with CNTs had a lower percolation threshold than those with GNSs (as shown in figure 2(c)), because the GNSs tended to aggregate and wrinkle, reducing their effective aspect ratio, and hindering the network formation. In fact, an ultra-low percolation threshold was achieved with CNTs (0.0025 vol%) because their high aspect ratio was maintained. On the other hand, graphene has the advantage of easier processability compared to CNTs as the viscosity of graphene composites is significantly lower than CNT-ones at the same loadings [10]. This means that a greater amount of filler can be introduced in the matrix, leading to higher ultimate values of conductivity.
Several theories have been developed for the evaluation of the effect of the number of interconnections on the conductivity of a random system but the number of variables affecting the conductivity of these systems makes the task challenging. Fournier et al used an analytical model which is based on the Fermi-Dirac distribution for the study of the transition from insulator to conductor [60]. In this case the conductivity is given by: where σ c , σ gr , σ m , are the conductivities of the composite, graphene and matrix respectively, p is the mass fraction, p c is the percolation threshold and b is an empirical parameter that leads to the change in electrical conductivity at p c . A fit of this equation against the experimental data will determine the percolation threshold. At no loading, the polymer is insulating, and remains insulating while the loadings are low. As the filler content is increased to the percolation threshold, a conductive network begins to form. At loadings greater than this, an efficient network has been formed and the conductivity is saturated. (b) Graphene enables a significantly lower percolation threshold in PET than graphite (adapted from [56], copyright 2010, with permission from Elsevier). (c) An example of the percolation threshold in high density polyethylene (HDPE) composites formed with CNT (black) and graphene nanosheets (red) (adapted from [57], copyright 2011, with permission from Elsevier).
The majority of reports in the literature use the classical percolation theory [61] in order to model the conductivity of composites. The equation that describes this theory is: where ψ gr is the volume fraction of graphene and ψ c is the percolation volume fraction, while σ c and σ gr are the conductivity of the composite and the graphene respectively. Moreover, t is the critical power law exponent and depends on the dimensionality of the network, normally taking values of ∼1.33 for 2D systems [62] (such as when confined in a coating) and ∼2 for 3D systems [63].

Modelling percolation in graphene composites
To model the electrically percolating behaviour in composites, one common approach is to use Monte Carlo methods. Here the fillers are represented by geometric objects that are placed randomly inside a constrained volume that represents the matrix. Once the objects are placed, the junctions between them are identified, and their junction resistance, R J is calculated. This is often done using the formula derived by Simmons [64]: (3) where V is the voltage across the junction, A J the contact area of the junction, J 0 an arbitrary constant, and e the charge on an electron. The constant K = 4π∆s √ 2m/h is a combination of Planck's constant h, the mass of an electron m, and the limits of the barrier at the Fermi level ∆s. Finally, ψ 1 is the mean tunnel barrier height and depends on the filler work function and the dielectric constant of the insulating matrix.
The representation of the filler is a key component of these methods. A common approach is the 'hard core, soft shell' structure. The hard core takes the dimensions of the physical graphene sheet, and the soft shell extends a few nanometres around the core to represent the tunnelling distance of the electrons. The cores cannot penetrate each other, but core and shells can overlap. When they do, a tunnelling junction is formed. These methods successfully modelled percolation thresholds in composites of graphene [65], and have also been applied to other nanofillers, such as CNTs [66]. In these models, the junction resistance is much larger than the resistance of the filler, and so the filler resistance is often ignored. This is par ticularly true for highly conductive fillers such as pristine graphene and CNTs. However, how the resistance of a less conductive filler, such as GO, could affect the overall network resistance has been less comprehensively studied.
These models also give insight into the ideal physical properties for the graphene filler. Hicks et al investigated the effect of filler aspect ratio on the con-ductivity of the resulting composite [65]. They found that large aspect ratios lead to a higher conductivity. This was particularly important at lower filler contents. This was because fillers with large aspect ratios form a network with fewer junctions than those with smaller aspect ratios, and fewer junctions reduces the total resistance of the network. In a similar way Zabihi et al found that fillers formed of larger sheets gave composites with higher conductivities [67].
It is also possible to predict the ideal dispersion of the filler. For example, models have shown that agglomeration of the filler will lead to lower percolation thresholds [68]. This result has been experimentally validated by using selective localisation, where the filler can be confined to only one of multiple matrices. This agglomeration effect is less important at higher loadings, where the probability of forming a conductive network is much higher. Another example is the alignment of the filler within the composite. Zabihi et al found that the flake-normal should be perpend icular to the electric field when the filler is disc-like [67]. This is to maximise the probability of forming a conductive network inside the composite.
Extensions to the model outlined above have helped understand other results from electrical composites, like the percolation of charge under alternating current (AC). Hashemi et al effectively modelled the response of a composite to AC current by including microcapacitors within the composite [69]. These microcapacitors represent fillers that are isolated by a thin layer of matrix, where charge can accumulate. Even better agreement (particularly at higher frequencies) was found by including frequency-affected tunnelling and Debye dielectric relaxation into the model, along with the microcapacitors [70]. Further developments on the temperature behaviour of composites have been made; in some cases, heating the composite causes the polymer to expand, which consolidates the conductive network, and so increases the composite's conductivity [71].

Conduction mechanisms in disordered graphene sheets
Different conduction mechanisms are observed when there is disorder in an insulator-conductor composite. Often charge will percolate via a hopping conduction mechanism [72], which involves the excitation of charge carriers into free, delocalised states [73]. Electrons then hop between the available states to transport charge. Hopping conduction mechanisms are split into two main mechanisms: nearest neighbour hopping and variable range hopping (VRH). Both mechanisms normally proceed simultaneously, but usually one is dominant. The general equation for hopping conduction is [74]: where T is temperature, σ 0 and T 0 are constants (although they do have some T dependence, it has a much weaker effect), and γ is the hopping exponent. The hopping exponent depends on the dominant hopping mechanism, which itself depends on the temperature of the system. At low temperatures variable range hopping is dominant.
The hopping exponent then depends on the available dimensions of the hopping as γ = 1/(D + 1) where D is the dimensionality. For 3D systems then, the equation yields: which is the same formalism as derived by Mott [75]. Mott VRH theory considers the density of states (DOS) as a constant value. However, the promotion of one electron to the unfilled state creates a hole in the occupied site. This electron-hole pair system modifies the DOS which vanishes at the Fermi energy due to long-range Coulomb interactions. This model is called Efros and Shklovskii (ES-VRH) and is described by a charge transport equation similar to the Mott model, where the exponent γ does not depend on system dimensions and is equal to 1/2.
While the electron transport properties of the single layer are studied in detail (e.g. ES-VRH model well describes the behaviour of rGO [76]), the conduction mechanisms involved in disordered graphene sheets are still not well established. In general, such systems are continuous thin films corresponding to a nonpercolative regime, which can be described as a dense mesh. rGO thin films can be used as an ideal test-bed system to understand the charge transport mechanisms in partially disordered systems; in the film, the single rGO sheets are placed all parallel to each other, and stacked together. This film yields a highly anisotropic conductive system, where the electric resistance at junctions between neighbouring sheets depends on the sheets stacking and overlap. The building block of such materials are single GO sheets which, due to their excellent processability in solution, can assemble in uniform and continuous thin (∼10 nm) films on large scale (tens of cm 2 ). Once fixed, the morphology of the thin film (i.e. position, number and shape of single sheets), the role of the structural order (i.e. amount of sp 2 regions and defects) and its relationship with the charge transport can be systematically studied by performing several reduction levels in a controllable way.
Vianelli et al [40] investigated the temperature dependence of charge transport properties of rGO thin films, tuning the sheet conductivity and number of defects by thermal reduction before the measurement (figure 3(a)). They characterized the electrical transport properties at intermediate reduction stages and in a wide temperature range (2-300 K), wide enough to assess the validity of different charge transport models.
The temperature dependence of the resistance R(T), as depicted in figure 3(b), fits well with equation (4) below the liquid nitrogen temperature, consistently with 2D-VRH model (γ = 1/3). At lower temperatures, a crossover to another regime, still to be identified, likely occurs. Hopping presumably occurs between the sp 2 regions by consecutive inelastic tunnelling processes. The scaling of the slope of the R(T) curves with increasing T ann , clearly indicates a gain in the electrical conductivity due to the restoration of π − π bonds and the decrease of the number of defects.
For a high degree of reduction of rGO (T ann = 700 °C and 940 °C), deviations are observed at high temper atures (T > 100 °C), ascribed to a crossover to an Arrhenius-like regime (γ = 1), where 3D thermally activated processes start to dominate the conduction.

Ideal composites
An ideal graphene-based composite system would comprise continuous graphene sheets with a large lateral size, that could be well dispersed within the matrix. CVD offers this as it produces high-quality graphene with large lateral dimensions. Vlassiouk et al [77] prepared composites of poly(methyl methacrylate) (PMMA) and CVD graphene using a layer-by-layer approach. As shown in figure 4(a), they formed two structures: a flat laminate (type 1) and a rolled scroll to form a fibre (type 2). The graphene loading in the flat laminate composite was only 0.13 vol%, however the conductivity was among the polycarbonate/CVD-graphene layered composites with a conductivity of 420 S m −1 at a filler content of 0.19 vol%, while the percolation threshold was as low as 0.003 vol% [78]. This is a highlypromising procedure for the production of advanced, multifunctional graphene-based nanocomposites, however producing these materials in bulk quantities is still challenging. A summary of the properties of these composites, and those discussed later, is presented in table 1.

Graphene nanoplatelets
Graphene nanoplatelets (GNPs) are among the most popular materials used for filling polymer matrices because of their inherent tensile strength, electrical and thermal conductivity, and their availability in bulk quantities. However, for GNPs, the composite preparation method is particularly important as they are prone to crumple, wrinkle, and roll during processing. This is particularly important for highshear processes such as melt mixing. In contrast, using solution blending can preserve the original form of the GNPs. An additional benefit of solution blending is that the viscosity of the nanocomposite is lower than those prepared by melt mixing, which allows the macromolecular chains to intercalate between the graphite layers during the preparation process [80].
As an example, Pang et al [81] used a solvent-assisted dispersion method followed by hot compression during the production of UHMWPE/GNP composites and they achieved a very low percolation threshold at only 0.07 vol% filler content. Other preparation methods have the advantage of distributing the flakes uniformly through the matrix, as exampled by in situ polymerization.
Kim et al conducted a comparison of the different effect of processing methods (solution blending, in situ polymerization, and melt compounding) on the conductivity of polyurethane/graphene nanocomposites [82]. They found that the highest conductivity values were obtained from composites prepared via solution blending: melt mixing lead to particle re-aggregation and particle attrition that reduced the lateral size of the graphene; and in situ polymerization caused covalent bonds to form between the matrix and the filler, which hindered direct contact between the fillers and reduced the effective aspect ratio. Similar results were also reported in the works of Kim et al [54] and Xu et al [83]. For all production methods, post-processing methods such as hot pressing or injection moulding can further affect the dispersion of the fillers and the ultimate conductivity of the composite.
The alignment of the filler within the matrix can also be affected by the processing conditions. The most industrially relevant preparation method is extrusion, and different types of extrusion can cause different orientation characteristics in the final composite. For example, the use of small-scale extrusion [84][85][86][87][88] and mono extrusion [89] have both been shown to cause a random orientation of GNPs within the polymer. On the other hand, multilayer coextrusion [89,90] and blown film extrusion [91,92] can give composites with orientated GNP flakes. Extrusion is very commonly followed by other processing steps, such as injection moulding or melt-mixing, in order to formulate the samples into well-defined specimens for testing, so post-processing can always alter the morphology of the flakes after extrusion [80,[93][94][95]. Therefore, there are other more reliable strategies for forcing the fillers to align, and these are discussed further in section 4.4.
Under the correct processing conditions, the selective-localization principle may also influence the conductivity of GNP-based composites. In this case, graphene is predominantly present in specific parts of the composite (often in distinct matrices), reducing the percolation threshold for the overall material. Qi et al prepared conductive graphene/polystyrene composites by solution mixing and reached percolation at a graphene content of 0.33 vol% [79] (figure 4(c)). These graphene composites had ultimate conductivity values of 2-4 orders of magnitude higher than those with CNTs produced in the same way. Qi et al then produced polymer blends where poly(lactic acid) (PLA) was incorporated in the graphene/PS composite. Graphene was found in these blended systems to localize in the PS-rich regions, resulting in a reduced percolation threshold of 0.075 vol%. Figure 4(e) shows schematically how selective localisation lowers the percolation threshold. When material is uniformly distributed, it cannot form a conductive network; if, however, the same amount of material is localised in a smaller volume, a network is more likely formed.
Moreover, functionalisation of the filler can help towards the improvement of solubility and processability, enhancing the interactions between the filler and the matrix, enabling a more homogeneous dispersion. However, the covalent interactions between the fillers and the matrix can sometimes disrupt the sp 2 hybridized carbon atom conductive pathways and reduce the electrical conductivity [96,97].

Graphene oxide and reduced derivatives
GO has a relatively low conductivity, but the functional groups on the graphene sheets can provide some advantages when used in composites. Most significantly, they can alter the van der Waals interactions between the GO and the matrix, leading to a better dispersion of the GO sheets within the polymers [29,41,[98][99][100][101]. For this reason, GO has attracted considerable attention as a nanofiller for polymer nanocomposites. The electrically insulating nature can be reversed to give a material more akin to graphene through either thermal or chemical reduction. A wide range of conductivities and percolation thresholds of GO and reduced GO composites depending on the processing method, polymer matrix and filler type have been reported. The electrical performance of both thermally reduced GO (TRGO)-and chemically reduced GO (CRGO)-based polymer nanocomposites are discussed in the next sections.

Thermally reduced GO (TRGO) based composites
Thermally reduced graphene oxide (TRGO) has been obtained by oxidation of graphite followed by thermal expansion [102], which increases the carbon percentage of the GO materials up to 97 wt.%, and increases their conductivity up to ∼10 5 S m −1 .
The use of TRGO to enhance the electrical performance of polymers has been successfully demonstrated for thermoplastics. Steurer et al melt compounded isotactic poly(propylene) (iPP), poly(styrene-co-acrylonitrile) (SAN), polyamide 6 (PA6), and polycarbonate (PC) in a twin-screw miniextruder together with TRGO [102]. They showed that TRGO with aspect ratios >200 can be uniformly dispersed during melt extrusion. As shown in figure 5(a), the authors observed electrical percolation at 4 wt.% TRGO loading into SAN, which gave a conductivity of 3.7 × 10 −8 S m −1 (as a comparison, at the same content of CB the conductivity was much lower, ≈10 −13 S m −1 ). By adding 12 wt.% of TRGO a specific conductivity of 2 S m −1 was measured, the same as that of the pure filler. For PC they measured a lower percolation threshold at a loading of 2.5 wt.% and yielding a conductivity of 7.7 × 10 −6 S m −1 .
In addition to thermoplastics, elastomers have also recently been combined with GO materials. For example, natural rubber (NR) latex nanocomposites have been reinforced with TRGO materials, and show electrical percolation at 3 phr, with measured conductivities around 10 −4 S m −1 [107]. In addition, a study on styrene-butadiene rubber (SBR) [103] has recently shown that TRGO exhibits superior electrical performance with respect to other carbon nanofillers such as carbon black, expanded graphite, and carbon nanotubes. The TRGO/SBR composite maintained good conductivity even at low electrical frequency, as shown in figure 5(b). Other groups have also found enhanced electrical conductivity in SBR/TRGO composites [117], and in natural rubber composites [118,119].
In all these cases, TRGO efficiently forms a percolated network. TRGO composites showed lower percolation thresholds than those containing CNTs, which was attributed to the higher aspect ratio of TRGO compared to 0D or 1D carbon nanoparticles [102,104]. This low percolation threshold is useful when producing nanocomposites with a low filler content. However, the maximum conductivity achieved Table 1. A summary of the conductive properties of graphene polymer nanocomposites, based on the graphene production method and the polymer processing method. Composites with CNT and CB fillers are included where the studies made direct comparisons.  with TRGO materials is about ten times lower than that of the equivalent CNT-based composites. This is likely due to the residual oxygen in TRGO, and the presence of defects, that both reduce its conductivity. This shows a key limitation of using functionalised graphene as a conductive filler. Chemical compatibilisation can improve the interfacial adhesion between the filler TRGO and the polymer: better adhesion leads to a better dispersion and results in a lower electrical percolation threshold. An example is the work reported by Shim et al, who coagulated SBR together with raw, carboxylated, and cetyltrimethylammonium bromide stabilized TRGO, and reported improvements on the electrical conductivity, showing percolation thresholds as low as 0.5 wt.% [117,120]. Another example is the addition of TRGO to maleated linear low-density polyethylene (LLDPE) and to its derivatives with pyridine aromatic groups by melt compounding [104]. In this report, low electrical percolation thresholds of between 0.5 and 0.9 vol.% were found [104] (as shown in figure 5(c)). However, preparing nanocomposites by melt mixing is difficult due to the low density and very high surface area of TRGO. Further, TRGO often requires special handling and safety procedures. An alternative to ther-mal reduction of GO is the chemical reduction of GO materials.

Chemically reduced GO (CRGO) based composites
Chemically reduced aqueous GO dispersions can be produced by exposing GO to strong reducing agents such as hydrazine. These treatments are not able to completely remove oxygen from the graphene sheets: the oxygen content of CRGO materials is similar to that of TRGO produced at 400 °C (≈15% oxygen). Agglomeration, where the graphene sheets restack after reduction, can be prevented either by using very low concentrations or by adding surfactants or polymers during the reduction step [121][122][123][124][125][126]. A number of strategies to combine CRGO materials with polymers can be found in the literature [103,118,[127][128][129][130][131][132][133][134][135]. For example, CRGO dispersions were blended together with a polymer latex to produce composites with good electrical properties (∼15 S m −1 at ∼1.8 wt.%), and low percolation thresholds (∼0.9 wt.%) [126,127]. Potts et al found similar electrical performances when including CRGO and TRGO into natural rubber (NR) composites, both of which were superior with respect to CB [118,119]. Schopp et al performed a comparative study on styrene-butadiene rubber (SBR) composites with the incorporation of different carbon nanofillers [103]. Processed under identical conditions the electrical conductivities of SBR composites for expanded graphite (EG), carbon black (CB) and CRGO were found to be four to five orders of magnitude smaller than those observed for TRGO, MLG and CNTs containing the same amount of carbon filler [103].

Graphene/CNT hybrid fillers
One area of increasing interest is the use of a combination of graphene and CNTs as fillers for conducting composites. In many cases the two carbon nanofillers have a synergistic effect, yielding composites that have better performance than those made from the individual components. These hybrid fillers have most successfully been applied in composites that require mechanical strength and electrical conductivity, such as for EMI shielding [136].
The synergistic effect has been seen in a paper formed of graphene and CNTs. The conductivity of the graphene paper was 1.8 × 10 5 S m −1 , but this increased to 2.7 × 10 5 S m −1 with the incorporation of CNTs [137]. For the case of polymer composites, a PU/GNP/CNT composite yielded a conductivity of 10 S m −1 after adding graphene and CNTs, which was higher than when adding them individually [136]. The origin of this increase is that the CNTs act as conductivity bridges between the GNP flakes, helping create a conducting network. Another contributing factor is that the CNTs can stop the graphene flakes from re-stacking, encouraging their dispersion in the matrix [138]. Finally, the graphene can also improve the rheological properties of the composite, again encouraging a good dispersion of the fillers [139] and reducing the brittleness of the final composite [140].

Functionalised graphene
Functionalised graphene materials are emerging as attractive alternatives to TRGO and CRGO as reinforcements for polymers. The incorporation of functional groups during dry grinding has been recently reported [105]. Nitrogen-containing functional groups can be incorporated by dry grinding in a nitrogen atmosphere, or oxygen groups if a CO 2 atmosphere is used. These functional groups promote interfacial coupling at the filler/epoxy (EP) interfaces, thus improving the dispersion in the host matrix. However, the functional groups did impair the electrical conductivity. As compared to TRGO/EP and N-TRGO/EP composites, the percolation threshold of dry-milled graphene (DMG) has been observed Figure 6. (a) An external electric field can encourage the orientation of a graphene filler, which results in a lower percolation threshold (b) (adapted from [146], CC BY 4.0). (c) Forming a graphene aerogel that displays a laminated structure (d), which can be used as the foundation for an epoxy composite. For this structure, the conductivity is greatest parallel to the alignment of the flakes (e) (adapted from [147], copyright 2015 American Chemical Society). (f) A diagram of a layer-by-layer structure on a substrate; (g) the electrical conductivity of (PEI/XNBR/PEI/GO)n films increases with an increasing number of deposition cycles (adapted from [108], copyright 2015, with permission from Elsevier). to increase, due to the significantly lower aspect ratios and lower viscosities of DMG/EP mixtures (as shown in figure 5(d)). Although TRGO materials currently show superior electrical performances, DMGs are still promising because they are cheaper than TRGO and higher filler contents are feasible without impairing processability.
One concept for increasing the electrical conductivity is doping graphene with heteroatoms such as nitrogen [141]. N-doped TRGO produced via thermolysis has been reported to improve the electrical conductivity of both graphene and the corre sponding polymer composites [109]. Increases in the nitrogen content from 0 to 12 wt.% were found to increase the electrical conductivity of the graphene material from 4800 to 47 000 S m −1 [109]. Nitrogen doping of the graphene filler has been reported to encourage the formation of a continuous conductive path in the host polymer of PA12 [109], polyethylene [142], polypropylene [110], polyurethanes [111], rubber [103], polystyrene [143,144] and polyamide [112] composites with respect to their TRGO analogues. The electrical percolation thresholds found for the highly exfoliated TRGO and N-TRGO in polymer composites are typically quite low (typically between 1 and 2.5 wt.%) in accordance with their efficient exfoliation and high aspect ratio [105].

Nanostructured architectures
The alignment of the conductive fillers also plays a role in the conductivity in composites [145]. Kim et al [93] prepared three sets of polycarbonate samples reinforced with few-layer graphene initially by melt blending and then moulded into three different geometries. They found that compression moulding encouraged the alignment of the platelet-shaped particles, but did prevent connectivity between them, and so ultimately reduced the conductivity. In another example, Wu et al [146] prepared epoxy/GNP composites where they induced an alignment of the flakes by applying an external electric field as shown in figure 6(a). The conductivities of the composites in the direction of alignment were 2-3 orders of magnitude higher than that along the transverse direction as seen in figure 6(b).
To encourage efficient contact between fillers, one possible strategy is to assemble the graphene sheets into layered structures [148]. For example, Yousefiu et al [128,149,150] have recently reported the synthesis of self-aligned, layered, CRGO/epoxy composites by simultaneous in situ reduction and self-alignment of GO sheets. Furthermore, the in situ polymerization procedure used enabled the formation of strong chemical bonds between the filler and the host epoxy matrix. The resulting materials displayed significantly improved electrical properties in the alignment direction, yielding a percolation threshold of 0.12 vol.%. A further increase in CRGO content beyond 2 wt.% resulted in an ultimate conductivity above 1 S m −1 , which is high enough for many practical applications, such as antistatic coatings, electromagnetic interference (EMI) shields and thermal conductors.
The electrical conductivities of graphene/ polymer nanocomposites prepared by direct dispersion of GNPs or CRGO sheets into the polymer (including those containing highly aligned and welldispersed ultra-large graphene sheets) are, however, commonly still below 10 S m −1 [150]. The formation of a 3D interconnected conductive network within the insulating polymer matrix has recently emerged as a more efficient way to improve the electrical conductivity [151][152][153]. As an example, Wang et al [147] recently reported the synthesis of graphene aerogels (GAs) through one-step chemical reduction and rational self-assembly of GO sheets using hydroiodic (HI) acid, followed by freeze-drying and vacuum-assisted epoxy infiltration (as shown in figure 6(c)). The porous microstructure of the resulting GA is clearly seen in the SEM image shown in figure 6(d). This prep aration method produced a composite with remarkable electrical conductivities of 4-25 S m −1 depending on the concentration of CRGO, as seen in figure 6(e). The percolation threshold for the GA/epoxy composites was found to be ∼0.25 wt.%, with the highest conductivity of 20 S m −1 obtained at 1.4 wt.% graphene content, which is 4 orders of magnitude higher than that found for 2.0 wt.% GNP [154], and nearly 2 orders of magnitude higher than those containing 3 wt.% CRGO [128] or 1 wt.% CNTs [155].
The formation of hydrogels from graphene/polymer composites has also begun to attract attention [156]. Here, the inclusion of GO into the polymer increases the polymer's ability to uptake water. This water uptake can be tuned, which in turn can lead to a tunable electrical conductivity. She et al investigated the conductivity of a PPy/GO composite after repeated cycling treatments of soaking in HCl and NaOH [157]. After HCl soaking the conductivity increased to ∼0.1 S m −1 , and after NaOH soaking it decreased to ∼1 × 10 −4 S m −1 . These graphene composites could be used in chemical sensing applications.

Layer-by-layer (LbL) assembly
Layer-by-layer (LbL) assembly has been widely used to fabricate graphene-containing, ultrathin films or graphene-based polymer composite films [158][159][160]. These highly ordered assemblies offer nanometre precision of filler placement and so could help create ideal nanocomposite structures. The LbL self-assembly strategy has recently been used for the development of graphene-based elastomer nanocomposites. For example, highly ordered free-standing multilayer films of alternating negatively charged GO sheets, positively charged polyethyleneimine (PEI), and negatively charged carboxylic acrylonitrile butadiene rubber (XNBR) latex have been successfully fabricated by electrostatic LbL self-assembly [108]. The structure of this LbL composite is shown in figure 6(f). The conductivity of these multilayer films was found to increase with increasing number of GO layers (from 6.5 × 10 −2 S m −1 to 0.82 S m −1 as their deposition cycles increased from 2 to 30), as shown in figure 6(g).
The fabrication of electrically conductive graphene/PS composite films by combining latex technology with LbL assembly has also been recently reported [129]. In this work, percolation in graphene/PS composites was observed with the assembly of only one layer of graphene when the filler concentration was around 0.2 vol.%. The electrical conductivities were found to increase dramatically from 7 × 10 −7 S m −1 to 0.02 S m −1 as the number of bilayers was increased from one to two (i.e. 0.9 vol.%), and to a value of 0.05 S m −1 with three bilayers (1.3 vol.%).

Effect of strain on percolated networks
A major application that exploits the electrical percolation behaviour of graphene nanocomposites is resistive strain sensors. Here strain on the composite causes a change in its microstructure and so causes a change in its electrical resistance [161]. A key metric of resistive strain sensors is the strength of this effect, called the gauge factor, and is defined as: where R 0 is the unstrained resistance, and R is the resistance at an applied strain ε (for low strains). Graphene was first applied as a thin film of graphene platelets spray cast onto a flexible plastic substrate [162]. The platelets overlapped in regions and they formed a conductive network, whose resistance changed under strain. They found that the GF could be tuned by controlling the degree of overlap, which was done by controlling the concentrations of the deposition. A maximum GF of 150 was recorded. An advantage of this method is that the flakes can be directly deposited onto any substrate, but the system was limited to only small strains.
Better performance over greater strains is possible by embedding graphene in a flexible nanocomposite. For example, graphene in a rubber matrix produced a GF = 35 that could operate up to strains of 800% [5]. More recently, combining graphene in a similar way with highly viscoelastic silicone polymer (commonly known as Silly Putty TM ) yielded a composite material with variable conductivity as described by classic percolation theory [163]. Figure 7(a) shows that the composite had a percolation threshold of 1.75 vol%, and (b) shows that a composite with 7.9 vol% loading had a gauge factor of 350 at low strain. At loadings closer to the percolation threshold the sensors were even more sensitive, with values up to 500 measured as shown in figure 7(c). This sensitivity allowed them to monitor  [163] with permission from AAAS). (e) A graphene/TPU composite sensor performs repeatedly over successive cycles (adapted from [164] with permission of The Royal Society of Chemistry). (f) The effect of strain on a percolating composite; filler connection are broken (and sometimes reformed), changing the composites conductivity (adapted from [165] with permission of The Royal Society of Chemistry). (g) Modelling the voltage drop across a composite's filler network as it is strained (adapted from [162], copyright 2012 American Chemical Society). breathing and pulse, as well as the movement of a spider (as shown in figure 7(d)).
Graphene sensors can also be formed from porous graphene structures that can then be filled with polymer. Using this method gauge factors up to 11 were achieved with a GNP-PDMS sensor [166]. The GNP network was formed by pressing the flakes onto a substrate and then allowing PDMS to diffuse into the network.
Structured graphene nanocomposites can also be formed after the matrix and filler have been mixed. A graphene-TPU mixture was freeze dried to give a structure with 90% porosity and a low density of 0.11 g cm −3 [164]. While these structures produced only low gauge factors (∼2) they performed well under successive compressive testing ( figure 7(e)).
The mechanism of resistance change is a combination of two processes, both of which depend upon the microstructure of the conducting network. The first is that as the composite stretches the number of conductive pathways is reduced as graphene flakes become separated [165] (figure 7(f)). The second mechanism involves disconnected but closely spaced graphene sheets that the charge carriers can tunnel between; as these sheets become more separated, the tunnelling current diminishes [165]. This can also be seen by modelling the voltage drop that occurs through the network as it is strained: figure 7(g) shows how the voltage drop is more pronounced as the material is strained because fewer conductive paths are available [162]. The process of breaking and forming junctions is often reversible and hence the graphene-polymer composites are able to perform many strain cycles (e.g. 400 cycles with 60% strain [167]) before failure.

Conclusions
In conclusion, graphene of various types has successfully been used as a filler to create conductive composites from insulating polymers. Of all the carbon nanofillers, GRMs remain the most promising candidates for electrically conductive polymer composites because of their large surface areas, large aspect ratios, and high electrical conductivities. They have yielded composites with very low percolation thresholds, and in addition, have shown to be easier to prepare at higher loadings, giving better ultimate conductivities.
In choosing a graphene filler, there is a careful balance between production and performance. The highest conducting graphene (that which has been mechanically exfoliated) cannot be produced in quantities needed for polymer fillers. However, solution-based techniques, which can produce the volumes required, tend to produce material with lower conductivity. However, recent years has seen this improve. There are now liquid exfoliation methods available that do not require covalent functionalisation of the graphene, and so produce material with very high intrinsic conductivities, albeit with some reduction in aspect ratio. At the same time, there has been consistent improvement in the ability to reduce GO to return its conductivity. This has perhaps yielded the most success in conductive composites, as the few remaining functional groups after reduction do play an important role in encouraging better dispersion in the matrix. Graphene production methods should prioritise producing larger, thinner sheets, that remain defect-free. Greater control of the functionalisation of the graphene sheets should also be explored. Careful tailoring of functional groups could yield an ideal conductive filler that disperses well while still maintaining a high conductivity.