Novel approach to percolation threshold on electrical conductivity of carbon nanotube reinforced nanocomposites

Abstract

To date, most analytical models used to calculate electrical conductivity in carbon nanotube (CNT) reinforced nanocomposites are not able to predict electrical properties for contents much higher than the percolation threshold. This is because these models do not take into account many critical factors, such as nanotube waviness, dispersion state and process parameters. In the present paper, a novel analytical model based on an equivalent percolation threshold concept, valid for all CNT contents, is developed for this approach. To achieve this, the influence of all these factors has been investigated and several experimental tests have been conducted in order to validate the model. The electrical conductivity varies by several orders of magnitude depending on the value of these parameters, increasing with carbon nanotube content and aspect ratio and decreasing with its waviness. From experimental data, it is found that the waviness increases with carbon nanotube content. Besides, functionalization also causes a local distortion of CNTs, producing more entanglement. When comparing two different dispersion procedures, calendering and toroidal milling, it is noticed that the first method has a greater stretching effect because the shear forces induced are much higher, causing the breakage of carbon nanotubes.

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1 Introduction

Since S. Iijima¹ discovered the first helical microtubules of graphitic carbon, the use of carbon nanotubes has increased due to their exceptional properties.^2–6 In fact, the addition of CNTs into an insulator resin enhances its electrical performance, providing conducting pathways, making it electrically conductive.^7–9 This has attracted the interest of many researchers, as their use in some applications like a lightning strike, or structural health monitoring due to self-sensing properties, can be made possible.^10–13

The determination of a percolation threshold, that is, the critical volume fraction at which a sharp increase in the electrical conductivity is observed when CNTs (or conductive fillers in general) form a network or an interconnected structure (conducting pathways), has attracted the interest of many researchers^14–20 as it is a key point for characterization of the electrical behavior of nanocomposites. It has been found that it depends essentially on the filler geometry¹⁶ but also there are any other factors influencing this value, such as the orientation,^17,18 waviness or distribution of carbon nanotubes.^19,20

To date, there is a lot of research focused on modelling the percolation threshold taking into account many of the parameters previously described.^21–24 Calculation of the percolation onset is essential in order to predict the electrical conductivity of the material, since many analytical models are based on a scaling rule which refers to the percolation threshold.^25,26

However, it has been demonstrated that this model is not able to predict conductivities properly when the content is much higher than the percolation threshold, as it does not take into account most of the parameters influencing the electrical behavior of the nanocomposite.^27,28 For this reason, there is a lot of research focused on developing numerical models including all of these parameters^29–31 but they are often very complicated, not time-effective and do not permit a general overview about the dispersion state of the material.

For all of these reasons, the aim of this study is to develop an analytical model for calculating electrical conductivity as simply as possible, but being able to include the effect of most of the parameters previously mentioned. This model would allow us to know the dispersion state which is essential not only for electrical applications, but also for mechanical ones. To achieve this purpose, a combined model between that proposed by J. Li et al.,²¹ which is able to estimate a percolation threshold according to the volume of agglomerates, and the classical scaling rule that allows the determination of electrical conductivity based on the percolation onset, will be used.

To validate the analytical model, several experimental measurements carried out on CNT–epoxy nanocomposites manufactured using different dispersion techniques and nanotube types have been conducted. The objective is to associate the electrical properties measured experimentally with those predicted by the analytical model depending on the dispersion state, waviness and filler geometry.

On the other hand, the correlation between experimental measurements and the analytical model will allow the estimation of the influence of each dispersion parameter on the electrical performance of the material, making it possible to optimize the dispersion processes.

2 Experimental procedure

2.1 Materials

Nanocomposites were manufactured with different multiwall carbon nanotubes (MWCNT) embedded in an epoxy resin.

The epoxy resin was a low viscosity DGBEA resin recommended for Resin Transfer Moulding (RTM), with a commercial formula called HexFlow RTM 6 supplied by Hexcel. This is a single-component system developed for service temperatures from −60 °C to 180 °C. Three types of MWCNTs (NC3100, NC3150, NC3152) supplied by Nanocyl were used, all with a purity of 95% and an average diameter of 9.5 nm. NC3100 and NC3150 are non-functionalized and have lengths up to 1.5 μm and 1 μm, respectively. The last one, NC3152, functionalized with amine groups, also has a length up to 1 μm.

2.2 Manufacturing

The manufacturing of nanocomposites was carried out in three different stages: (1) addition of the nanoreinforcement into the epoxy matrix and mechanical dispersion, (2) degasification of the mixture and (3) curing.

First of all, a masterbatch at the maximum content which allows a toroidal flow (that is, 1 wt% for NC3100, 0.8 wt% for NC3150 and 4 wt% for NC3152) was manufactured by toroidal stirring for 15 min using a Dispermat® dissolver at 6000 rpm with a rotating disc of 50 mm diameter . Then, the mixture was diluted with more epoxy resin in order to achieve the desired concentration. This new mixture was stirred mechanically and then dispersed into a EXAKT 80E three roll mill machine from EXAKT Technologies Inc. The calendering dispersion was optimized in previous studies,^32,33 and consisted of a progressive reduction of the gaps between rolls (the minimum distance being 5 μm) at a constant rotating speed of 250 rpm for the first roll.

After the dispersion process, it was necessary to use a degasification step to evacuate the air entrapped in the epoxy mixture. Before that, the mixture was heated up to 80 °C in order to facilitate this process. Then, degasification was done under vacuum for 15 minutes. For the manufacturing of nanocomposites, the doped resins were cured with an isothermal cycle at 180 °C for 120 min.

2.3 Characterization of nanocomposites

A microstructural study of the cured nanocomposites has been carried out in order to determine the influence of the different stages of dispersion. To analyze the microstructure, materials were fractured at cryogenic conditions. First, a pre-crack was caused with the material being immersed in N₂ for 3 minutes. Then, the sample was fractured and coated with a thin layer of platinum (5–6 nm). This microstructural study was carried out by TEM, using a Philips Tecnai 20–200 kV apparatus, and FEG-SEM, using a Nova NanoSEM FEI 230 apparatus from Philips.

DC volume conductivity was evaluated according to ASTM D257 using a Source Measurement Unit (SUM) of Keithley Instrument Inc. (mod. 2410) connected through an interface GPIB to a PC. The electrical resistance was determined by the slope of the current–voltage curve, from which the electrical conductivity can be obtained, taking into account the specimen geometry. Three specimens (10 × 10 × 1 mm) were measured for each sample. Silver paint was used to ensure good contact with a 10 × 1 mm section of the electrodes. The applied voltage was within the range of 0–20 V and 0–250 V for nanocomposites with a high or low level of conductivity, respectively.

3 Theoretical analysis

3.1 Calculation of electrical conductivity based on an interparticle distance (IPD) and classic scaling rule

Electrical conductivity is calculated by using the classical model based on the scaling rule:

σ = σ₀(ϕ − ϕ_c)^t, (1)

in which ϕ is the content of the carbon nanotubes, ϕ_c is the percolation threshold, t is a critical exponent determined experimentally, fixed at a value of around 1.6–2 for 3D systems^27,31,34 (1.7 in this study), and σ₀ is a parameter which depends on the intrinsic conductivity of CNTs (σ_CNT) and the aspect ratio (Λ):³⁵

σ₀ = σ_CNT × 10^{0.85{log(Λ)−1}} (2)

To date, most of the research focused on the analysis of the electrical conductivity of nanocomposites defines the percolation threshold as an invariable parameter. However, theoretical results obtained using these models do not fit the experimental ones very accurately because they do not consider most of the parameters that could affect the formation of an electric percolation network in the material. The aim of this study is to create a novel method for modelling electrical conductivity which allows the evaluation of the most relevant characteristics of the material, such as the dispersion state or filler geometry.

As some researchers have shown,²¹ the percolation threshold depends on the dispersion state and the aspect ratio:

in which L^eq = l cos β is the length of the equivalent element (β being the angle between the carbon nanotube and the preferential direction, and l being the length of the carbon nanotube), d is the diameter of the CNT, ε is the localized volume content of carbon nanotubes in an agglomerate and ξ is the volume fraction of agglomerated CNTs in the nanocomposite, defined by these expressions:

with V_CNT being the volume of an individual nanotube, N being the number of CNTs presented in an agglomerate and D being the diameter of the equivalent element.

It is noticed that the dispersion state of the nanofiller is not constant, as it depends on the filler content, so a new parameter, named the equivalent percolation threshold, ϕ^*_c, must be defined. This new percolation onset will take into account the dispersion state, so it will be different depending on the nanotube content. By computing these parameters, the classical model is redefined by this method:

σ = σ₀(ϕ − ϕ^*_c)^t (5)

Thus, it is necessary to know the value of ϕ^*_c as a function of nanotube content. In order to achieve that, a correlation between the dispersion state, average orientation and nanofiller content must be defined.

A correlation between average orientation and dispersion state can be easily found. Some research shows that the more agglomerated the dispersion state, the more random the orientation is.³⁶ Then, it can be supposed that in the absence of agglomerates, all the nanotubes are aligned in the flow direction (that is, with a random orientation in the x–y plane, with x being the direction of preferential orientation). On the other hand, when nanotubes are agglomerated, there is no preferential orientation, that is, they are randomly distributed. Therefore, the length of the equivalent cubic element L_eq would be defined as:

L_eq = (1 − ξ)L_dis + ξL_agl = (1 − ξ)l cos(β_dis) + ξl cos(β_agl), (6)

with and being the average projection of agglomerated and dispersed CNTs with the x axis, respectively, supposing that all nanotubes are distributed in a random way. Therefore, combining eqn (3) and (6), the expression for the percolation threshold can be rewritten as follows:

In order to make the calculations easier, it is going to be supposed that ξ = ε. Once the dispersion parameters are defined, the statistical percolation threshold, that is, the lowest value of percolation obtained when all the nanotubes are well dispersed in the absence of agglomerates, (ϕ_c0), can be calculated:

in which ξ_c0 and ε_c0 are the dispersion parameters at the statistical percolation threshold. In the absence of agglomerates, ξ_c0 = 0, so eqn (8) can be rewritten as follows:

Knowing ϕ_c0, it is possible to estimate the average number of particles in an equivalent cubic element at ϕ_c (N^eq_c):

with N_agl being the number of nanoparticles in an agglomerate, which is . For a content different from the percolation threshold, eqn (10) can be rewritten as follows:

in which N^eq is the total number (volume fraction) of nanoparticles for a given content in an equivalent element and ξ* is the dispersion parameter at this content. n is calculated using this formula:

where all the contents are expressed as a volume fraction of the total nanocomposite:

with ρ^N and ρ^M being the densities of carbon nanotube and matrix, respectively, and w^N being the weight fraction of CNTs. The density of carbon nanotubes is estimated by using the expressions given by Ch. Laurent et al.,³⁷ depending on the average diameter and number of walls, which were measured by TEM (Fig. 1), and is similar for the three types of CNTs tested.

Fig. 1 TEM image of the nanotubes.

From these expressions, it is possible to calculate the value of the equivalent percolation threshold, given any content:

At the statistical percolation threshold (ϕ_c0), the carbon nanotubes are going to be considered straight in order to make the calculations easier. However, at other contents, the carbon nanotubes may not be actually straight, with waviness being a key factor in the filler geometry and also in the elastic properties.^38,39 Considering each wavy carbon nanotube englobed within a perfect cylinder of an aspect ratio Λ_cyl, it is found that:

in which is the effective length of the wavy nanotube and a is the amplitude of waviness, as can be seen in Fig. 2(a). Rewriting eqn (15), it is possible to obtain the value of the wave amplitude as follows:

Fig. 2 Geometry of the (a) wavy and (b) equivalent carbon nanotube.

Therefore, defining the waviness ratio λ = a/L_eff and combining this with eqn (16), it is possible to calculate the waviness of CNTs:

To calculate the percolation threshold and electrical conductivity, an equivalent carbon nanotube will be defined (Fig. 2(b)) with the same volume and a length equal to L_eff, so the effective diameter d_eff and aspect ratio Λ_eff will be:

Therefore, eqn (2), (5) and (14) will be rewritten as follows, in which Λ = Λ_eff:

In order to validate the theoretical calculations, this study is also going to be focused on estimating the influence of two different dispersion methods, toroidal stirring and calendering, on the dispersion and geometry parameters defined previously. The aim is to set up a correlation between the electrical conductivity and dispersion state.

The two dispersion methods are based on shear forces in order to disaggregate the agglomerates. However, sometimes these shear forces can lead to the breaking of carbon nanotubes depending on their characteristics,⁴⁰ reducing their aspect ratio. Shear forces are estimated as a result of the bending stress induced by the fluid over the carbon nanotubes; leading to this formula in the case of straight CNTs:

in which η is the resin viscosity, [small gamma, Greek, dot above] = v/h is the shear rate (v being the linear velocity and h the gap), E is the Young’s Modulus of the CNT and Λ* is the resulting aspect ratio. This new effective aspect ratio Λ* will be used as the nominal aspect ratio of carbon nanotubes, in order to calculate their waviness ratio.

Therefore, using all of these expressions, it is possible to know the dispersion state given by the dispersion parameters (correlated to the content ϕ), the actual aspect ratio Λ* and waviness factor λ for the two dispersion methods described above. Reciprocally, it will be possible to calculate the electrical conductivity accurately by knowing an estimation of these parameters.

3.2 Parametric study

Parametric studies have been conducted in order to evaluate the effect of the different parameters defined previously on the electrical conductivity of nanocomposites. In that particular case, this study has been focused on the influence of the aspect ratio, given by the geometry of carbon nanotubes and the shear rate of the dispersion process, waviness and content. To achieve this purpose, both the aspect ratio and content have been graphically represented versus waviness.

Fig. 3 describes the influence of waviness and content on the electrical conductivity at three different aspect ratios (50, 200 and 500). It is observed that conductivity decreases with waviness while it increases with content, which is obvious. For CNTs with an aspect ratio of 50, the nanocomposite is not electrically conductive for waviness values higher than 0.4 and contents lower than 0.2 wt%. In the case of CNTs with an aspect ratio of 200, the nanocomposite is not electrically conductive for waviness values higher than 0.8 and contents lower than 0.3 wt% and finally, in the last particular case (aspect ratio of 500), the nanocomposite is electrically conductive for all waviness and contents values given.

Fig. 3 Effect of the content and waviness on electrical conductivity for aspect ratios of (a) 50 (b) 200 and (c) 500.

In addition, it is possible to know the equivalent percolation threshold as a function of waviness and filler content, given by the points at which the nanocomposite becomes electrically conductive. Fig. 4 shows the correlation between ϕ^*_c and the mentioned parameters for three aspect ratio values. It is noticed that the equivalent percolation onset increases with waviness due to the negative effect that it has on the electrical conductivity. On the other hand, the aspect ratio has a positive effect on ϕ^*_c. It is also observed that ϕ^*_c is very sensitive to CNT waviness, especially for lower values of aspect ratio.

Fig. 4 Effect of nanotube waviness on filler content at percolation threshold for three different values of aspect ratio.

On the other hand, Fig. 5 describes the influence of the aspect ratio and waviness on electrical conductivity at three different contents (0.1, 0.3 and 1 wt%). It is observed that the correlation between waviness and aspect ratio is similar to that between waviness and content, as could be expected. It is particularly interesting to observe the importance of waviness in electrical conductivity, as a slight variation in this parameter has a similar effect to an appreciable reduction in aspect ratio. This could be expected because the effective aspect ratio used to calculate the electrical conductivity depends on the waviness factor. This sensitivity is larger for lower contents as the difference between the content and percolation threshold is very small, so the electrical conductivity is very sensitive to small variations in aspect ratio.

Fig. 5 Effect of the aspect ratio and waviness on electrical conductivity for contents of (a) 0.1% (b) 0.3% and (c) 1%.

By observing these graphs it is possible to obtain a complete mapping of electrical conductivity as a function of the main characteristics of carbon nanotube dispersion; that is, content, waviness and aspect ratio. This also makes it possible to develop an efficient dispersion method in order to achieve higher values of electrical conductivity.

4 Experimental analysis

As has been described previously, two dispersion methods have been conducted in order to validate the analytical model: toroidal stirring and a combination of toroidal stirring plus three roll mill calendering. The experimental results for electrical conductivity are given in Fig. 6.

Fig. 6 Experimental values of electrical conductivity (a) as a function of nanofiller content after calendering process and (b) at maximum content for both toroidal and three roll mill processes, and comparison to theoretical values as a function of content and waviness for aspect ratios of (c) 67.5 (NC3100 and NC3150) and (d) 63.5 (NC3152).

Using this experimental data it is possible to calculate the value of dispersion parameters and waviness with the analytical model developed. In order to achieve this, it has been supposed that at statistical percolation, fixed at ϕ_c0, the waviness ratio λ is 0 (all the nanotubes are straight). The Young’s modulus is estimated as 1 TPa for non-functionalized nanotubes and 0.9 TPa for functionalized ones, due to the effect of covalent bonds.^6,41,42 With these values of Young’s Modulus and the shear rate, which was calculated from the processing conditions (2250 rpm of the fastest roll and a h value of 5 μm) as approximately 3 × 10⁶ s⁻¹ in the case of the three roll mill process, and using eqn (20), the shear forces in calendering lead to a nominal aspect ratio Λ* of 67.5 for NC3100 and NC3150 and 63.5 for NC3152. Therefore, this indicates that there is a breakage of carbon nanotubes, as it has been stated in previous studies.^40,43

Fig. 6 also shows the calculated electrical conductivity as a function of CNT content and waviness for the actual aspect ratio after the three roll mill process, that is, 67.5 for NC3100 and NC3150 (Fig. 6(c)) and 63.5 for NC3152 (Fig. 6(d)), similar to the images in Fig. 5. It is observed that the electrical conductivity from experimental results is much lower than the maximum that can be achieved when the nanotubes are totally straight. So, this indicates that the CNTs are curved.

By applying the model it can be concluded that CNT waviness increases with carbon nanotube content. This could be explained by nanotubes tending to be more entangled when they are part of an agglomerate, due to the greater interactions between carbon nanotubes, causing a reduction in their effective aspect ratio.⁴⁴ On the other hand, when they are well dispersed, they tend to be aligned in the flow direction. Fig. 7(a) shows the dependence between waviness and the content of carbon nanotubes.

Fig. 7 Correlation between content and (a) waviness and (b) ϕ^*_c for CNTs after the three roll mill process.

This greater entanglement with content can be appreciated in the TEM images of Fig. 8(a)–(c) for 0.2, 0.3 and 0.5 wt% CNT contents. However, a quantitative study of waviness ratio has been conducted for comparison to the theoretical values calculated. Fig. 8(d) shows that experimental values obtained by image analysis are in good agreement with those predicted by the analytical model. On the other hand, a higher entanglement and waviness ratio is observed when the CNTs are part of an agglomerate, which increases with CNT content, as is shown in these images.

Fig. 8 TEM images of carbon nanotubes at (a) 0.2 (b) 0.3 and (c) 0.5 wt% and (d) comparison between experimental results from image analysis and theoretical predictions.

Nevertheless, it is important to highlight again the difficulty in obtaining an accurate value of waviness ratio from image analysis, and the values obtained only allow validation of the tendency predicted by the analytical model.

Going back to the theoretical calculations in Fig. 7(a), in the particular case of NC3100, it is noticed that at 0.1% content, the waviness ratio is nearly 0 and it suddenly increases when the content is 0.2%. This could happen because the theoretical model is very sensitive to small variations in contents near the percolation threshold, and a slight variation in carbon nanotube content may result in a noticeable variation of electrical conductivity and therefore a remarkable variation of waviness ratio. It is important to notice the consequence that process parameters have on the final electrical properties of the material.

In addition, it is observed that NC3152 is more entangled than NC3150 and NC3100, which show similar values. This is probably due to the effect of amino-functionalization, which is not very favorable for achieving good dispersion. Moreover, the presence of amino groups induces local distortions along the radial direction of carbon nanotube sidewalls,^42,45 which could also explain the increase in the waviness ratio in this case.

On the other hand, Fig. 7(b) shows the correlation between the equivalent percolation threshold and the content of carbon nanotubes for the three types of CNTs. It is noticed that ϕ^*_c increases approximately proportionally to the CNT content. In addition, it is observed that values of ϕ^*_c are similar for the three cases, being slightly higher for NC3152 because they are more entangled than NC3100 and NC3150.

For toroidal stirring, the process parameters (6000 rpm and a h value of 7.5 mm) lead to a shear rate of approximately 3000 s⁻¹. Therefore, shear forces induced by toroidal stirring are much lower than for the calendering process and using eqn (20), it is noticed that there is no carbon nanotube breakage in these particular cases.

Fig. 9 shows that the values of waviness ratio for toroidal stirring are much higher than for the calendering process. This could be explained because the three roll mill tends to stretch the carbon nanotubes due to the higher shear forces induced. In addition, it is observed that the reduction in waviness due to calendering process is higher in the case of NC3100 than NC3150, because the stretching effect induced by the three roll mill is more prevalent for longer nanotubes. On the other hand, the reduction in waviness for NC3152 is the highest, which is probably due to a larger stretching effect by the three roll mill on the covalent bonds.

Fig. 9 Values of waviness ratio for toroidal stirring and toroidal stirring + three roll mill calendering processes.

In order to analyze the effect of the dispersion process on agglomeration, it is necessary to focus not only on the dispersion parameters, but also on the average number of particles per unit volume. That is because dispersion parameters only allow a general overview of dispersion quality. Due to the varying effective aspect ratio for different contents, the average number of particles per cubic element, N, supposing that ε = 1, will also change. The following expression allows us to calculate N:

in which L_eq is the length of the equivalent cubic element for straight carbon nanotubes, and L^*_eff is the effective length for wavy CNTs.

As can be observed in the FEG-SEM images of Fig. 10, the average size of larger agglomerates is drastically reduced after the three roll mill method compared to toroidal stirring (Fig. 10(a)–(c)), as can be expected due to calendering improving the dispersion quality (Fig. 10(d)–(f)). The calculated average number of particles per cubic element (which can be considered as the average size of agglomerates) listed in Table 1 shows that the experimental qualitative results (Fig. 10) and the theoretical predictions are in good agreement, with a drastic reduction of larger agglomerates after the calendering process. Therefore, this model allows us to have a quantitative knowledge of the dispersion state.

Fig. 10 Images of the dispersion obtained by toroidal stirring for (a) NC3100, (b) NC3150, and (c) NC3152 and by toroidal stirring + three roll calendering for (d) NC3100, (e) NC3150, and (f) NC3152. The highlighted areas indicate the presence of larger agglomerates.

Table 1 Average size of agglomerates for the two dispersion processes at maximum carbon nanotube contents

Type of nanotube	NT	NT+C
NC3100	31 950	3610
NC3150	10 275	3400
NC3152	37 240	6335

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From the theoretical results, it is possible to determine that this reduction in agglomerate size after the calendering process is probably more drastic in the case of long CNTs, because of their greater degree of entanglement by toroidal stirring.

From these results, a very appreciable reduction on the aspect ratio of carbon nanotubes after the three roll mill process is noticed, especially for NC3100. This is probably due to the separation effect induced by the three roll mill, which is more pronounced for the CNTs with a larger aspect ratio. On the other hand, there is a huge reduction in agglomerate size, especially for longer nanotubes. The waviness is also reduced due to the stretching effect induced by the three roll mill. This reduction is more prevalent in the case of NC3152 due to the effect that shear forces have on the covalent bonds.

Therefore, this model is able to predict some average dispersion and waviness parameters which could also be correlated to mechanical properties, giving a more detailed mapping of the nanocomposite, especially for contents much higher than the percolation threshold, in which the classic model is not able to predict electrical conductivity properly. By applying this model, it is noticed that calendering has a stretching effect on the nanotubes, causing the disaggregation of larger agglomerates. On the other hand, toroidal stirring induces lower shear forces but the resulted carbon nanotubes are wavier, reducing the potential of this process for achieving the best electrical properties.

5 Conclusions

A novel analytical model has been developed in order to calculate the electrical conductivity in CNT reinforced nanocomposites. For this purpose, a redefinition of the classical scaling rule has been made by using an equivalent percolation threshold concept based on the dispersion state of the nanofiller.

Three parameters have been identified as the most influential on the electrical properties: content, waviness and aspect ratio, so their influence on the electrical conductivity has been studied.

It has been observed that electrical conductivity increases with the aspect ratio and content and decreases with waviness. Knowing the relationship between these three parameters has made it possible to estimate the equivalent percolation threshold more accurately than using other methods previously developed.

This analytical model has been validated by experimental measurements. It has been observed that the waviness ratio increases with nanofiller content, as would be expected. In addition, the dispersion parameters increase with content, which implies a higher equivalent percolation threshold with content, which is also expected due to more entanglement.

On the other hand, the influence of dispersion procedures has also been investigated. The main effects of the three roll mill process have been identified as the stretching of CNTs due to shear forces induced by the three roll mill and a breakage of carbon nanotubes, leading to a drastic reduction of the maximum agglomerate size in comparison to toroidal stirring, which is not able to stretch the carbon nanotubes, causing more entanglement. In contrast, the shear forces induced by this method are much lower than in the calendering process.

Filler geometry and functionalization also have a strong influence on the electrical properties. By using three types of carbon nanotubes, it can be concluded that functionalization causes more entanglement with an increase in the waviness ratio, due to the effect of amino groups. In addition, it has been noticed that the stretching effect of the three roll mill method is more prevalent in the case of longer CNTs. On the other hand, after the calendering process, the waviness ratio is similar for longer and shorter nanotubes. Therefore, from the experimental results, it can be concluded that waviness has a role in governing the electrical conductivity, since the effective aspect ratio depends on it. This effect is particularly pronounced in contents near the percolation threshold because the redefined ϕ^*_c depends, essentially, on waviness.

In conclusion, this novel analytical model is able to correlate three critical factors, namely the waviness, aspect ratio and content of carbon nanotubes, to the electrical conductivity of nanocomposites. This allows the estimation of the electrical properties for contents much higher than the percolation threshold more accurately than the classical model and, in addition, it makes it possible to know the dispersion state of nanofillers more accurately.

Acknowledgements

This work was supported by the Ministerio de Economía y Competitividad of Spanish Government [Project MAT2013-46695-C3-1-R] and Comunidad de Madrid Government [Project P2013/MIT-2862].

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