1 Introduction

In general, the electric conduction of most polymer composites is quite low because the surrounding polymer acts as an insulator. The development of high-performance materials for microelectronic and biomedical applications presents one of the priority tasks of modern applied science [1]. In this case, the resistivity of polymer is drastically reduced by adding highly conductive fillers such as graphene nanoplatelets (GNPs) and carbon nanotubes (CNTs) [2, 3]. The fillers are able to form conductive paths that impart conductivity to the insulating polymer by electron hopping effect that happens by charge transfer at electrical contacts between them [4]. However, nanoparticles of metal alloys added to composites could increase conductivity due to optimal balance of electric and magnetic properties, but residual stresses in the coating can lead to a decrease in the electromagnetic properties [5]. Combination of different carbon nanofillers is a promising way to improve the electronic properties with enhanced mechanical characteristics. The development of such carbon nano-form materials and interest in them is for their physicochemical and high electrical conductivity properties which candidates them for electromagnetic interference shielding that provides protection from the electromagnetic fields [6, 7]. The possibility of their usage in various applications such as biomedicine and microelectronics mainly depends on type, dispersion, volume fraction and interfacial interaction between polymer matrix and fillers [8, 9] and it is relevant to minimize the amount of added fillers because carbon fillers are not free from some disadvantages. One of which is their low resistance to the harsh influence of environmental factors such as electromagnetic radiation. Thus, the introduction of high quantity of such fillers deteriorates the physical advantage of polymer matrices [10].

GNP has received much attention due to its two-dimensional structure and large specific surface area which makes it a candidate for practical applications as an electrically conductive additive [11, 12]. This outstanding electronic property of GNP offers good potential for applications such as energy devices, flexible electronics, sensors for applications like health and fitness monitoring [13, 14]. The addition of GNP improves inter-filler contacts and aids the transport of electrons over long distances [15, 16]. The thicknesses of GNPs are usually in the range of nanometers, while their diameter may be in the micrometer scale, which results in high specific surface area and high aspect ratios of GNPs [17].

Due to the good electrical conductivity along with high aspect ratio, CNT attracts the attention for improving the conductivity of an insulating matrix [18]. Polymer-based materials have resistivity in the order of \(10^{9}\) to \(10^{12}\) \(\left( {{\Omega }.{\text{m}}} \right)\) which can be significantly reduced by incorporating CNTs [7]. The tunneling resistance plays an important role in electrical conductivity of polymer nanocomposites. The tunneling effects are ignored if the polymer thickness between two CNTs is larger than the cutoff distance 1.8 nm which is larger than the distance at which the transmission probability is ensured to be less than \({10}^{-6}\) [19].

The addition of conductive fillers to a polymer matrix modifies its electrical properties through a percolation process [20, 21]. This behavior is related to the formation of continuous electron paths in the nanocomposite with the filler volume fraction above the percolation threshold. An electrical percolation threshold is defined as a critical volume fraction when the conductivity of the composite increases by several orders of magnitude [22, 23]. For such an increase to happen, the conductive nanofillers need to be close to each other to reduce resistivity between fillers which is high when the nanofillers are separated by more than a few nanometers [24, 25]. Percolation threshold prediction allows determining the minimum amount of nanofillers needed to have a good conductivity [26]. Above percolation threshold, conductive networks are the main participants to the composite’s conductivity [27, 28].

Using two different nanofillers can remarkably improve the electrical properties of the hybrid nanocomposites due to synergistic enhancement of physical properties and increased interaction between nanofillers [29]. Using these fillers allows controlling the geometry of the resulting nanostructure with high accuracy and obtaining nanostructures of various geometries and a large aspect ratio [30]. In a system with different nanofillers, shorter nanofiller may form conductive paths between the longer nanofillers, and percolation behavior enhances the mechanism of the conductivity. Kinetic network formation processes may be responsible for this enhancement [31]. CNT/GNP polymer hybrid nanocomposites can be represented as heterogeneous materials, caused many researchers to express several models to calculate tunneling resistivity [32, 33]. Li et al. [34] developed an analytical method based on average inter-particle distance to estimate the percolation threshold of conducting polymer nanocomposites containing disk-shaped nanoscale platelets with large aspect ratios. 2D and 3D randomly distributed platelets were included in their model to evaluate the influence of geometric shape of platelets on the percolation threshold of nanocomposite. Their parametric studies suggested that the percolation threshold of nanocomposite increased linearly with increasing the platelet thickness and decreased nonlinearly with increasing the filler diameter. Mora et al. [35] developed a computational tool to quantify the appropriate size of RVE for different CNT loadings and to estimate the amount of CNTs contributing to the conductive network. They proposed a simplified formula for this purpose and found that increasing the aspect ratio of CNTs generally results in more efficient conductive networks. Gbaguidi et al. [36] utilized a 2D Monte Carlo percolation model to quantify the effect of CNT agglomeration on the percolation threshold of CNT/GNP hybrid nanocomposites. They generated the microstructure of tunneling network with different agglomerate contents and morphologies to consider the effect of adding GNP as second filler. Their results indicated that high level of agglomeration leads to an increase in the percolation threshold with a decrease in electrical conductivity. Mutlay et al. [37] examined the percolation behavior of GNP-filled nanocomposites theoretically and experimentally. They formulated a power law-based percolation theory to predict the percolation threshold. They found 0.3 vol.% as the best value of percolation threshold and 10 S/m as the maximum conductivity. Perets et al. [38] studied the concentration dependences of nanocomposite electrical conductivity with the GNPs, CNTs and hybrid nanocomposites. They concluded that the addition of nanoscale carbon particles to the low-viscosity polymer results in formation of two percolation transitions. They showed that the combination of two nanofillers resulted in an enhancement of the conductivity up to 20 times. This phenomenon can be related to a thinner polymer layer between the nanofillers. Xu et al. [39] prepared the GNP-reinforced PMMA nanocomposites by polymerization of MMA in the presence of GNPs with the aid of sonication and heating. The nanocomposite electrical conductivity exhibited an insulator–conductor transition at a very small percolation threshold for GNPs. Their study demonstrated that the percolation threshold of polymer nanocomposites was about 1 wt%. Also, their results indicated that the difference in nanofiller geometry (aspect ratio and surface area) has a significant influence on forming a conducting network in the PMMA matrix and can improve the electrical properties.

An improved analytical model is proposed in this study for predicting the electrical percolation threshold of hybrid nanocomposites containing two different conductive nanofillers, including GNPs and CNTs. In the first stage, the geometric representation of CNTs is generated by considering the orientation, aspect ratio and volume fraction of CNTs. Then, an advanced systematic formulation is expressed for the conductivity of CNT-polymer nanocomposite. In second stage, based on an improved percolation-like model, a novel equation for predicting the electrical conductivity of hybrid CNT/GNP polymer nanocomposites was developed. The predicted results for resistivity are compared with those of experimental data to check the accuracy of the developed model [40]. Parametric studies were conducted while being tested and validated with experimental data from previously published papers. The parametric investigations clearly showed that the nanofiller geometry and volume fraction are the dominant factors which affect the resistivity of hybrid nanocomposites. The remainder of this paper is organized as follows. In Sect. 2, a methodology is presented to consider geometric representation of CNT network effects on the tunneling resistance of CNT-polymer nanocomposite. Section 3 explains the evaluation of resistivity of hybrid nanocomposite by adding GNPs into the single filler nanocomposite as a function of average separation distance between nanoplatelets. Section 4 presents the prediction results and compares them with the experimental results, including the effects of average tunneling distance and aspect ratio on the resistivity of hybrid nanocomposites. Finally, Sect. 5 contains concluding remarks.

2 Analytical formulation of nanocomposite

Adding electrically conductive nanofillers into a polymer forms a conductive path that reduces resistivity. Tunneling is responsible for the formation of the conductive paths. The CNTs are considered as a source of electrons and the polymer layer as a potential barrier. The electrical resistance of CNT-polymer nanocomposites can be represented by the resistance of CNT percolating networks inside the composite systems [41]. In CNT-filled nanocomposites, current could be transformed by direct electron tunneling from one CNT to another. Direct contact between CNTs can also provide a conductive path for current flow. Consider a uniformly random distribution of CNTs in a representative volume element (RVE) of a CNT-polymer nanocomposite with side length \(L\), as shown in Fig. 1b. Each CNT in the RVE is assumed straight with starting point \(\vartheta_{i}^{0} = L \times {\text{rand}}\), where \(\vartheta\) could be any of \(x,y\) or \(z\). The dimensions of ending point of CNT are represented by Eq. (1).

$$ \begin{gathered} x_{i}^{1} = x_{i}^{0} + l_{i} {\text{sin}}\theta_{i} {\text{cos}}\varphi_{i} \hfill \\ y_{i}^{1} = y_{i}^{0} + l_{i} {\text{sin}}\theta_{i} {\text{sin}}\varphi_{i} \hfill \\ z_{i}^{1} = z_{i}^{0} + l_{i} {\text{sin}}\theta_{i} \hfill \\ \end{gathered} $$
(1)
Fig. 1
figure 1

a Schematic of hybrid nanocomposite containing CNTs and GNPs b schematic of the 3D representation of CNTs generated in representative volume element with polar and azimuthal angles

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where \(l_{i}\), \(\varphi_{i}\) and \(\theta_{i}\) are the length, the azimuthal and polar angles of \(i\) th CNT which are random variables depicted in Fig. 1b.

After determination of the position of CNTs, the minimum distance between walls of the paired CNTs must be calculated. If \(i\) th and \(j\) th CNTs’ vectors (\(v_{i}\) and \(v_{j}\)) are connected by a vector \(v_{ij}\), the minimum distance between them is defined as \(d_{ij} = \frac{{v_{ij} .\left( {v_{i} \times v_{j} } \right)}}{{v_{i} \times v_{j} }}\). If the minimum distance is less than the maximum effective distance of tunneling effect (cutoff distance), the CNT pair is considered in contact. Walls of contacted CNTs are separated by van der Waals distance \(d_{{{\text{vdw}}}}\), due to the effect of the van der Waals interaction.

Generally, the intrinsic resistance along CNTs with two nearest intersections can be evaluated easily [42].

$$ R_{f} = \frac{4l}{{\pi \sigma_{f} D^{2} }} $$
(2)

where \(l\) is length of the CNT segment between the contact points and \(\sigma_{f}\) and \(D\) are intrinsic electrical conductivity and diameter of CNT. The contact region between two CNTs is at the nanoscale. Since the electron density associated with a single state in a conductor (CNT) of length \(l\) is \(1/l\), the current–voltage characteristics can be written by the Landauer-Büttiker formula, such that [43].

$$ I = \frac{2e}{h}\mathop \int \limits_{0}^{\infty } \tau M\left[ {\frac{1}{{e^{{\frac{E - \mu - eV}{{k_{B} T}}}} + 1}} - \frac{1}{{e^{{\frac{E - \mu }{{k_{B} T}}}} + 1}}} \right]dE $$
(3)

Here, \(M\) is the total number of conduction channels. Assuming that the number of \(M\) is constant over the energy range \(\mu_{2} < E < \mu_{1}\) (460 for multi-walled CNTs), \(\tau\) denotes the average probability that an electron at one end of the conductor (CNT) will transmit to the other end. The constants \(T\), \(V\) and \(\mu\) represent the respective temperature, voltage and the potential energy (cutoff energy) of a CNT. Other constants are the electron charge \(e\), Planck’s constant \(h\) and Boltzmann constant \(k_{B}\) [44].

Dealing with a large number of nanotubes makes it difficult to evaluate tunneling resistivity between each pair of CNTs. This limitation leads to the development of the percolation model by averaged properties that aid the computation. The term \(\mathop \int \nolimits_{0}^{\infty } \tau M\frac{1}{{e^{{\frac{E - \mu }{{k_{B} T}}}} + 1}}{\text{d}}E\) is rewritten by change of the variable \(\frac{E - \mu }{{k_{B} T}} = \chi\) and using the integral \(\mathop \int \nolimits_{0}^{\infty } \frac{\chi }{{e^{\chi } + 1}}{\text{d}}\chi = \frac{{\pi^{2} }}{12}\) as depicted in Eq. (4).

$$ \mathop \int \limits_{0}^{\infty } \tau M\frac{1}{{e^{{\frac{E - \mu }{{k_{B} T}}}} + 1}}{\text{d}}E \approx \mathop \int \limits_{0}^{\mu } \tau M{\text{d}}E + \frac{{\pi^{2} }}{6} \times \left( {k_{B} T} \right)^{2} \frac{{{\text{d}}\left\{ {\tau M} \right\}}}{{{\text{d}}E}}\left| {\begin{array}{*{20}c} {E = \mu } \\ \\ \end{array} } \right. $$
(4)

Using Sommerfeld expansion with respect to the temperature \(T\) leads to Eq. (5) [43].

$$ I = \frac{2e}{h}\left\{ {\mathop \int \limits_{\mu }^{\mu + eV} \tau M{\text{d}}E + \frac{{\pi^{2} }}{6}\left( {k_{B} T} \right)^{2} \times \frac{{{\text{d}}\left\{ {\tau M} \right\}}}{{{\text{d}}E}} + O\left( {\frac{{k_{B} T}}{\mu }} \right)^{4} } \right\} $$
(5)

Here, \(O\left( {\frac{{k_{B} T}}{\mu }} \right)^{4}\) represents the remaining expansion terms to be ignored. Under low voltage \(V\), it can be estimated that \(\frac{{{\text{d}}\left\{ {\tau M} \right\}}}{{{\text{d}}E}}\left| {\begin{array}{*{20}c} {\mu + eV} \\ \mu \\ \end{array} \approx \frac{{eVM{\text{d}}^{2} T}}{{{\text{d}}E^{2} }}\left| {\begin{array}{*{20}c} \mu \\ \\ \end{array} } \right.} \right.\). Considering an integer for the channel number, tunneling resistance can be evaluated as drafted in Eq. (6).

$$ I = \frac{2e}{h}\left\{ {\mathop \int \limits_{\mu }^{\mu + eV} \tau M{\text{d}}E + \frac{{\pi^{2} }}{6}\left( {k_{B} T} \right)^{2} \times \frac{{eVM{\text{d}}^{2} T}}{{{\text{d}}E^{2} }}\left| {\begin{array}{*{20}c} \mu \\ \\ \end{array} } \right.} \right\} $$
(6)

Since \(k_{B} T \approx 0.026 {\text{eV}} \ll {\Delta }E \approx 5 {\text{eV}}\), therefore Eq. (6) can be approximated by the first-order Taylor expansion of the term \(\tau M\), as exhibited in Eq. (7).

$$ I \approx \frac{2e}{h}\left\{ {\mathop \int \limits_{\mu }^{\mu + eV} \tau MdE} \right\} $$
(7)

In this equation, \(\frac{2e}{h}\left( {\sim 80\frac{{{\text{nA}}}}{{{\text{meV}}}}} \right)\) is the current carried per unit energy by a single state.

The transmission probability is estimated by using the Schrodinger equation with a rectangular potential barrier as depicted in Eq. (8) considering \({\Delta }E\) as the difference of work functions between the CNT and polymer [43].

$$ \tau = {\text{exp}}\left( {\frac{{ - \left( {d - D} \right)}}{{h/\sqrt {8m_{e} {\Delta }E} }}} \right){ } D + d_{{{\text{vdW}}}} < d \le D + d_{{{\text{cutoff}}}} $$
(8)

Here, \(d\) shows the minimum distance between two lines of two neighbor CNTs, where the probability of tunneling is high. This distance should be no less than the van der Waals separation distance \(d_{{{\text{vdW}}}} \approx 0.4\, {\text{nm}}\). If the distance is below \(D + d_{{{\text{cutoff}}}}\) at the closest point of the approach of the two lines of two CNTs, the neighboring CNTs are considered to be in contact. Symbol \(d_{{{\text{cutoff}}}}\) denotes the maximum distance between two CNTs and the maximum possible thickness of the medium separating two adjacent CNTs that allow the tunneling penetration of electrons. Higher value of \(d_{{{\text{cutoff}}}}\) does not lead to any observable consequences as the tunneling probability decays exponentially with \(d\), but it brings very high computation costs. Therefore, an upper cutoff on the integral is necessary and the tunneling effect can be safely ignored beyond this distance. The symbol \({\Delta }E\) depicts the potential barrier height which corresponds to energy level for conduction and is assumed to have a rectangular shape at the equilibrium state in which do not cause any transfer of electrons or current flowing through insulating layer. The electrons must surmount the potential barrier by the electric tunnel effect to flow through the insulating region.

Using Sommerfeld expansion and solving the Landauer-Buttiker formula with a rectangular potential barrier, the tunneling resistance of CNT junction can be evaluated as depicted in Eq. (9).

$$ R_{{{\text{tun}}}} = \frac{h}{{2e^{2} M\tau }} $$
(9)

Substituting transmission probability from Eq. (8) into Eq. (9) leads to Eq. (10).

$$ R_{{{\text{tun}}}} = \frac{h}{{2e^{2} }}\frac{1}{{M.\exp \left( { - \frac{{d_{a} \sqrt {8m_{e} \Delta E} }}{h}} \right)}} = \frac{h}{{2e^{2} }}\frac{{\exp \left( {2\pi .2d_{a} .\frac{{\sqrt {2m_{e} \Delta E} }}{2\pi \hbar }} \right)}}{M} $$
(10)

Here, \(d_{a} = d - D\) and \(\hbar = 2\pi h\) is Planck’s constant. The contact area at the contact surface could be approximated by the area of CNT cross section which is multiplied to top and bottom of division line in Eq. (10).

$$ R_{{{\text{tun}}}} = \frac{h}{{2e^{2} }}\left( {\frac{{\pi d_{a}^{2} /4}}{a}} \right)\frac{{{\text{exp}}\left( {\frac{{4\pi d_{a} \sqrt {2m_{e} {\Delta }E} }}{h}} \right)}}{M} $$
(11)

Considering Eq. (11), \(M\) as an integer, and Ln \(\tau = \frac{{ - d_{a} }}{{h/\sqrt {8m_{e} {\Delta }E} }}\) one can write Eq. (12).

$$ R_{{{\text{tun}}}} = \frac{{d_{a} h^{2} }}{{ae^{2} \sqrt {2m_{e} {\Delta }E} }}{\text{exp}}\left( {\frac{{4\pi d_{a} \sqrt {2m_{e} {\Delta }E} }}{h}} \right) $$
(12)

Above the percolation threshold, although CNTs are still not physically in contact to each other, they connect electrically and form a conductive network. Two main sources of electron transport mechanisms are the intrinsic conductance of the nanotubes and the conductance from electron tunneling between sufficiently close nanotubes. Considering both tunneling and direct contact resistances, resistivity between two CNTs is given by Eq. (13).

$$ \rho = R_{f} + R_{{{\text{tun}}}} = \frac{l}{\sigma a} + \frac{{d_{a} h^{2} }}{{ae^{2} \sqrt {2m_{e} {\Delta }E} }}{\text{exp}}\left( {\frac{{4\pi d_{a} \sqrt {2m_{e} {\Delta }E} }}{h}} \right) $$
(13)

3 Hybrid nanocomposites

A schematic sketch of hybrid nanocomposite containing CNTs and GNPs is shown in Fig. 1a. GNPs are modeled as thin circular platelets, which are homogeneously distributed within the matrix. Electrical conductivity of CNT-polymer medium containing conducting GNP is described by the percolation theory. Percolation theory proposes that below a critical volume fraction, conductive fillers are individually isolated in the medium. As the volume fraction of filler increases, percolation probability of the fillers also increases until a conductive network of the fillers is formed. In the case of GNPs, geometry is defined by their aspect ratio \(\alpha = 2r/t\), where \(r\) and \(t\) are radius and thickness of GNP, respectively. The total resistance of GNP-reinforced nanocomposites is considered as expressed in Eq. (14).

$$ R = R_{f} + R_{{{\text{tun}}}} $$
(14)

where \(R_{f}\) and \(R_{{{\text{tun}}}}\) denote the intrinsic resistances of GNP and tunneling regions, respectively. The effect of aspect ratio on the intrinsic electrical resistivity can be explained by Eq. (15).

$$ R_{f} = \frac{2r}{{t^{2} \sigma_{f} }} $$
(15)

where \(\sigma_{f}\) is GNP intrinsic electrical conductivity. Considering the matrix material, the tunneling resistance of this medium includes the resistances of GNP and matrix, which can be expressed by Eq. (16).

$$ R_{{{\text{tun}}}} = \frac{1}{{\delta \sigma_{f} }} + \frac{\rho \gamma }{{\delta^{2} }} $$
(16)

Here, \(\rho\) is CNT-polymer resistivity, which was calculated in Eq. (13), the symbol \(\gamma\) is defined as the average separation distance which decreases with the increase in the GNP volume fraction [45] and symbol \(\delta\) is the tunneling diameter, which can be expressed by Eq. (17).

$$ \delta = \left( {\frac{{\pi r^{2} t}}{{V_{fp} }}} \right)^{1/3} $$
(17)

Li and Kim [34] considered a 3D random distribution of platelets into their model and formulated the percolation threshold of GNPs as a function of geometry and average separation distance of GNPs dispersed in the matrix. Here, \(V_{fp}\) is the critical volume fraction of 3D random platelets distributed in nanocomposite systems for gaining electrical conductivity, which is a function of aspect ratio and average separation distance indicating geometrical sizes needed to from critical conductive network paths leading to the percolation threshold and can be given by Eq. (18).

$$ V_{fp} = \frac{{27\pi r^{2} t}}{{\left( {2r + \gamma } \right)^{3} }}. $$
(18)

Substituting \(R_{f}\) and \(R_{{{\text{tun}}}}\) into Eq. (14) gives the total resistance of GNP.

$$ R = \frac{1}{{\delta \sigma_{f} }} + \frac{\rho \gamma }{{\delta^{2} }} + \frac{2r}{{t^{2} \sigma_{f} }} $$
(19)

Now, the conductivity of an extended GNP assuming \(\gamma \ll r\) (\(\gamma\) in nm and \(r\) in \({\mu m}\) scales) is predicted by Eq. (20).

$$ \sigma_{e} = \frac{2r + \gamma }{{t^{2} R}} \cong \frac{2r}{{\left( {\frac{{t^{2} }}{{\delta \sigma_{f} }} + \frac{{\rho \gamma t^{2} }}{{\delta^{2} }} + \frac{2r}{{\sigma_{f} }}} \right)}} $$
(20)

Above the percolation threshold, the GNPs will form very long conductive paths with equivalent conductivity. By assuming that those equivalent GNPs are randomly oriented, a simple model was suggested by Deng and Zheng [46] to evaluate the electrical conductivity of polymer nanocomposite considering the volume fraction of filler \(V_{f}\) as follows.

$$ \sigma = \sigma_{0} + \frac{{\xi V_{f} \sigma_{e} }}{3} $$
(21)

Not all GNPs form part of the percolated network. The symbol \(\xi\) is the probability of percolated GNPs and will increase from 0 to 1. The percolation probability depends on GNP aspect ratio as depicted in Eq. (22).

$$ \xi = \frac{{V_{f}^{1/3} - V_{fp}^{1/3} }}{{1 - V_{fp}^{1/3} }} $$
(22)

Here, \(V_{f}\) is the volume fraction of GNP and \(V_{fp}\) is percolation threshold. The probability of percolated GNPs is mainly dependent on the dimension and volume fraction of GNPs. Smaller \(\xi\) means fewer numbers of GNPs cooperated in the percolated network. Thus, smaller \(\xi\) value results in more GNPs that are needed to form critical conductive network paths, leading to a higher percolation threshold.

The electrical conductivity of nanocomposite is evaluated by replacing Eq. (20) in Eq. (21).

$$ \sigma = \sigma_{0} + \frac{{\left( {2/3} \right)\xi V_{f} r\delta^{2} \sigma_{f} }}{{t^{2} \delta + \rho \gamma t^{2} \sigma_{f} + 2r\delta^{2} }} $$
(23)

Figure 2 shows the flowchart of the process of calculating the hybrid nanocomposite resistivity. First, a 3D model of composite containing randomly distributed CNTs was constructed in the RVE as step 1 of 1st stage. RVE was defined as a cube with side length \(L\). For the sake of simplicity, CNTs were assumed straight. Generated CNTs which may exceed the RVE replaced with a new one. The distance between two CNTs is calculated as step 2 of 1st stage. In calculating the minimum distance between two CNTs, the geometrical method of calculating distance between two lines in space was used. The new CNT is removed if any of the \(d_{1} , \ldots ,d_{i - 1} \) was less than van der Waals distance as step 3 of 1st stage. According to the tunneling mechanism, 1.8 nm is adopted as tunnel effect distance in this study. This distance is checked to ignore the tunneling effect if the polymer thickness between two CNTs is larger than cutoff distance as step 4 of 1st stage. The tunneling regions between neighboring CNTs were assumed to express the whole resistance of CNT-polymer nanocomposite which is calculated as step 5 of 1st stage. After checking the desired volume fraction in step 6, the resistivity of CNT-polymer nanocomposite is calculated as step 7 of 1st stage. Next, the established model connects the resistivity obtained in stage 1 with the percolation-like model of GNP. The percolation-like model based on percolation onset and GNP dimensions generates assigned values for the GNP such as an averaged radius \(r\), thickness \(t\), intrinsic electrical conductivity \({\sigma }_{f}\) and volume fraction \({v}_{f}\) as step 1 of 2nd stage. Then, \({v}_{fp}\) is calculated using Eq. (18) considering a favorable average separation distance in range of 1 < γ < 2 nm for GNPs as step 2 of 2nd stage. After evaluation of tunneling diameter as step 3 of 2nd stage, the tunneling resistivity of GNPs network is calculated by Eq. (16) as step 4 of 2nd stage, using tunneling diameter δ derived from step 3. Finally, electrical conductivity of hybrid nanocomposite is computed via Eq. (23), using ξ derived from Eq. (22), having ρ from first stage.

Fig. 2
figure 2

Flowchart of different calculation steps of the two-stage analytical model based on the present methodology

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4 Results and discussion

The effects of several key parameters on the resistivity of hybrid nanocomposites are examined considering the van der Waals separation distance as \(d_{{{\text{vdW}}}} = 0.4\, {\text{nm}}\) [47]. When the distance between two CNTs is at the nanoscale and smaller than the cutoff distance \(d_{{{\text{cutoff}}}} \approx 1.8\, {\text{nm}}\), electron tunneling between CNTs occurs. Also, the nanofillers are well dispersed in the matrix without agglomeration which leads to better properties [26]. Figure 3 shows schematic of CNT dispersion with different aspect ratios, maximum orientation angles and volume fractions.

Fig. 3
figure 3

Schematic of different CNT dispersion states of a 0.1 vol.% of highly aligned CNTs with \(\alpha = 250\) b 0.25 vol.% of highly aligned CNTs with \(\alpha = 250\) c 0.1 vol.% aligned CNTs with \(\alpha = 250\) d 0.25 vol.% aligned CNTs with \(\alpha = 100\) e 0.1 vol.% randomly oriented CNTs with \(\alpha = 250\) f 0.1 vol.% randomly oriented CNTs with \(\alpha = 450\)

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The developed theory is applied to examine a set of experimental data on the percolation threshold and resistivity of the GNP/CNT polypropylene nanocomposites. To compare our results with the experimental data [40], the barrier potential, intrinsic electrical conductivity and dimensions of GNP and CNT are taken in accordance with the experimental conditions, as depicted in Table 1 [40, 48,49,50].

Table 1 Geometrical and morphological properties of different phases
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Figure 4 shows electrical resistivity as a function of GNP volume fraction for different CNT volume fractions in comparison with Al-saleh experimental data. In agreement with experimental findings, the model predicted that the composite attains lower percolation threshold with higher CNT content [40]. These findings were attributed to the more facile formation of percolation network in the presence of CNTs. Both the analytical and experimental studies suggested that by incorporating more GNPs, the electrical resistivity declines. When GNP volume fraction is low, the distance between the GNPs is so large that the contact resistance is close to the resistance of the polymer. The decrease in the distance between GNPs leads to formation of new conductive networks. Once GNPs make the conductive networks, the resistivity of the nanocomposite sharply decreases due to the significant reduction of contact resistance. Figure 4 shows that a combination of CNT and GNP led to a sudden change in resistivity onset resulting from the large difference between intrinsic electrical conductivity of different phases and the creation of homogeneous electrically conductive pathways by electron hopping [51].

Fig. 4
figure 4

Comparison of electrical resistivity between present model and experimental data for different CNT volume fractions

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Figure 5 presents the effects of the CNT volume fraction and aspect ratio on the resistivity of the hybrid nanocomposites versus GNP volume fraction in comparison with Al-saleh experimental data. It is observed that there is a good agreement between predicted results and experimental data. As expected, the resistivity decreased with the increase in CNT aspect ratio. The resistivity of nanocomposite is dependent to the aspect ratio of the CNTs because higher aspect ratio for the same CNT volume fraction means more CNT length, and therefore more crossings. Thus, when the CNT aspect ratio increases, the CNTs begin to contact and connect to each other attributable to more crossing. Also, the long nanostructure of CNT has more permanent interactions with neighbor CNTs, which is beneficial to form new conductive networks. Figure 5 also demonstrates that the percolation threshold of the hybrid nanocomposite gradually moved toward left with increasing CNT aspect ratio which means that longer CNTs can make conductive network paths at a lower volume fraction. Below percolation threshold, CNTs with smaller aspect ratio cause higher resistivity because they cannot reduce the insulating gap. It can be seen from Fig. 5 that when the GNP volume fraction is low, the values of the resistivity of the nanocomposite decrease slightly with increasing the CNT volume fraction. After the GNP volume fraction passes the percolation threshold value, the change of the resistivity of the nanocomposite does not depend anymore on CNT volume fraction. The volume fraction of CNT should be kept low because increasing the volume fraction may cause defects and low percentage of this filler is preferred for mechanical purposes [51].

Fig. 5
figure 5

Electrical resistivity of hybrid nanocomposite versus GNP volume fraction as a function of CNT aspect ratio and volume fraction

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To represent the influence of CNT maximum orientation angle, a series of analytical evaluation is carried out as depicted in Fig. 6. Also, the variation of the resistance of composite versus GNP volume fraction as a function of CNT maximum orientation angle is compared with experimental data of Liu et al. [52] and Bai et al. [53] in Fig. 6. As observed in Fig. 6, present study results exhibit some discrepancies from the experimental data. Resistivity predictions are lower than the experimental data at all filler volume fractions due to the formation of percolated networks at lower filler volume fractions. One possible reason for this discrepancy could be that this study concerns uniform dispersion for CNTs, while most experiments could not fabricate composites with uniform disparity. Highly aligned CNTs had an orientation angle in the range of \(0<{\theta }_{\mathrm{max}}<10\) [54] in which \({\theta }_{\mathrm{max}}\) is the angle between the line where the CNT is located and the y-axis. The difference in the percolation threshold in Fig. 6 is very small due to weak synergistic interaction of GNPs with CNTs before the formation of conducting bridges. This is also related to low CNT volume fraction and superior electrical conductivity properties of GNP on the formation of network. When the CNTs are perfectly aligned, they are almost parallel and connective paths are less likely to be formed. Therefore, nanocomposite with highly aligned CNTs has a higher resistivity onset, while composite with randomly oriented CNTs demonstrates lower resistivity onset due to the bridging phenomenon, which appears earlier related to the decrease in the size of the gap between CNTs and increase in electron exchange interactions.

Fig. 6
figure 6

Electrical resistivity of hybrid nanocomposite versus GNP volume fraction as a function of orientation level of CNT

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Figure 7 demonstrates the effect of GNP aspect ratio on the percolation threshold and resistivity of hybrid nanocomposites. The results are also compared with experimental data of Liu et al. and Bai et al. The model predictions at different GNP aspect ratios showed good agreement with corresponding experimental data. However, there are some discrepancies between predictions and experimental data of Bai et al. related to high sensitivity of electrical resistivity to the GNP separation distance and dimensions which is a major challenge to be measured properly by experimental tools. It is seen that there is a strong connection between the GNP aspect ratio and the percolation threshold due to GNP aspect ratio effect on electric field concentration far below the intrinsic resistivity of the matrix [55]. GNP with smaller \(\alpha \) cannot form new conductive networks, because its smaller diameter to thickness ratio makes it difficult to connect to neighboring GNPs that leads to the formation of segregated structures with weak affinity between them. Thus, smaller aspect ratio of GNP is not beneficial for the enhancement of multiple interfaces and bridging among GNPs because the thick randomly distributed rigid two-dimensional GNPs are unable to form a percolation network as effective as thin GNPs due to the interspaces between them. Thin GNPs lead to longer conductive network due to higher overlap ratio between fillers. The percolation threshold value of the nanocomposite is lower at higher GNP aspect ratio. Above the percolation threshold, as the GNP aspect ratio increases, the resistivity of hybrid nanocomposite decreases because of the enhanced number of GNP intersections and the additional internal electric field creation due to an increase in the number of intersections capable of charge transformation.

Fig. 7
figure 7

Electrical resistivity of single filler and hybrid nanocomposite versus GNP volume fraction as a function of GNP aspect ratio

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Figure 8 shows the effect of height of barrier potential on the percolation threshold and resistivity of the hybrid nanocomposite. It is proved that the barrier height of the matrix is an important factor affecting the resistivity of hybrid nanocomposites and the energy required for the electron transmission [51]. Barrier height can be explained as the difference of the work functions between the CNT and polymer defining enough energy to transfer the charge [56]. As shown in Fig. 8, percolation threshold increased with the increase in barrier potential. Larger height of barrier potential decreased the tunneling probability \(\tau \) caused the percolation paths to disappear, which led to the increase in percolation threshold and resistivity. Also, both curves related to composite with mixed fillers indicated conductivity gains. The need for GNP is considerably reduced with the introduction of a small amount of CNT due to the formation of co-supporting conductive networks.

Fig. 8
figure 8

Electrical resistivity of single filler and hybrid nanocomposite versus GNP volume fraction as a function of height of barrier potential

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Figure 9 shows the effect of average separation distance of GNPs on the resistivity of hybrid nanocomposite. The results are compared with experimental data, and a good agreement between the model results and the experimental data was achieved. The tunneling effect is ignored if the polymer thickness between two CNTs is larger than the cutoff distance 1.8 nm beyond which the transmission probability is less than \({10}^{-6}\). Between cutoff distance range 1–2 nm, the tunneling resistance can vary several orders of magnitude. The results revealed that average separation distance has an important effect on both the resistance and percolation threshold of the composite. Because average separation distance acts as an insulation gap, reducing the homogeneity of the electrically conductive paths and decreasing sufficient conductors that reserve for the formation of the new conductive network [57]. It can be inferred from Fig. 9 that the volume fraction of GNP demanded for creating conductive networks increased as average separation distance increased. This can be attributed to the fact that the increased separation distance removed the electrically conductive paths consisting of GNP due to a gap which is large enough to obliterate charge transformation [58]. With \(\gamma \) as the critical parameter that determines the percolation threshold, a smaller value of average separation distance reduces the percolation threshold. This indicates that the tunneling relies on the overlap of the filler to form conductive network and decreasing average separation distance allows more GNPs to be connected in order to form conductive networks.

Fig. 9
figure 9

Electrical resistivity of hybrid nanocomposite versus GNP volume fraction as a function of GNP’s average separation distance and CNT volume fraction

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Figure 10 presents the electrical conductivity of the hybrid nanocomposite as functions of GNP intrinsic electrical conductivity and CNT volume fraction in comparison with experimental data. Results shown in Fig. 10 indicate a good agreement between present model predictions and experimental data. However, there are also some discrepancies between the predictions and experimental data. These discrepancies are related to the uniform dispersion state assumed in the model and the lack of geometrical-structural knowledge for different experimental measures such as the length and diameter of the CNTs and separation distance and thickness of the GNPs. Model predicted lower electrical resistivity for higher intrinsic electrical conductivity of GNP. This happening can be attributed to the fact that GNPs with higher intrinsic electrical conductivity have lower volume resistivity. This is related to formation of more conductive paths due to more conductive GNPs’ bridges. High electrical conductivity of GNP increased the synergistic effect in terms of electrical conductivity, since the densities of electrical contacts are higher if the GNPs are more conductive causing more penetration between fillers [56, 59]. The change in electrical conductivity with GNP’s intrinsic electrical conductivity is also directly related to the change in the density of electron charge that can lead to a change in the ballistic nature and the number of transferred charges [57].

Fig. 10
figure 10

Electrical resistivity of hybrid nanocomposite versus GNP volume fraction as a function of GNP’s intrinsic electrical conductivity and CNT volume fraction

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5 Conclusion

An analytical model was developed to determine the resistivity of hybrid CNT/GNP polymer composites. This model could successfully capture the quantitative behavior of hybrid nanocomposite filled with GNP and CNT. The phenomenon of percolation threshold was also investigated. Resistivity evaluation exhibited that added nanofillers such as CNT and GNP could provide very low percolation threshold for hybrid nanocomposites. When the GNP volume fraction was lower, there were lower numbers of the overlaps between the fillers and the electrical conductive network could not be effectively formed, leading to high resistivity of the hybrid nanocomposites. The hybrid nanocomposite presented conductive percolation behavior with increasing of the nanofiller aspect ratio due to enhancement of overlaps between the GNPs owing to the relatively larger aspect ratio. The percolation threshold value was higher when the matrix barrier height was higher as expected by the decreasing tunneling probability. With increasing the GNP intrinsic electrical conductivity, the percolation threshold of the hybrid nanocomposite decreased due to the increase in the density of electron charge that can lead to a change in the ballistic nature and the number of transferred electron charges. To check the efficiency of the present model, the predicted results were compared and validated with experimental data successfully.