
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

72698
10.1038/s41598-024-72698-0
Article
A novel approach to predict the electrical conductivity of nanocomposites by a weak interphase around graphene network
Zare Yasser y.zare@aut.ac.ir

1
Munir Muhammad Tajammal 2
Rhee Kyong Yop rheeky@khu.ac.kr

3
1 https://ror.org/02f71a260 grid.510490.9 Biomaterials and Tissue Engineering Research Group, Department of Interdisciplinary Technologies, Breast Cancer Research Center, Motamed Cancer Institute, ACECR, Tehran, Iran
2 https://ror.org/02gqgne03 grid.472279.d 0000 0004 0418 1945 College of Engineering and Technology, American University of the Middle East, Egaila, 54200 Kuwait
3 https://ror.org/01zqcg218 grid.289247.2 0000 0001 2171 7818 Department of Mechanical Engineering (BK21 four), College of Engineering, Kyung Hee University, Yongin, Republic of Korea
14 9 2024
14 9 2024
2024
14 2151426 5 2024
10 9 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
Herein, we offer a model for estimating the tunneling conductivity of polymer-graphene nanocomposites based on interfacial properties, the proportion of networked graphene, and the wettability value between the polymer medium and the filler. The interfacial properties are influenced by the minimum diameter of the nanosheets (Dc), whose conductivity can be transferred to the medium via interfacial conduction (τ). These parameters impact the actual aspect ratio and the volume proportion of the filler, which, in turn, control the onset of percolation and the proportion of nanosheets in the network. We apply all these parameters to develop a novel model for estimating the conductivity of graphene systems. The predictions obtained from this model across different parameter ranges are discussed. Additionally, experimental measurements are employed to evaluate the proposed equations. High filler conductivity enhances the nanocomposite’s conductivity by a strong interfacial conduction. However, the conductivity cannot be transferred to the polymer medium under condition of weak interfacial conduction. Furthermore, a robust interphase and a small Dc contribute to increased conductivity. Ultimately, the developed equations accurately predict the onset of percolation and conductivity, validated by real experimental data.

Keywords

Conductivity
Polymer nanocomposites
Graphene
Interfacial properties
Tunneling effect
Subject terms

Engineering
Materials science
issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

The advanced applications of polymer nanocomposites in electronic devices and sensors require electrical conductivity1–7. However, polymers are typically insulators, and nanofillers can be incorporated into polymer mediums to render these polymers conductive. Among nanomaterials, graphene is distinguished by a range of excellent properties, including mechanical rigidity, physical dimensions, and electrical conductivity8–13. Consequently, polymer-graphene nanocomposites offer numerous advantages, such as outstanding electrical properties. Initial researches on graphene products have concentrated on preparation techniques and low percolation threshold14–16. A minimal percolation threshold is attainable due to the large aspect ratio and substantial surface area of graphene nanosheets17. However, certain undesirable phenomena, such as folding and problematic networking, diminish the effectiveness of graphene in nanocomposites18.

Poor interfacial properties generally arise between the polymer matrix and nanoparticles in polymer nanocomposites due to the inadequate compatibility between these components19,20. These suboptimal interfacial properties compromise the effectiveness of the nanoparticles within the nanocomposites, as complete interfacial adhesion is essential for the transfer of specific beneficial properties from the nanoparticles to the polymer matrix21,22. Additionally, the interfacial properties can influence the conductivity of the composites, given that the conductivity of the nanoparticles needs to be conveyed to the insulated polymer medium. While a robust interface can facilitate the transmission of filler conductivity to the matrix, a weak interface fails to do so. Nevertheless, previous investigations into the conductivity of graphene-enhanced products have largely overlooked this critical aspect.

The interphase typically dictates the properties of polymer nanocomposites due to the extensive surface areas of the nanofillers23–28. It can establish continuous networks, thereby expediting the onset of percolation and cultivating extensive conductive networks29–31. Thus, interphase regions significantly impact the conductivity of such nanocomposites. Additionally, electron tunneling between adjacent nanoparticles affects the conductivity of polymer nanocomposites32–36. Electrons are also capable of traversing the contact zones between nanosheets. As a result, both the onset of percolation and the conductivity of the product are governed by the tunneling effect, which does not necessitate physical connections for electron transfer between nanosheets within the networks. Nevertheless, many researchers have overlooked the aspects of incomplete interfacial adhesion, the tunneling effect, and interphase in their studies on polymer-graphene nanocomposites, despite these factors being primary determinants of the percolation threshold and conductivity. Moreover, there has been a lack of a straightforward model capable of predicting the effects of interface properties, interphase, and the tunneling effect on conductivity.

In this study, interfacial properties such as interfacial conduction (τ) are assumed based on the minimum diameter of graphene nanosheets that can completely transfer the nanofiller’s conductivity to the medium (Dc). These parameters modify the aspect ratio and filler volume fraction of the nanocomposites. Furthermore, the onset of percolation and the filler’s share in the networks are described using the mentioned factors. A model is then developed to estimate the conductivity of polymer-graphene nanocomposites, accounting for imperfect interfacial bonding, interphase, tunneling effect, and filler wettability in relation to the polymer medium. The model’s validity is assessed by comparing its predictions across a range of parameters with experimental outcomes. In fact, the impact of these parameters on conductivity is utilized to confirm the accuracy of the proposed model.

Development of model

A weak interface cannot withstand significant shear stress due to loading, leading to yielding or debonding. As a result, interfacial shear stress causes poor generation of normal stress in the particles, necessitating a substantial space for the normal stress to achieve filler strength [37]. In this scenario, a large area of filler is only partially engaged because of the weak interfacial adhesion, which diminishes the stiffening efficiency of the nanoparticles.

The same approach is used to determine the role of a weak interface on conductivity. When \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 \leqslant x \leqslant {D_c}$$\end{document} (i.e., in the first case), the entire nanosheet does not achieve the filler conduction (σf); however, in the second case (i.e.,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${D_c} \leqslant x \leqslant D/2$$\end{document}), the normal conductivity (σ) can touch σf, and the extreme filler conductivity is transferred to the insulated medium.

Dc represents the minimal diameter of the sheets necessary to transfer the complete conductivity of graphene to the polymer host, as detailed below:1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${D_c}=\frac{{{\sigma _{{f_{}}}}t}}{{2\tau }}=\frac{{{\sigma _{{f_{}}}}{\alpha _{}}D}}{{2\tau }}$$\end{document}

where D and t represent the width and thickness of the nanosheets, respectively, α is the inverse aspect ratio defined as α = t/D, and the interfacial conduction is denoted by τ.

The average normal conductivity (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline {\sigma }$$\end{document}) is equal to σf when a perfect interface is formed, but \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline {\sigma }$$\end{document} is less than σf under incomplete interfacial adhesion, which lessens the actual nanosheet diameter (Deff) as follows:2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline {\sigma } D={\sigma _f}{D_{eff}}$$\end{document}

According to this equation, weak interfacial bonding decreases the actual converse aspect ratio (αeff) and volume portion (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi _{eff}}$$\end{document}) of the graphene.

Moreover, αeff and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi _{eff}}$$\end{document} are expressed as37:3 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha _{eff}}=\alpha \left( {\frac{{8D_{c}^{2}}}{{{D^2}}}+1} \right)$$\end{document}

4 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi _{eff}}={\varphi _f}\left[ {\frac{1}{2}+\left( {\frac{{D - 2{D_c}}}{{{D^2}}}} \right)(D - {D_c})} \right]$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi _f}$$\end{document} is the filler volume portion.

Substituting Dc from Eq. 1 into the above equations results in5 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha _{eff}}=\alpha \left( {\frac{{2\sigma _{f}^{2}{\alpha ^2}}}{{{\tau ^2}}}+1} \right)$$\end{document}

6 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi _{eff}}={\varphi _f}\left[ {\frac{1}{2}+\left( {1 - \frac{{{\sigma _{{f_{}}}}{\alpha _{}}}}{\tau }} \right)\left( {1 - \frac{{{\sigma _{{f_{}}}}{\alpha _{}}}}{{2\tau }}} \right)} \right]$$\end{document}

The percolation beginning of the graphite-filled samples is expressed as follows38:7 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi _p}=\frac{{27\pi {D^2}t}}{{4{{(D+d)}^3}}}$$\end{document}

Here, d is the tunneling size.

Furthermore, D > > d abridges this equation to:8 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi _p}=\frac{{27\pi t}}{{4D}}$$\end{document}

As mentioned, the interphase modifies the percolation threshold with a minor filler addition. Furthermore, the tunneling distance can facilitate network formation by closely situated nanosheets. Thus, both tunnels and interphase can reduce the onset of percolation as follows:9 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi _p}=\frac{{27\pi {t^2}}}{{4tD+2(D{t_i}+Dd)}}$$\end{document}

where ti is the interphase deepness.

The converse aspect ratio (α = t/D) in Eq. 9 suggests the following:10 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi _p}=\frac{{27\pi t\alpha }}{{4t+2{t_i}+2d}}$$\end{document}

Substituting αeff from Eq. 5 into the above equation yields:11 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi _p}=\frac{{27\pi t\alpha \left( {\frac{{2\sigma _{f}^{2}{\alpha ^2}}}{{{\tau ^2}}}+1} \right)}}{{4t+2{t_i}+2d}}$$\end{document}

Only conductive networks affect conductivity39; therefore, the presence of networked sheets in the nanocomposites is more crucial than the overall filler volume fraction.

The networked fractions post-percolation can be evaluated as40:12 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f=\frac{{\varphi _{f}^{{1/3}} - \varphi _{p}^{{1/3}}}}{{1 - \varphi _{p}^{{1/3}}}}$$\end{document}

Here, f can be enhanced by the incomplete interface, depth of interphase, and tunneling size, assuming the filler fraction and percolation threshold from Eqs. 6 and 11 are considered in Eq. 12 as follows:13 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f=\frac{{\varphi _{{eff}}^{{1/3}} - \varphi _{p}^{{1/3}}}}{{1 - \varphi _{p}^{{1/3}}}}$$\end{document}

The net volume share is calculated as:14 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _{N}^{{}}=f\varphi _{f}^{{}}$$\end{document}

The actual values of f and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _{f}^{{}}$$\end{document} from Eqs. 13 and 6 can be used to modify the above equation as follows:15 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _{N}^{{}}=f\varphi _{{eff}}^{{}}$$\end{document}

The aforementioned parameters can be used in a model to estimate the conductivity of graphene-filled products.

Taherian41 proposed an equation for calculating the conductivity of nanocomposites as follows:16 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ={\sigma _m}+\frac{{A{\sigma _f}}}{{\alpha +B\alpha \exp \left( { - \frac{{roundness}}{{\cos \theta }}} \right)}}$$\end{document}

where σm is the conduction of the polymer medium, which can be ignored, A and B are constant factors, and cos(θ) is the wettability between the polymer and the nanofiller.

The roundness is determined to be in the range of 0–141 as follows:17 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$roundness=\frac{{1000 - \frac{1}{\alpha }}}{{1000}}$$\end{document}

Moreover, the surface energies of the constituents determine wettability and filler dispersion42.

The wettability was suggested by cos (θ) (θ is wetting angle)41 as:18 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cos \theta =\frac{{{\gamma _f} - {\gamma _{pf}}}}{{{\gamma _p}}}$$\end{document}

where γf, γp, and γpf represent the surface energies of the particle, polymer, and polymer-particle interface, in that order.

Additionally, γpf is expressed as follows41:19 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\gamma _{pf}}={\gamma _p}+{\gamma _f} - 2{({\gamma _f}{\gamma _p})^{1/2}}$$\end{document}

However, the Taherian model overlooks certain critical parameters that influence the conductivity of polymer-graphene nanocomposites. Numerous studies have demonstrated that the conductivity of a nanocomposite correlates with its filler content15,43. This correlation exists because the conductivity of the filler, which exceeds that of the polymer, enhances the overall conductivity of the nanocomposite. Additionally, the width and thickness of graphene nanosheets range from a few microns to 1–5 nm, respectively, resulting in an aspect ratio exceeding 1000. By modifying Eq. 17, it’s possible to achieve a maximum aspect ratio of 5000 as follows:20 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$roundness=\frac{{5000 - \frac{1}{\alpha }}}{{5000}}$$\end{document}

Additionally, the tunneling effect, which diminishes as tunnel size increases, significantly impacts conductivity40. Moreover, d is inversely related to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi _f}$$\end{document} as follows44:21 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d \propto \varphi _{f}^{{ - 1/3}}$$\end{document}

Conductivity is nonlinearly related to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi _f}$$\end{document} owing to the percolation of the filler in the samples.

By applying these equations, the Taherian model (Eq. 16) can be refined and simplified to estimate the conductivity of graphene-based products as follows:22 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma =\frac{{\varphi _{N}^{2}{\sigma _f}}}{{\alpha {{\left( {\frac{d}{z}} \right)}^6}\exp \left[ { - \frac{{roundness}}{{\cos (\theta )}}} \right]}}=\frac{{{{(f{\varphi _{eff}})}^2}{\sigma _f}}}{{\alpha {{\left( {\frac{d}{z}} \right)}^6}\exp \left[ { - \frac{{\frac{{5000 - \frac{1}{\alpha }}}{{5000}}}}{{\frac{{{\gamma _f} - {\gamma _{pf}}}}{{{\gamma _p}}}}}} \right]}}$$\end{document}

where z is the tunneling factor. The effect of an incomplete interface on the conductivity of the product can be determined by substituting f, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _{{eff}}^{{}}$$\end{document}, and α from Eqs. 13, 6, and 5 into the suggested model. This model is capable of delineating the impacts of various factors, including graphene characteristics, network structures, interfacial properties, wettability, interphase depth, and tunnel sizes, on the product’s conductivity.

Results and discussion

Analysis of percolation onset

Equation 11 is used to analyze the roles of various factors on the percolation onset. Contour plots show the effects of two factors on the percolation onset, while other parameters are constant assumed as t = 1 nm, d = 5 nm, D = 2 μm, ti = 5 nm, τ = 400 S/m and σf = 105 S/m.

Figure 1a reveals the effects of grapehen size on the percolation onset. The lowest percolation onset is observed by the thinnest and biggest nanosheets, while the thickest and shortest nanosheets maximize the percolation onset. This trend is reasonable, since thin and big nanosheets enhance the aspect ratio reducing the percolation onset. In fact, thin and long nanosheets provide more contacts facilitating the creation of network. However, thick and short nanosheets have small number of contacts increasing the percolation onset. Consequently, Eq. 11 properly links the percolation onset to graphene dimensions.

Figure 1b shows the percolation onset as a function of Dc and ti. The maximum percolation onset is shown at the highest Dc and lowest ti, but percolation onset reduces by shorter Dc and higher ti. This means that small Dc and thick interphase are essential to minimize the percolation onset. A low Dc indicating the high interfacial properties declines the inverse aspect ratio, which positively changes the percolation onset. In fact, a lower Dc reveals the higher interfacial features and bigger aspect ratio of nanosheets reducing the percolation onset. Moreover, a thick interphase reduces the space among the nanosheets, which grows the number of contacts decreasing the percolation onset. In contrast, a thinner interphase cannot change the space among the nanosheets increasing the percolation onset. Accordingly, Eq. 11 properly expresses the percolation onset by Dc and ti.

Fig. 1 Dependence of percolation onset (Eq. 11) on (a) grapehen size, (b) Dc and ti and (c) tunneling size and interfacial conduction.

Figure 1c correlates the percolation onset to tunneling size and interfacial conduction. The minimum level of percolation onset is observed at small tunneling size and high interfacial conduction, while the lowest extents of interfacial conduction and tunneling size maximize the percolation onset. A bigger tunnel reduces the percolation onset, because the network is created by a low number of nanosheets. Conversely, shorter tunnels increase the percolation onset, since a higher number of nanosheets are required to form the network. Additionally, the interfacial conduction inversely manages the percolation onset, because a higher interfacial conduction reveals the bigger and stronger interphase reducing the percolation onset. In contrast, a poorer interfacial conduction discloses the poorer interphase, which cannot fall the percolation onset. Hence, Eq. 11 correctly handles the impacts of both tunneling size and interfacial conduction on the percolation onset.

Influences of various factors on conductivity

The proposed model can evaluate the influence of various factors on the conductivity of the product. Two- and three-dimensional representations show the effects of two specific factors on conductivity, while considering average levels of other parameters (t = 1 nm, d = 5 nm, D = 2 μm, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _{f}^{{}}$$\end{document}= 0.01, ti = 5 nm, τ = 400 S/m, γp = 40 mN/m, σf = 105 S/m, γf = 50 mN/m, and z = 0.5 nm).

Figure 2 illustrates the effects of τ and σf on the conductivity of the product using the refined equation. The highest conductivity, 0.12 S/m, is achieved with τ = 800 S/m and σf = 2.5 × 105 S/m, while no conductivity is observed when τ = 200 S/m and σf = 2.5 × 105 S/m. Thus, conductivity is directly related to τ, whereas σf plays a different role. Actually, the proposed model clearly explains the influence of τ (interfacial conduction) and σf (filler conduction) on the conductivity of nanocomposites.

A large τ leads to increased conductivity due to enhanced interfacial conduction. Specifically, a larger τ indicates the stronger interfacial bonds that are capable of transferring the filler conductivity to the insulated medium. A higher τ results in lower and higher values of the actual inverse aspect ratio and filler concentration, respectively, thus elevate the product conductivity. Conversely, a small τ fails to augment the conductivity as a weak interface is incapable of conveying the graphene conductivity to the polymer matrix. Accordingly, τ as interfacial conduction directly manages the conductivity of graphene composites.

Fig. 2 Differences in the conductivity obtained using by Eq. 22 for different τ and σf values: (a) three- and (b) two-dimensional designs.

A low σf (graphene conductivity) cannot produce high conductivity but influences the conductivity differently across varying levels of τ. High filler conductivity cannot be transferred to the polymer medium under the weak interfacial conditions; however, with a high τ, this conductivity can be transferred, thereby enhancing the conductivity of the nanocomposite. Moreover, an increased σf raises Dc (as per Eq. 1), which in turn lowers the actual inverse aspect ratio and filler volume. Given that polymer matrices are typically insulative, a significant increase in σf substantially impacts the product’s conductivity. Therefore, augmentations in both σf and τ lead to enhanced conductivity, as strong interfacial bonding is crucial for the transfer of graphene conductivity to the polymer medium. Conversely, a combination of low τ and high σf does not influence the product conductivity, because the filler conductivity cannot be effectively transferred to the insulated medium. These observations validate that the proposed model accurately assesses the impact of σf on the conductivity of graphene nanocomposites.

Figure 3 illustrates the effects of thickness (t) and width (D) on the calculated conductivity. High thickness and low width result in negligible conductivity, whereas the smallest thickness and largest width achieve the highest conductivity of the product. Specifically, the maximum conductivity, at 0.14 S/m, occurs at t = 1 nm and D = 4 μm. Hence, thin and wide nanosheets lead to nanocomposites with high conductivity, while thick and narrow nanosheets do not. These findings indicate that the dimensions of graphene nanosheets significantly impact the conductivity of the products.

Thinner and wider nanosheets result in a lower inverse aspect ratio, thereby increasing the interfacial area. A low α leads to a decrease in Dc, enhances the actual inverse aspect ratio, actual filler volume, onset of percolation, and the proportion of networked sheets. Specifically, a smaller α is associated with improved interfacial properties and larger networks, thus boosting the conductivity of the product. In contrast, higher inverse aspect ratio causes the poorer interfacial properties and smaller network weakening the conductivity. These findings confirm that the model can accurately assess the impacts of t and D on the conductivity of nanocomposites. Previous research has explored the influence of the aspect ratio of graphene on the onset of percolation15,43, yet the effect of aspect ratio on conductivity has not been extensively documented.

Fig. 3 Forecasted conductivity for different values of t and D: (a) three- and (b) two-dimensional patterns.

Figure 4 clarifies the association of conductivity to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _{f}^{{}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _{p}^{{}}$$\end{document} obtained using the proposed model. The maximum conductivity of approximately 2.2 S/m is achieved when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _{f}^{{}}$$\end{document} = 0.03 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _{p}^{{}}$$\end{document} = 0.001, whereas no conductivity is achieved when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _{f}^{{}}$$\end{document} < 0.013 or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _{f}^{{}}$$\end{document} < 0.017 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _{p}^{{}}$$\end{document} > 0.005. Consequently, the conductivity of the nanocomposite increases with an increase in filler volume and a decrease in the percolation threshold.

A significant filler volume portion markedly enhances the product conductivity, as the conductivity of the graphene surpasses that of the polymer matrices. Furthermore, an increased filler volume can raise the f, leading to the formation of bulky and dense network within the composite. Since the big network of conductive nanosheets improves the conductivity, there is a direct relation between the nanocomposite conductivity and graphene volume fraction. On the other hand, a low amount of graphene cannot enhance the product conductivity, because it produces the small network limiting the electron transferring. Numerous experimental and theoretical studies have established a direct relationship between conductivity and filler concentration16,45, although a large filler volume does not alter the nanocomposite’s conductivity due to the constant dimensions of the networks41. Instead, a reduced percolation threshold enhances f (as per Eq. 13), resulting in larger networks and, consequently, increased product conductivity. In conclusion, the novel model can accurately determine the powers of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _{f}^{{}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _{p}^{{}}$$\end{document} on the nanocomposite conductivity.

Fig. 4 (a) Three- and (b) two-dimensional plans of the conductivity based on\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _{f}^{{}}$$\end{document}and\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _{p}^{{}}$$\end{document}.

Figure 5 illustrates the ti and Dc effects on the conductivity. The maximum conductivity of 0.08 S/m is shown at ti = 10 nm and Dc = 100 nm, but an insulative product is observed at Dc > 500 nm. A large ti and small Dc result in high conductivity; however, a large Dc cannot increase conductivity. Therefore, a thicker interphase and smaller Dc produce the more conductivity in the nanocomposite, whereas only a high Dc substantially decreases the conductivity.

Fig. 5 Conductivity under different ti and Dc values: (a) three- and (b) two-dimensional designs.

A substantial interphase creates a wide interphase zone within the products, leading to a decrease in percolation threshold and an increase in the network fraction. More precisely, the interphase contributes to the network formation, meaning that a significant interphase around the nanosheets fosters the development of large conductive networks. Instead, a thin interphase cannot enlarge the network worsening the conductivity. So, the depth of the interphase directly influences the product conductivity.

A small Dc enhances the interfacial properties, facilitating the transfer of filler conductivity to the host. Consequently, a small Dc elevates the inverse aspect ratio and filler volume, resulting in increased conductivity. On the other hand, a large Dc diminishes the actual inverse aspect ratio and filler volume, thus lowering the product conductivity. Specifically, a large Dc indicates poor interfacial properties, preventing the transfer of filler conductivity to the polymer medium. A substantial Dc may also impair the effectiveness of filler conductivity in enhancing the product’s conductivity due to suboptimal interfacial properties. These findings affirm that the new model effectively evaluates the impact of Dc on conductivity.

Figure 6 displays the powers of γp and γf on the conductivity attained using the proposed model. A large γp and small γf yield good conductivity, whereas the conductivity reduces owing to a small γp and large γf. Therefore, γp and γf directly and conversely affect conductivity, in that order. Consequently, a polymer medium possessing high surface energy, combined with graphene of low surface energy, exhibits high conductivity.

Fig. 6 Influences of γp and γf on the conductivity: (a) three- and (b) two-dimensional designs.

A high γp and a low γf reduce the wettability between the polymer medium and the nanoparticles, as indicated by Eq. 18. Under these condition, graphene tends to agglomerate because poor wettability leads to an uneven distribution of nanoparticles within the polymer medium41. The graphene agglomerates promote the contacts among the nanoparticles. Actually, a reduced wettability results in the larger network, thereby enhancing the product conductivity. In contrast, more wettability of nanosheets by polymer chains enhances the extent of dispersed nanosheets weakening the contact number. As a result, a low γp and a high γf decline the network size and nanocomposite conductivity. Similar observations were documented in a previous study41. These findings suggest that the roles of γp and γf in affecting conductivity are accurately captured by the proposed model, even though these factors only modestly tune the conductivity.

Figure 7 presents the conductivity at various tunneling distances (d) and tunneling parameters (z). The peak conductivity of 200 S/m is reached with d = 2 nm and z = 1.6 nm, whereas conductivity significantly drops when d exceeds 4 nm or z falls below 1 nm. Thus, having short tunnels and a substantial tunneling parameter is essential for achieving high conductivity. According to Eq. 11, a small tunneling size negatively impacts the onset of percolation in such nanocomposites. A higher percolation threshold is noted with smaller tunneling sizes because widely spaced nanosheets are unable to establish conductive networks. In effect, larger tunnels are ineffective at facilitating electron transport.

The maximum tunnel size in nanocomposites is limited to 10 nm46; hence, a tunneling distance exceeding 10 nm fails to produce the tunneling effect. Reducing the tunneling size can enhance the conductivity, since narrow tunnels have poor tunneling resistance providing more electron conduction. Additionally, the parameter z has a direct impact on the product’s conductivity, representing the characteristics of the tunneling regions within nanocomposites. A smaller z reflects less favorable tunneling conditions, whereas a larger z suggests more optimal conditions for conductivity. Electron transport is influenced by tunnel characteristics such as contact area, tunneling size, and contact resistance. It should be noted that even for larger z values, the tunneling effects are not observed within tunnel sizes below 10 nm weakening the nanocomposite conductivity. Therefore, the refined model is capable of assessing the effects of d and z on conductivity.

Fig. 7 Conductivity under different d and z values (Eq. 22): (a) three- and (b) two-dimensional patterns.

Evaluation of the equations based on experimental results

Certain examples from the literature validate the proposed equations using preliminary data. Six reported samples, along with their properties, are summarized in Table 1. It is worth noting that reduced graphene oxide (rGO) exhibits randomness in size and shape. Additionally, factors such as the degree of reduction during the reduction process and crystallinity significantly impact the graphene conductivity. Hence, it is essential to obtain the exact values of graphene size and conductivity calculating the nanocomposite conductivity by the proposed model. Actually, the average values of graphene dimensions and conductivity are considered in the proposed model to estimate the product conductivity. By comparing the experimental onset of percolation with Eq. 11, a good agreement is shown and the average values for ti, d, τ, and Dc are derived, as detailed in Table 1. The polyimide (PI)-graphene sample demonstrates the thickest interphase, longest tunnels, and highest interfacial conduction, in contrast, the poly(vinylidene fluoride) (PVDF) sample shows the lowest interfacial conduction and the greatest Dc among the samples presented. The calculations mentioned above substantiate the significance of tunnels and interphase in the composite structure. Moreover, the characteristics of interfacial properties at the onset of percolation clearly play a pivotal role in influencing conductivity.

Table 1 Some samples reported in previous studies and their properties, as well as the calculations of various parameters using the developed equations.

No.	Samples [Ref.]	t (nm)	D (µm)	γp (mN/m)	\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi _p}$$\end{document}	ti (nm)	d (nm)	τ (S/m)	Dc (nm)	z (nm)	
1	PVDF/graphene16	1	2	31	0.0030	3	3	200	250.0	0.19	
2	Epoxy/graphene47	2	2	42	0.0050	6	7	680	128.2	0.38	
3	PI/graphene48	3	5	40	0.0015	30	12	700	214.0	3.50	
4	ABS/graphene43	1	4	40	0.0013	3	3	300	167.0	0.38	
5	PET/graphene17	2	2	42	0.0050	5	9	600	166.7	2.30	
6	PS/graphene49	1	2	41	0.0010	9	10	500	100.0	2.98	

These calculated parameters are utilized to estimate the conductivity using the proposed model, with the average levels of γf = 50 mN/m and σf = 105 S/m. Figure 8 illustrates both the experimental and predicted data for all the nanocomposite samples, demonstrating the model’s accuracy in forecasting conductivity trends. The congruence between the experimental and predicted results verifies the accuracy of the model. Thus, this novel model is validated for predicting conductivity in samples, factoring in incomplete interfacial properties, tunneling size, depth of interphase, and wettability values of the fillers in relation to the polymer medium.

Fig. 8 Experimental and predicted conductivity by Eq. 22 for graphene nanocomposites containing (a) PVDF16, (b) epoxy47, (c) PI48, (d) ABS43, (e) PET/graphene17 and (f) PS49.

The z values are detailed in Table 1. The highest and lowest z values correspond to the PI and PVDF graphene nanocomposites, respectively. In summary, the proposed equation for determining the onset of percolation (Eq. 11) and the model for estimating conductivity yield precise results that align closely with experimental data.

Conclusions

A novel model has been developed to assess the conductivity of graphene-based systems, taking into account factors such as incomplete interfacial properties, interphase region, the tunneling effect, the extent of networked nanosheets, and wettability values. The model primarily considers Dc and τ as the interfacial parameters influencing the aspect ratio and filler volume. The thinnest and biggest nanosheets, the lowest Dc, the thickest interphase, the smallest tunnels and the highest interfacial conduction produce the minimum percolation onset. It was discovered that the conductivity of nanocomposites is intrinsically linked to interfacial conduction. Enhanced filler conductivity leads to higher conductivity of the nanocomposite due to robust interfacial conduction; however, this transfer of filler conductivity to the polymer medium is impeded by weak interfacial conduction. Thin and wide nanosheets are conducive to high conductivity, attributed to their expansive interface and substantial aspect ratio. Furthermore, a compact interphase and a minimal Dc contribute to increased conductivity, whereas a large Dc, indicative of inferior interfacial properties, adversely affects the conductivity. The presence of low-surface-energy graphene in a polymer medium with high surface energy is beneficial for conductivity, as it optimizes filler distribution and compacts the filler networks. The necessity for short tunnels and significant tunneling parameters for high conductivity is also underscored. Experimental validations confirm the model’s efficacy in accurately predicting conductivity, underscoring its potential applicability in designing graphene-based conductive materials.

Author contributions

Y.Z. and K.Y.R. prepared the paper. M.T.M. revised the paper.

Data availability

The data that support the findings of this study are available on request from corresponding author.

Declarations

Competing interests

The authors declare no competing interests.

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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References

1. Ma Q Xing D Hao B Ma P-C Interfacial engineering strategy to improve the piezoresistive performance of polymer nanocomposites via chemical bonding of carbon nanotubes on graphene Compos. Part A: Appl. Sci. Manufac. 2023 174 107730 10.1016/j.compositesa.2023.107730
Ma, Q., Xing, D., Hao, B. & Ma, P-C. Interfacial engineering strategy to improve the piezoresistive performance of polymer nanocomposites via chemical bonding of carbon nanotubes on graphene. Compos. Part A: Appl. Sci. Manufac. 174, 107730 (2023).10.1016/j.compositesa.2023.107730
2. Naghib SM Behzad F Rahmanian M Zare Y Rhee KY A highly sensitive biosensor based on methacrylated graphene oxide-grafted polyaniline for ascorbic acid determination Nanatechnol. Reviews 2020 9 1 760 767 10.1515/ntrev-2020-0061
Naghib, S. M., Behzad, F., Rahmanian, M., Zare, Y. & Rhee, K. Y. A highly sensitive biosensor based on methacrylated graphene oxide-grafted polyaniline for ascorbic acid determination. Nanatechnol. Rev. 9 (1), 760–767 (2020).10.1515/ntrev-2020-0061
3. Mohammadpour-Haratbar, A., Zare, Y. & Rhee, K. Y. Electrochemical biosensors based on polymer nanocomposites for detecting breast cancer: recent progress and future prospects. Adv. Colloid Interface Sci. 102795. (2022).
4. Haghgoo M Ansari R Hassanzadeh-Aghdam MK Jang S-H Nankali M Analytical modeling of synergistic carbon nanotube/carbon black effects on the sensitivity of nanocomposite strain sensors Compos. Part A: Appl. Sci. Manufac. 2023 173 107711 10.1016/j.compositesa.2023.107711
Haghgoo, M., Ansari, R., Hassanzadeh-Aghdam, M. K., Jang, S-H. & Nankali, M. Analytical modeling of synergistic carbon nanotube/carbon black effects on the sensitivity of nanocomposite strain sensors. Compos. Part A: Appl. Sci. Manufac. 173, 107711 (2023).10.1016/j.compositesa.2023.107711
5. Haghgoo M Ansari R Hassanzadeh-Aghdam MK Jang S-H Nankali M Simulation of the role of agglomerations in the tunneling conductivity of polymer/carbon nanotube piezoresistive strain sensors Compos. Sci. Technol. 2023 243 110242 10.1016/j.compscitech.2023.110242
Haghgoo, M., Ansari, R., Hassanzadeh-Aghdam, M. K., Jang, S-H. & Nankali, M. Simulation of the role of agglomerations in the tunneling conductivity of polymer/carbon nanotube piezoresistive strain sensors. Compos. Sci. Technol. 243, 110242 (2023).10.1016/j.compscitech.2023.110242
6. Zare Y Modeling of tensile modulus in polymer/carbon nanotubes (CNT) nanocomposites Synth. Met. 2015 202 68 72 10.1016/j.synthmet.2015.02.002
Zare, Y. Modeling of tensile modulus in polymer/carbon nanotubes (CNT) nanocomposites. Synth. Met. 202, 68–72 (2015).10.1016/j.synthmet.2015.02.002
7. Arjmandi SK Khademzadeh Yeganeh J Zare Y Rhee KY Development of Kovacs model for electrical conductivity of carbon nanofiber–polymer systems Sci. Rep. 2023 13 1 7 10.1038/s41598-022-26139-5 36593230
Arjmandi, S. K., Khademzadeh Yeganeh, J., Zare, Y. & Rhee, K. Y. Development of Kovacs model for electrical conductivity of carbon nanofiber–polymer systems. Sci. Rep. 13 (1), 7 (2023).36593230 10.1038/s41598-022-26139-5
8. Ghanbari S Ahour F Keshipour S An optical and electrochemical sensor based on l-arginine functionalized reduced graphene oxide Sci. Rep. 2022 12 1 1 14 10.1038/s41598-022-23949-5 34992227
Ghanbari, S., Ahour, F. & Keshipour, S. An optical and electrochemical sensor based on l-arginine functionalized reduced graphene oxide. Sci. Rep. 12 (1), 1–14 (2022).34992227 10.1038/s41598-022-23949-5
9. Ji J-H Lee G Koh J-H Synthesis of a nitrogen doped reduced graphene oxide based ceramic polymer composite nanofiber film for wearable device applications Sci. Rep. 2022 12 1 15583 10.1038/s41598-022-19234-0 36114221
Ji, J-H., Lee, G. & Koh, J-H. Synthesis of a nitrogen doped reduced graphene oxide based ceramic polymer composite nanofiber film for wearable device applications. Sci. Rep. 12 (1), 15583 (2022).36114221 10.1038/s41598-022-19234-0
10. Zare Y Rhee KY Effect of contact resistance on the electrical conductivity of polymer graphene nanocomposites to optimize the biosensors detecting breast cancer cells Sci. Rep. 2022 12 1 1 10 10.1038/s41598-022-09398-0 34992227
Zare, Y. & Rhee, K. Y. Effect of contact resistance on the electrical conductivity of polymer graphene nanocomposites to optimize the biosensors detecting breast cancer cells. Sci. Rep. 12 (1), 1–10 (2022).34992227 10.1038/s41598-022-09398-0
11. Zare Y Rhee KY Park S-J Progressing of a power model for electrical conductivity of graphene-based composites Sci. Rep. 2023 13 1 1596 10.1038/s41598-023-28232-9 36709238
Zare, Y., Rhee, K. Y. & Park, S-J. Progressing of a power model for electrical conductivity of graphene-based composites. Sci. Rep. 13 (1), 1596 (2023).36709238 10.1038/s41598-023-28232-9
12. Hosseini SR Alavi Nikje MM Synthesis and characterization of novel epoxy-urethane coating and its graphene nanocomposites Polym. Compos. 2023 44 5 2794 2803 10.1002/pc.27280
Hosseini, S. R. & Alavi Nikje, M. M. Synthesis and characterization of novel epoxy-urethane coating and its graphene nanocomposites. Polym. Compos. 44 (5), 2794–2803 (2023).10.1002/pc.27280
13. Mamallan S Narayanan V Demystifying the role of graphene nanoplatelets percentage and sonication duration on the mechanical properties of the glass fabric/graphene nanoplatelets hybrid nano-composite Polym. Compos. 2022 43 11 8170 8180 10.1002/pc.26985
Mamallan, S. & Narayanan, V. Demystifying the role of graphene nanoplatelets percentage and sonication duration on the mechanical properties of the glass fabric/graphene nanoplatelets hybrid nano-composite. Polym. Compos. 43 (11), 8170–8180 (2022).10.1002/pc.26985
14. Vo NH Dao TD Jeong HM Electrically conductive Graphene/Poly (methyl methacrylate) composites with ultra-low percolation threshold by Electrostatic Self-Assembly in Aqueous Medium Macromol. Chem. Phys. 2015 216 7 770 782 10.1002/macp.201400560
Vo, N. H., Dao, T. D. & Jeong, H. M. Electrically conductive Graphene/Poly (methyl methacrylate) composites with ultra-low percolation threshold by Electrostatic Self-Assembly in Aqueous Medium. Macromol. Chem. Phys. 216 (7), 770–782 (2015).10.1002/macp.201400560
15. Tu Z A facile approach for preparation of polystyrene/graphene nanocomposites with ultra-low percolation threshold through an electrostatic assembly process Compos. Sci. Technol. 2016 134 49 56 10.1016/j.compscitech.2016.08.003
Tu, Z. et al. A facile approach for preparation of polystyrene/graphene nanocomposites with ultra-low percolation threshold through an electrostatic assembly process. Compos. Sci. Technol. 134, 49–56 (2016).10.1016/j.compscitech.2016.08.003
16. He L Tjong SC Low percolation threshold of graphene/polymer composites prepared by solvothermal reduction of graphene oxide in the polymer solution Nanoscale Res. Lett. 2013 8 1 132 10.1186/1556-276X-8-132 23522102
He, L. & Tjong, S. C. Low percolation threshold of graphene/polymer composites prepared by solvothermal reduction of graphene oxide in the polymer solution. Nanoscale Res. Lett. 8 (1), 132 (2013).23522102 10.1186/1556-276X-8-132
17. Zhang H-B Electrically conductive polyethylene terephthalate/graphene nanocomposites prepared by melt compounding Polymer 2010 51 5 1191 1196 10.1016/j.polymer.2010.01.027
Zhang, H-B. et al. Electrically conductive polyethylene terephthalate/graphene nanocomposites prepared by melt compounding. Polymer. 51 (5), 1191–1196 (2010).10.1016/j.polymer.2010.01.027
18. Du J Comparison of electrical properties between multi-walled carbon nanotube and graphene nanosheet/high density polyethylene composites with a segregated network structure Carbon 2011 49 4 1094 1100 10.1016/j.carbon.2010.11.013
Du, J. et al. Comparison of electrical properties between multi-walled carbon nanotube and graphene nanosheet/high density polyethylene composites with a segregated network structure. Carbon. 49 (4), 1094–1100 (2011).10.1016/j.carbon.2010.11.013
19. Zare Y Daraei A Vatani M Aghasafari P An analysis of interfacial adhesion in nanocomposites from recycled polymers Comput. Mater. Sci. 2014 81 612 616 10.1016/j.commatsci.2013.08.041
Zare, Y., Daraei, A., Vatani, M. & Aghasafari, P. An analysis of interfacial adhesion in nanocomposites from recycled polymers. Comput. Mater. Sci. 81, 612–616 (2014).10.1016/j.commatsci.2013.08.041
20. Rostami M Mohseni M Ranjbar Z An attempt to quantitatively predict the interfacial adhesion of differently surface treated nanosilicas in a polyurethane coating matrix using tensile strength and DMTA analysis Int. J. Adhes. Adhes. 2012 34 24 31 10.1016/j.ijadhadh.2011.12.005
Rostami, M., Mohseni, M. & Ranjbar, Z. An attempt to quantitatively predict the interfacial adhesion of differently surface treated nanosilicas in a polyurethane coating matrix using tensile strength and DMTA analysis. Int. J. Adhes. Adhes. 34, 24–31 (2012).10.1016/j.ijadhadh.2011.12.005
21. Zare Y The roles of nanoparticles accumulation and interphase properties in properties of polymer particulate nanocomposites by a multi-step methodology Compos. Part A: Appl. Sci. Manufac. 2016 91 127 132 10.1016/j.compositesa.2016.10.003
Zare, Y. The roles of nanoparticles accumulation and interphase properties in properties of polymer particulate nanocomposites by a multi-step methodology. Compos. Part A: Appl. Sci. Manufac. 91, 127–132 (2016).10.1016/j.compositesa.2016.10.003
22. Zare Y Study on interfacial properties in polymer blend ternary nanocomposites: role of nanofiller content Comput. Mater. Sci. 2016 111 334 338 10.1016/j.commatsci.2015.09.053
Zare, Y. Study on interfacial properties in polymer blend ternary nanocomposites: role of nanofiller content. Comput. Mater. Sci. 111, 334–338 (2016).10.1016/j.commatsci.2015.09.053
23. Chipara M Baibarac M Compagnini G Gao J From interface to interphase Surf. Interfaces 2023 42 103435 10.1016/j.surfin.2023.103435
Chipara, M., Baibarac, M., Compagnini, G. & Gao, J. From interface to interphase. Surf. Interfaces. 42, 103435 (2023).10.1016/j.surfin.2023.103435
24. Kamae T Drzal LT Carbon fiber/epoxy composite property enhancement through incorporation of carbon nanotubes at the fiber-matrix interphase–part II: mechanical and electrical properties of carbon nanotube coated carbon fiber composites Compos. Part A: Appl. Sci. Manufac. 2022 160 107023 10.1016/j.compositesa.2022.107023
Kamae, T. & Drzal, L. T. Carbon fiber/epoxy composite property enhancement through incorporation of carbon nanotubes at the fiber-matrix interphase–part II: mechanical and electrical properties of carbon nanotube coated carbon fiber composites. Compos. Part A: Appl. Sci. Manufac. 160, 107023 (2022).10.1016/j.compositesa.2022.107023
25. Zare Y Estimation of material and interfacial/interphase properties in clay/polymer nanocomposites by yield strength data Appl. Clay Sci. 2015 115 61 66 10.1016/j.clay.2015.07.021
Zare, Y. Estimation of material and interfacial/interphase properties in clay/polymer nanocomposites by yield strength data. Appl. Clay Sci. 115, 61–66 (2015).10.1016/j.clay.2015.07.021
26. Zare Y Rhee KY Park S-J Predictions of micromechanics models for interfacial/interphase parameters in polymer/metal nanocomposites Int. J. Adhes. Adhes. 2017 79 111 116 10.1016/j.ijadhadh.2017.09.015
Zare, Y., Rhee, K. Y. & Park, S-J. Predictions of micromechanics models for interfacial/interphase parameters in polymer/metal nanocomposites. Int. J. Adhes. Adhes. 79, 111–116 (2017).10.1016/j.ijadhadh.2017.09.015
27. Zare Y Rhee KY Prediction of tensile modulus in polymer nanocomposites containing carbon nanotubes (CNT) above percolation threshold by modification of conventional model Curr. Appl. Phys. 2017 17 6 873 879 10.1016/j.cap.2017.03.010
Zare, Y. & Rhee, K. Y. Prediction of tensile modulus in polymer nanocomposites containing carbon nanotubes (CNT) above percolation threshold by modification of conventional model. Curr. Appl. Phys. 17 (6), 873–879 (2017).10.1016/j.cap.2017.03.010
28. Zare Y Modeling the strength and thickness of the interphase in polymer nanocomposite reinforced with spherical nanoparticles by a coupling methodology J. Colloid Interface Sci. 2016 465 342 346 10.1016/j.jcis.2015.09.025 26704592
Zare, Y. Modeling the strength and thickness of the interphase in polymer nanocomposite reinforced with spherical nanoparticles by a coupling methodology. J. Colloid Interface Sci. 465, 342–346 (2016).26704592 10.1016/j.jcis.2015.09.025
29. Qiao R Brinson LC Simulation of interphase percolation and gradients in polymer nanocomposites Compos. Sci. Technol. 2009 69 3 491 499 10.1016/j.compscitech.2008.11.022
Qiao, R. & Brinson, L. C. Simulation of interphase percolation and gradients in polymer nanocomposites. Compos. Sci. Technol. 69 (3), 491–499 (2009).10.1016/j.compscitech.2008.11.022
30. Baxter SC Robinson CT Pseudo-percolation: critical volume fractions and mechanical percolation in polymer nanocomposites Compos. Sci. Technol. 2011 71 10 1273 1279 10.1016/j.compscitech.2011.04.010
Baxter, S. C. & Robinson, C. T. Pseudo-percolation: critical volume fractions and mechanical percolation in polymer nanocomposites. Compos. Sci. Technol. 71 (10), 1273–1279 (2011).10.1016/j.compscitech.2011.04.010
31. Zare Y Rhee KY Development and modification of conventional Ouali model for tensile modulus of polymer/carbon nanotubes nanocomposites assuming the roles of dispersed and networked nanoparticles and surrounding interphases J. Colloid Interface Sci. 2017 506 283 290 10.1016/j.jcis.2017.07.050 28738279
Zare, Y. & Rhee, K. Y. Development and modification of conventional Ouali model for tensile modulus of polymer/carbon nanotubes nanocomposites assuming the roles of dispersed and networked nanoparticles and surrounding interphases. J. Colloid Interface Sci. 506, 283–290 (2017).28738279 10.1016/j.jcis.2017.07.050
32. Du F Nanotube networks in polymer nanocomposites: rheology and electrical conductivity Macromolecules 2004 37 24 9048 9055 10.1021/ma049164g
Du, F. et al. Nanotube networks in polymer nanocomposites: rheology and electrical conductivity. Macromolecules. 37 (24), 9048–9055 (2004).10.1021/ma049164g
33. Ryvkina N Tchmutin I Vilčáková J Pelíšková M Sáha P The deformation behavior of conductivity in composites where charge carrier transport is by tunneling: Theoretical modeling and experimental results Synth. Met. 2005 148 2 141 146 10.1016/j.synthmet.2004.09.028
Ryvkina, N., Tchmutin, I., Vilčáková, J., Pelíšková, M. & Sáha, P. The deformation behavior of conductivity in composites where charge carrier transport is by tunneling: Theoretical modeling and experimental results. Synth. Met. 148 (2), 141–146 (2005).10.1016/j.synthmet.2004.09.028
34. Ambrosetti G Solution of the tunneling-percolation problem in the nanocomposite regime Phys. Rev. B 2010 81 15 155434 10.1103/PhysRevB.81.155434
Ambrosetti, G. et al. Solution of the tunneling-percolation problem in the nanocomposite regime. Phys. Rev. B. 81 (15), 155434 (2010).10.1103/PhysRevB.81.155434
35. Hu N Karube Y Yan C Masuda Z Fukunaga H Tunneling effect in a polymer/carbon nanotube nanocomposite strain sensor Acta Mater. 2008 56 13 2929 2936 10.1016/j.actamat.2008.02.030
Hu, N., Karube, Y., Yan, C., Masuda, Z. & Fukunaga, H. Tunneling effect in a polymer/carbon nanotube nanocomposite strain sensor. Acta Mater. 56 (13), 2929–2936 (2008).10.1016/j.actamat.2008.02.030
36. Mohammadpour-Haratbar A Zare Y Rhee KY Simulation of electrical conductivity for polymer silver nanowires systems Sci. Rep. 2023 13 1 5 10.1038/s41598-022-25548-w 36593261
Mohammadpour-Haratbar, A., Zare, Y. & Rhee, K. Y. Simulation of electrical conductivity for polymer silver nanowires systems. Sci. Rep. 13 (1), 5 (2023).36593261 10.1038/s41598-022-25548-w
37. Zare Y Effects of imperfect interfacial adhesion between polymer and nanoparticles on the tensile modulus of clay/polymer nanocomposites Appl. Clay Sci. 2016 129 65 70 10.1016/j.clay.2016.05.002
Zare, Y. Effects of imperfect interfacial adhesion between polymer and nanoparticles on the tensile modulus of clay/polymer nanocomposites. Appl. Clay Sci. 129, 65–70 (2016).10.1016/j.clay.2016.05.002
38. Li J Kim J-K Percolation threshold of conducting polymer composites containing 3D randomly distributed graphite nanoplatelets Compos. Sci. Technol. 2007 67 10 2114 2120 10.1016/j.compscitech.2006.11.010
Li, J. & Kim, J-K. Percolation threshold of conducting polymer composites containing 3D randomly distributed graphite nanoplatelets. Compos. Sci. Technol. 67 (10), 2114–2120 (2007).10.1016/j.compscitech.2006.11.010
39. Takeda T Shindo Y Kuronuma Y Narita F Modeling and characterization of the electrical conductivity of carbon nanotube-based polymer composites Polymer 2011 52 17 3852 3856 10.1016/j.polymer.2011.06.046
Takeda, T., Shindo, Y., Kuronuma, Y. & Narita, F. Modeling and characterization of the electrical conductivity of carbon nanotube-based polymer composites. Polymer. 52 (17), 3852–3856 (2011).10.1016/j.polymer.2011.06.046
40. Feng C Jiang L Micromechanics modeling of the electrical conductivity of carbon nanotube (CNT)–polymer nanocomposites Compos. Part A: Appl. Sci. Manufac. 2013 47 143 149 10.1016/j.compositesa.2012.12.008
Feng, C. & Jiang, L. Micromechanics modeling of the electrical conductivity of carbon nanotube (CNT)–polymer nanocomposites. Compos. Part A: Appl. Sci. Manufac. 47, 143–149 (2013).10.1016/j.compositesa.2012.12.008
41. Taherian R Experimental and analytical model for the electrical conductivity of polymer-based nanocomposites Compos. Sci. Technol. 2016 123 17 31 10.1016/j.compscitech.2015.11.029
Taherian, R. Experimental and analytical model for the electrical conductivity of polymer-based nanocomposites. Compos. Sci. Technol. 123, 17–31 (2016).10.1016/j.compscitech.2015.11.029
42. Rittigstein P Torkelson JM Polymer–nanoparticle interfacial interactions in polymer nanocomposites: Confinement effects on glass transition temperature and suppression of physical aging J. Polym. Sci., Part B: Polym. Phys. 2006 44 20 2935 2943 10.1002/polb.20925
Rittigstein, P. & Torkelson, J. M. Polymer–nanoparticle interfacial interactions in polymer nanocomposites: Confinement effects on glass transition temperature and suppression of physical aging. J. Polym. Sci., Part B: Polym. Phys. 44 (20), 2935–2943 (2006).10.1002/polb.20925
43. Gao C Graphene networks with low percolation threshold in ABS nanocomposites: Selective localization and electrical and rheological properties ACS Appl. Mater. Interfaces 2014 6 15 12252 12260 10.1021/am501843s 24969179
Gao, C. et al. Graphene networks with low percolation threshold in ABS nanocomposites: Selective localization and electrical and rheological properties. ACS Appl. Mater. Interfaces. 6 (15), 12252–12260 (2014).24969179 10.1021/am501843s
44. Maiti S Suin S Shrivastava NK Khatua B Low percolation threshold in polycarbonate/multiwalled carbon nanotubes nanocomposites through melt blending with poly (butylene terephthalate) J. Appl. Polym. Sci. 2013 130 1 543 553 10.1002/app.39168
Maiti, S., Suin, S., Shrivastava, N. K. & Khatua, B. Low percolation threshold in polycarbonate/multiwalled carbon nanotubes nanocomposites through melt blending with poly (butylene terephthalate). J. Appl. Polym. Sci. 130 (1), 543–553 (2013).10.1002/app.39168
45. Kim SY Noh YJ Yu J Prediction and experimental validation of electrical percolation by applying a modified micromechanics model considering multiple heterogeneous inclusions Compos. Sci. Technol. 2015 106 156 162 10.1016/j.compscitech.2014.11.015
Kim, S. Y., Noh, Y. J. & Yu, J. Prediction and experimental validation of electrical percolation by applying a modified micromechanics model considering multiple heterogeneous inclusions. Compos. Sci. Technol. 106, 156–162 (2015).10.1016/j.compscitech.2014.11.015
46. Li J Correlations between percolation threshold, dispersion state, and aspect ratio of carbon nanotubes Adv. Funct. Mater. 2007 17 16 3207 3215 10.1002/adfm.200700065
Li, J. et al. Correlations between percolation threshold, dispersion state, and aspect ratio of carbon nanotubes. Adv. Funct. Mater. 17 (16), 3207–3215 (2007).10.1002/adfm.200700065
47. Li Y Mechanical, electrical and thermal properties of in-situ exfoliated graphene/epoxy nanocomposites Compos. Part A: Appl. Sci. Manufac. 2017 95 229 236 10.1016/j.compositesa.2017.01.007
Li, Y. et al. Mechanical, electrical and thermal properties of in-situ exfoliated graphene/epoxy nanocomposites. Compos. Part A: Appl. Sci. Manufac. 95, 229–236 (2017).10.1016/j.compositesa.2017.01.007
48. Xu L Chen G Wang W Li L Fang X A facile assembly of polyimide/graphene core–shell structured nanocomposites with both high electrical and thermal conductivities Compos. Part A: Appl. Sci. Manufac. 2016 84 472 481 10.1016/j.compositesa.2016.02.027
Xu, L., Chen, G., Wang, W., Li, L. & Fang, X. A facile assembly of polyimide/graphene core–shell structured nanocomposites with both high electrical and thermal conductivities. Compos. Part A: Appl. Sci. Manufac. 84, 472–481 (2016).10.1016/j.compositesa.2016.02.027
49. Stankovich S raphene-based Compos Mater. Nat. 2006 442 7100 282 286
Stankovich, S. et al. Graphene-based Compos. Mater. Nat. 442(7100), 282–286 (2006).
