Predicting of tunneling resistivity between adjacent nanosheets in graphene

Scientific Reports volume 13, Article number: 12455 (2023) Cite this article

149 Accesses

Metrics details

In this work, the tunneling resistivity between neighboring nanosheets in grapheme–polymer nanocomposites is expressed by a simple equation as a function of the characteristics of graphene and tunnels. This expression is obtained by connecting two advanced models for the conductivity of graphene-filled materials reflecting tunneling role and interphase area. The predictions of the applied models are linked to the tested data of several samples. The impressions of all factors on the tunneling resistivity are evaluated and interpreted using the suggested equation. The calculations of tunneling resistivity for the studied examples by the model and suggested equation demonstrate the same levels, which confirm the presented methodology. The results indicate that the tunneling resistivity decreases by super-conductive graphene, small tunneling width, numerous contacts among nanosheets and short tunneling length.

Graphene-filled products can be utilized in different grounds like electronics, electromagnetic shielding, sensing, energy devices and diodes, because graphene displays ideal electrical, mechanical, thermal and chemical properties1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18. The higher aspect ratio and bigger surface area of graphene nanosheets compared to CNT cause lower percolation inception and more conductivity19. So, the investigators have widely focused on the polymer graphene nanocomposites to optimize their performances. Most recent studies on polymer graphene nanocomposites have tried to prepare the samples with little percolation inception and great conductivity by low filler amount20,21,22. The percolation inception inversely connects to the aspect ratio of nanofiller as the ratio of diameter to thickness23, 24. Therefore, many parameters such as the dimensions, dispersion quality and aggregation/agglomeration of nanoparticles handle the percolation inception and thus, the nanocomposite’s conductivity.

Some new parameters attributed to nanoscale including tunneling effect and interphase can also govern the percolation inception. The tunneling tool mainly controls the conductivity of nanocomposites (abbreviated as conductivity here), since the electrons can be easily transported over small tunnels between neighboring particles25,26,27,28. In fact, the conductivity does not need the physical joining of nanoparticles and so, the tunneling effect changes the percolation inception in nanocomposites. However, only few investigators have concentrated on the tunneling conductivity in CNT-based products29,30,31. Moreover, the interphase, owing to the large external area of nanoparticles can efficiently decrease the percolation inception. The interphase is the depleted polymer layer at the filler–polymer interface32, 33. The interphase areas covering the nanoparticles can join together and construct the nets in the samples34,35,36,37,38. This attractive topic has been studied for mechanical behavior of polymer nanocomposites39,40,41,42,43, but the role of interphase in the conductivity was negligibly studied.

Various equations were advanced for the conductivity of CNT-filled examples assuming the factors for CNT such as amount, waviness, conduction and aspect ratio44,45,46,47. Also, few studies have reported the significances of tunneling effect and interphase on the conductivity of CNT products34, 47, 48. However, the modeling works on the conductivity of graphene-based systems are really incomplete. The former works usually correlated the percolation inception to filler aspect ratio and judged the conductivity by conventional power-law equation49,50,51. In summary, the foregoing studies have not considered the interphase and tunnels in the percolation inception and conductivity, while these factors mainly control the mentioned terms.

In this study, two models for the conductivity of graphene-based samples are coupled to describe the tunnel resistivity by the characteristics of graphene and tunnels. The predictability of the applied models is weighed by the experimented conductivity of some samples from literature. Furthermore, the novel model and the submitted equation are used to get the tunnel resistivity. Additionally, the novel equation is utilized to evaluate and validate the effects of different factors on the tunnel resistivity.

A polymer nanocomposite includes the nanofiller, surrounding interphase and tunnels between neighboring nanoparticles. Figure 1 displays the mentioned components in a graphene nanocomposite in which the interphase covers the nanosheets and the tunneling zones form between nearby nanosheets.

The components in polymer graphene nanocomposites.

The conductivity in polymer nanocomposites needs the networking of conductive nanofiller, which occurs above an essential filler amount as percolation inception20, 52.

The percolation inception in polymer graphite nanocomposites was formulized 24 by:

where “D” and “t” show the diameter and thickness of sheets and “λ” is tunneling length. Equation 1 was derived for graphite nanoparticles with high aspect ratio (D/t). Since both graphene and graphite have high aspect ratio, we used Eq. (1) for graphene-filled samples. Actually, very thin and long platelets (high aspect ratio) are similar to graphene nanosheets with high aspect ratio. Both thickness and diameter of platelets/nanosheets play the main role in the percolation onset, as mentioned in Eq. (1).

Assuming D >  > λ, Eq. (1) is simplified as:

The latter equation can be advanced assuming tunneling and interphase zones as:

where “ti” is interphase deepness. This equation represents that the percolation inception links to filler size, interphase deepness and tunneling length. Equation 3 is true, because the percolation onset adversely depends on the filler aspect ratio23, 24 and also this equation suggests a dimensionless parameter. Equation 3 is valid for only graphene-filled samples and cannot predict the percolation if the shape of the fillers is different.

The interphase areas also grow the efficiency of nanofiller in nanocomposites, because both nanofiller and interphase produce the networks. The total volume fraction of interphase in polymer graphene nanocomposites2 is given by:

where “\(\varphi_{f}\)” is filler volume share. The interphase areas can add to the filler nets; so, the effective volume fraction of nanofiller includes the fractions of graphene and surrounding interphase as:

Also, the percentage of nanosheets contributing to the conductive networks53 can be estimated by:

Assuming “\(\varphi_{eff}\)” (Eq. 5) develops the “f” to:

expressing that both graphene and interphase parts affect the dimensions of conductive networks.

To consider the role of tunnels between nanosheets in the conductivity, the extended nanosheets are suggested. In other words, an extended nanosheet contains the graphene and tunneling zones.

The total intrinsic resistance of a prolonged nanosheet is proposed by:

where “Rf” and “Rc” are the basic resistances of graphene and tunnels, in that order.

“Rf” is expressed as:

where “σf” is the conduction of graphene.

The tunnel resistance also consists of the resistances of graphene and insulated polymer within the tunnels. Accordingly, the tunnel resistance is stated by the resistances of graphene (Rg) and polymer (Rt) in the contact zones as:

“Rg” and “Rp” can be suggested54 by:

where “d” is tunnel diameter, “ρ” is tunnel resistivity by insulated polymer and “S” shows tunnel area (S ≈ d2).

Replacing of Eqs. (11) and (12) into Eq. (10) results in the intrinsic resistance of tunnels as:

which offers the whole intrinsic resistance of prolonged nanosheets (Eq. 8) as:

The total resistance of prolonged nanosheets assuming tunnels is applied to foresee the conductivity.

The total conductivity of extended nanosheets (S/m) can be expressed by reversing “Rext” as:

Two researchers45 obtained an equation for conductivity (random CNT) as:

where “σ0” is the conductivity of polymer medium (10−13–10−15 S/m), which can be unnoticed. Equation (16) can be applied to guesstimate the conductivity in graphene-filled samples.

When the operative filler amount (Eq. 5) and the conductivity of prolonged sheets (Eq. 15) are reflected in Eq. (16), the conductivity is given by:

which correlates the conductivity to graphene, interphase and tunnel properties.

Weber and Kamal55 also formulated the longitudinal resistivity for fiber-based composites by:

where “Af” shows filler cross-section zone, “l” is fiber length, “ρf” is the resistivity of fiber and “θ” is angle among fibers and current path. In addition, “X” is links to contact number (m) by:

where the highest “m” is 1555.

The composite’s conductivity can be gotten by inverse “ρl” as:

which can be advanced for the graphene systems.

The cross-section zone of graphene can be obtained as:

Also, “l” is replaced by “D” and σf = 1/ρf. Additionally, for 3-D haphazard spreading of nanoparticles in the samples56, it is considered that:

So, Eq. (20) can be expressed for graphene systems by operative filler amount (Eq. 5) as:

Moreover, the tunneling length mainly impresses the conductivity, since it manages the electron transferring at tunnels (Fig. 1). The tunneling length (λ) correlates to \(\varphi_{f}^{ - 1/3}\)53, 57. Since Eq. (20) suggests a linear link amid conductivity and “\(\varphi_{f}^{{}}\)”, the conductivity connects to “λ−3”.

Based on these remarks, the latter equation can consider the tunneling length as:

where “z” is tunneling characteristics length, which is considered as 0.1 nm for graphene-filled samples. This equation signifies a model for conductivity of products by graphene dimensions, graphene conduction, number of contacts, graphene amount in the nets, tunnel diameter, interphase deepness and tunneling length.

Now, Eqs. (17) and (24) can be combined to express an equation for tunnel resistivity (ρ).

Equations (17) and (24) are coupled as:

which expresses the tunnel resistivity as:

According to this equation, the tunnel resistivity in polymer graphene systems associates to the features of graphene and tunnels. This equation clearly demonstrates the significances of each parameter on the tunnel resistivity.

The predictability of both models is confirmed by the experimented levels of examples from previous articles. Also, the tunnel resistivity for the samples is calculated and compared. Four graphene examples including polyimide (PI) (\(\varphi_{p}^{{}}\) = 0.0015, D ≈ 5 μm, t = 3 nm)58, polystyrene (PS) (\(\varphi_{p}^{{}}\) = 0.001, D ≈ 2 μm, t = 1 nm)59 (No. 1), poly (ethylene terephthalate) (PET) (\(\varphi_{p}^{{}}\) = 0.005, D ≈ 2 μm, t = 2 nm)60 and PS (\(\varphi_{p}^{{}}\) = 0.0005, D ≈ 4 μm, t = 1 nm)61 (No. 2) were chosen from earlier researches. The values o f (ti, λ) can be analyzed by fitting the percolation inception to Eq. (3). Using this equation, (ti, λ) levels of (7, 9), (10, 9), (3, 4) and (7, 10) nm are obtained for PI, PS (No. 1), PET and PS (No. 2) nanocomposites, correspondingly. These calculations reveal the foundation of unlike interphase and tunnels in the nanocomposites. The densest interphase and the largest tunnels are shown in PS/graphene (No. 1) and PS/graphene (No. 2) samples, while the least levels of interphase deepness and tunneling length are observed in PET/graphene sample. It is obvious that a deep interphase and a big tunnel decrease the percolation inception. So, assumption of interphase deepness and tunneling length in the percolation inception is necessary, because these parameters largely manipulate the percolation inception in nanocomposites. Using these calculations, it is possible to calculate the conductivity by the developed models. Graphene conductivity and cos2 (θ) are reflected as 105 S/m and 1/3, in that order.

Figure 2 reveals the experimented and theoretical ranks of conductivity for the examples. As observed, the calculations demonstrate fine matching with the experimented data of examples. As a result, both models suitably visualize the conductivity for the samples, which confirm their predictability for all polymer graphene nanocomposites. The values of (m, d) (d in nm) by Eq. (24) are obtained as (5, 10), (3, 10), (4, 5) and (120, 400) for PI, PS (No. 1), PET and PS (No. 2) nanocomposites, in that order. These results display the different ranges of contact number and tunnel diameter in the nanocomposites. The most desirable levels of these parameters are obtained for PS/graphene (No. 2) sample. Conferring to the experimented results in Fig. 2, this sample displays the most level of conductivity. So, it is concluded that the number of contacts and the tunnel diameter mainly affect the conductivity. Also, Eq. (17) calculates the tunnel resistivity (ρ) as 6.04, 7.8, 0.67 and 21.4 Ω.m for PI, PS (No. 1), PET and PS (No. 2) examples, correspondingly. The utmost and the lowermost levels of tunnel resistivity are shown in PS/graphene (No. 1) and PET/graphene nanocomposites, respectively. The same values of tunnel resistivity are also obtained for the reported samples using the suggested equation (Eq. 26). In fact, both Eqs. (17) and (26) present the same levels for tunnel resistivity based on the experimental results of conductivity and percolation inception. These outputs support the new equation for tunnel resistivity in nanocomposites. In other words, Eq. (26) can be used to foresee the tunnel resistivity.

The experimented conductivity and the calculations by Eqs. (17) and (24) for (a) PI58, (b) PS59 (No. 1), (c) PET60 and (d) PS (No. 2)61 graphene products.

In this section, the inspirations of all factors on the tunnel resistivity are evaluated using the novel equation (Eq. 26).

Figure 3 illustrates the roles of “t” and “σf” in the tunnel resistivity at λ = 5 nm, m = 10, d = 200 nm and z = 0.1 nm. The lowest tunnel resistivity as 10 Ω.m is observed at σf > 1.65*105 S/m, but the tunnel resistivity increases to 45 Ω.m at σf = 0.5*105 S/m. As a result, the graphene conduction inversely influences the tunnel resistivity, while the thickness of graphene nanosheets cannot affect it. In fact, a super-conductive nanofiller can mainly decrease the tunnel resistivity in nanocomposites, which promotes the conductivity. However, the graphene thickness is an ineffective factor, which cannot change the tunnel resistivity.

The impressions of “t” and “σf” on the tunnel resistivity (Eq. 26) by (a) 3-D and (b) contour designs.

The graphene nanosheets cover the tunnels in the nanocomposites. Undoubtedly, the conductive graphene significantly decreases the tunnel resistivity, because it can facilitate the transference of electrons through tunnels. On the other hand, poor-conductive graphene cannot reduce the tunnel resistivity, because it cannot affect the resistance of tunnels containing insulated polymer film and graphene nanosheets. So, it is meaningful to observe an inverse relation between tunnel resistivity and graphene conduction. Additionally, the thickness of graphene cannot control the tunnel resistivity, because its role in the tunnels is negligible. Conclusively, the size of graphene nanosheets cannot affect the tunnel resistivity, but the conduction of graphene positively decreases it.

Figure 4 also shows the dependency of tunnel resistivity on “d” and “θ” at t = 2 nm, λ = 5 nm, σf = 105 S/m, m = 10 and z = 0.1 nm. The smallest level of tunnel resistivity as about 0 is obtained by θ < 60°, while the highest tunnel resistivity is calculated at the highest ranges of both “d” and “θ”. The calculations demonstrate that the maximum tunnel resistivity of 350 Ω.m is detected at d = 300 nm and θ = 80°. Consequently, the diameter of contact area between adjacent nanosheets and the filler angle directly change the tunnel resistivity. In fact, it is important to decrease the levels of tunnel diameter and filler angle to obtain a poor tunnel resistivity.

Expression of tunnel resistivity (Eq. 26) by “d” and “θ”: (a) 3-D and (b) contour schemes.

The tunnel diameter between nanosheets determines the extent of tunnels, because two neighboring nanosheets form the tunneling space. In other words, a big tunnel diameter causes a large tunnel in the nanocomposites, whereas less “d” forms a small tunneling zone. Since the tunnels mainly contains the insulated polymer matrix, a large tunnels produces a high tunnel resistivity. On the other hand, the less value of tunnel diameter between adjacent nanosheets yields a small tunnel. In this condition, the tunnel resistivity mainly decreases, because of the small tunnels. Moreover, the nanosheets oriented to the current direction can easily and effectively transfer the charges and improve the conductivity, while a high angle between nanosheets and electron current limits the tunneling conductivity. Accordingly, a low “θ” significantly increases the electron transportation and decreases the tunnel resistivity. However, a high “θ” produces an improper implementation of nanosheets in the nanocomposite, which grows the tunnel resistivity. Therefore, both tunnel diameter and orientation angle logically influence the tunnel resistivity in nanocomposites.

The stimuli of “m” and “λ” on the tunnel resistivity are observed in Fig. 5. The uppermost tunnel resistivity of 220 Ω.m is obtained in m = 2 and λ = 10 nm, but the tunnel resistivity largely declines to 0 at λ < 3 nm or m > 15 and λ < 5 nm. Accordingly, abundant contacts among sheets and a minor tunneling length attain a deprived tunnel resistivity. In contrast, a less quantity of contacts and long tunnel negatively raise the tunnel resistivity.

(a) 3-D and (b) contour designs for the variation of tunnel resistivity (Eq. 26) at altered series of “m” and “λ”.

A high number of contacts reduce the tunnel resistivity, but the limited ranges of contact number increase it. In fact, the large links between conductive graphene nanosheets produce the conductive tunneling zones in nanocomposites weakening the tunnel resistivity. However, the low extent of contacts demonstrates the presence of insulated polymer matrix between nanosheets, which considerably increases the tunnel resistivity. As a result, the suggested equation (Eq. 26) properly predicts the correlation between tunnel resistivity and contact number. In addition, a high tunneling length between nanosheets shows the existence of a thick polymer layer in the tunnels. Since the insulated polymer matrix weakens the transportation of electrons in the tunnels, it is reasonable to obtain a high tunnel resistivity in this condition, due to the high amount of insulated material. Nevertheless, a small tunneling length displays a thin polymer film between neighboring nanosheets, which causes a poor tunnel resistivity. The former articles reported that the large tunnel decreases the tunneling conductivity, due to the restricted electron shifting over tunnels or high tunnel resistivity2, 62. Based on these reasons, the suggested equation reasonably indicates the role of tunneling length in the tunnel resistivity.

The tunnel resistivity in polymer graphene nanocomposites was defined as a function of the characteristics of graphene and tunnels. This equation was extracted by extending the graphene nanosheets and upgrading a conventional model. The estimates of these models demonstrate fine arrangements with the experimented data. Also, the tunnel resistivity of the reported samples calculated by the developed model and the suggested equation give similar values, which approve the presented equations. σf > 1.65*105 S/m produces the tunnel resistivity of 10 Ω.m, but the tunnel resistivity grows to 45 Ω.m at σf = 0.5*105 S/m. Consequently, the graphene conduction inversely handles the tunnel resistivity, nonetheless the thickness of graphene nanosheets cannot affect it. The smallest level of tunnel resistivity as about 0 is also obtained by θ < 60°, while the highest tunnel resistivity is calculated by the highest ranges of both “d” and “θ”. As a result, the diameter of contact area between nanosheets and the filler angle directly govern the tunnel resistivity. In addition, the highest tunnel resistivity of 220 Ω.m is gotten by m = 2 and λ = 10 nm, nevertheless the tunnel resistivity mostly declines to around 0 at λ < 3 nm or m > 15 and λ < 5 nm. Therefore, plentiful contacts among nanosheets and a small tunneling length achieve a low tunnel resistivity in nanocomposites.

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

Farouq, R. Functionalized graphene/polystyrene composite, green synthesis and characterization. Sci. Rep. 12(1), 21757 (2022).

Article ADS CAS PubMed PubMed Central Google Scholar

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).

Article Google Scholar

Zare, Y. & Rhee, K. Y. An innovative model for conductivity of graphene-based system by networked nano-sheets, interphase and tunneling zone. Sci. Rep. 12(1), 1–9 (2022).

Article Google Scholar

Azizi-Lalabadi, M. & Jafari, S. M. Bio-nanocomposites of graphene with biopolymers; fabrication, properties, and applications. Adv. Colloid Interface Sci. 292, 102416 (2021).

Article CAS PubMed Google Scholar

Ikram, R. et al. Recent advances in chitin and chitosan/graphene-based bio-nanocomposites for energetic applications. Polymers 13(19), 3266 (2021).

Article CAS PubMed PubMed Central Google Scholar

Fatima, N. et al. Recent developments for antimicrobial applications of graphene-based polymeric composites: A review. J. Ind. Eng. Chem. 100, 40–58 (2021).

Article CAS Google Scholar

Bahrami, S., Baheiraei, N. & Shahrezaee, M. Biomimetic reduced graphene oxide coated collagen scaffold for in situ bone regeneration. Sci. Rep. 11(1), 1–10 (2021).

Article Google Scholar

Cong, R. et al. Characteristics and electrochemical performances of silicon/carbon nanofiber/graphene composite films as anode materials for binder-free lithium-ion batteries. Sci. Rep. 11(1), 1–11 (2021).

Article Google Scholar

Khosrozadeh, A., Rasuli, R., Hamzeloopak, H. & Abedini, Y. Wettability and sound absorption of graphene oxide doped polymer hydrogel. Sci. Rep. 11(1), 1–11 (2021).

Article Google Scholar

Mousavi, H. & Grabowski, M. Nonlinear electron transport across short DNA segment between graphene leads. Solid State Commun. 279, 30–33 (2018).

Article ADS CAS Google Scholar

Mohammadpour-Haratbar, A., Boraei, S. B. A., Zare, Y., Rhee, K. Y. & Park, S.-J. Graphene-based electrochemical biosensors for breast cancer detection. Biosensors 13(1), 80 (2023).

Article CAS PubMed PubMed Central Google Scholar

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. 2022:102795.

Asadzadeh Patehkhor, H., Fattahi, M. & Khosravi-Nikou, M. Synthesis and characterization of ternary chitosan–TiO2–ZnO over graphene for photocatalytic degradation of tetracycline from pharmaceutical wastewater. Sci. Rep. 11(1), 1–17 (2021).

Article Google Scholar

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).

Article Google Scholar

Mohammadi, A. A. et al. Comparative removal of hazardous cationic dyes by MOF-5 and modified graphene oxide. Sci. Rep. 12(1), 1–12 (2022).

Article Google Scholar

Aliya, M., Zare, E. N., Faridnouri, H., Ghomi, M. & Makvandi, P. Sulfonated starch-graft-polyaniline@ graphene electrically conductive nanocomposite: Application for tyrosinase immobilization. Biosensors 12(11), 939 (2022).

Article CAS PubMed PubMed Central Google Scholar

Mohammadpour, Z., Abdollahi, S. H. & Safavi, A. Sugar-based natural deep eutectic mixtures as green intercalating solvents for high-yield preparation of stable MoS2 nanosheets: application to electrocatalysis of hydrogen evolution reaction. ACS Appl. Energy Mater. 1(11), 5896–5906 (2018).

Article CAS Google Scholar

Mohammadpour, Z. & Majidzadeh-A, K. Applications of two-dimensional nanomaterials in breast cancer theranostics. ACS Biomater. Sci. Eng. 6(4), 1852–1873 (2020).

Article CAS PubMed Google Scholar

Xie, S., Liu, Y. & Li, J. Comparison of the effective conductivity between composites reinforced by graphene nanosheets and carbon nanotubes. Appl. Phys. Lett. 92(24), 243121 (2008).

Article ADS Google Scholar

Haghgoo, M., Ansari, R. & Hassanzadeh-Aghdam, M. Synergic effect of graphene nanoplatelets and carbon nanotubes on the electrical resistivity and percolation threshold of polymer hybrid nanocomposites. Eur. Phys. J. Plus 136(7), 1–20 (2021).

Article Google Scholar

Cho, J. et al. Enhanced electrical conductivity of polymer nanocomposite based on edge-selectively functionalized graphene nanoplatelets. Compos. Sci. Technol. 189, 108001 (2020).

Article CAS Google Scholar

Folorunso, O., Hamam, Y., Sadiku, R., Ray, S. S. & Adekoya, G. J. Statistical characterization and simulation of graphene-loaded polypyrrole composite electrical conductivity. J. Market. Res. 9(6), 15788–15801 (2020).

CAS Google Scholar

Berhan, L. & Sastry, A. Modeling percolation in high-aspect-ratio fiber systems. I. Soft-core versus hard-core models. Phys. Rev. E 75(4), 41120 (2007).

Article ADS CAS Google Scholar

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).

Article CAS Google Scholar

Razavi, R., Zare, Y. & Rhee, K. Y. A two-step model for the tunneling conductivity of polymer carbon nanotube nanocomposites assuming the conduction of interphase regions. RSC Adv. 7(79), 50225–50233 (2017).

Article ADS CAS Google Scholar

Zare, Y. & Rhee, K. Y. Advanced model for conductivity estimation of graphene-based samples considering interphase effect, tunneling mechanism, and filler wettability. J. Ind. Eng. Chem. 108, 81–87 (2022).

Article CAS Google Scholar

Zare, Y., Rhim, S. S. & Rhee, K. Y. Development of Jang-Yin model for effectual conductivity of nanocomposite systems by simple equations for the resistances of carbon nanotubes, interphase and tunneling section. Eur. Phys. J. Plus. 136(7), 1–15 (2021).

Article Google Scholar

Mousavi, H. & Bamdad, M. The transport properties of poly (G)-poly (C) DNA oligomers in the Harrison’s model. J. Mol. Graph. Model. 112, 108138 (2022).

Article CAS PubMed Google Scholar

Fang, C., Zhang, J., Chen, X. & Weng, G. J. A Monte Carlo model with equipotential approximation and tunneling resistance for the electrical conductivity of carbon nanotube polymer composites. Carbon 146, 125–138 (2019).

Article CAS Google Scholar

Prabhakar, R. et al. Tunneling-limited thermoelectric transport in carbon nanotube networks embedded in poly (dimethylsiloxane) elastomer. ACS Appl. Energy Mater. 2(4), 2419–2426 (2019).

Article CAS Google Scholar

Megha, R. et al. Enhancement in alternating current conductivity of polypyrrole by multi-walled carbon nanotubes via single electron tunneling. Diam. Relat. Mater. 87, 163–171 (2018).

Article ADS CAS Google Scholar

Zare, Y. & Rhee, K. Y. Expansion of Takayanagi model by interphase characteristics and filler size to approximate the tensile modulus of halloysite-nanotube-filled system. J. Mark. Res. 16, 1628–1636 (2022).

CAS Google Scholar

Hassanzadeh-Aghdam, M. K., Mahmoodi, M. J. & Ansari, R. Creep performance of CNT polymer nanocomposites—An emphasis on viscoelastic interphase and CNT agglomeration. Compos. B Eng. 168, 274–281 (2019).

Article CAS Google Scholar

Zare, Y. & Rhee, K. Y. Significances of interphase conductivity and tunneling resistance on the conductivity of carbon nanotubes nanocomposites. Polym. Compos. 41(2), 748–756 (2020).

Article CAS Google Scholar

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).

Article ADS CAS PubMed Google Scholar

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).

Article CAS Google Scholar

Zare, Y. & Rhee, K. Y. Development of a model for modulus of polymer halloysite nanotube nanocomposites by the interphase zones around dispersed and networked nanotubes. Sci. Rep. 12(1), 1–12 (2022).

Article Google Scholar

Zare, Y. & Rhee, K. Y. Crucial interfacial shear strength to consider an imperfect interphase in halloysite-nanotube-filled biomedical samples. J. Market. Res. 19, 3777–3787 (2022).

CAS Google Scholar

Shin, H., Yang, S., Choi, J., Chang, S. & Cho, M. Effect of interphase percolation on mechanical behavior of nanoparticle-reinforced polymer nanocomposite with filler agglomeration: A multiscale approach. Chem. Phys. Lett. 635, 80–85 (2015).

Article ADS CAS Google Scholar

Qiao, R. & Brinson, L. C. Simulation of interphase percolation and gradients in polymer nanocomposites. Compos. Sci. Technol. 69(3), 491–499 (2009).

Article CAS Google Scholar

Zare, Y. & Rhee, K. Y. A multistep methodology for calculation of the tensile modulus in polymer/carbon nanotube nanocomposites above the percolation threshold based on the modified rule of mixtures. RSC Adv. 8(54), 30986–30993 (2018).

Article ADS CAS PubMed PubMed Central Google Scholar

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).

Article ADS Google Scholar

Mohammadpour-Haratbar, A., Zare, Y. & Rhee, K. Y. Development of a theoretical model for estimating the electrical conductivity of a polymeric system reinforced with silver nanowires applicable for the biosensing of breast cancer cells. J. Market. Res. 18, 4894–4902 (2022).

CAS Google Scholar

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).

Article CAS Google Scholar

Deng, F. & Zheng, Q.-S. An analytical model of effective electrical conductivity of carbon nanotube composites. Appl. Phys. Lett. 92(7), 071902 (2008).

Article ADS Google Scholar

Taherian, R. Experimental and analytical model for the electrical conductivity of polymer-based nanocomposites. Compos. Sci. Technol. 123, 17–31 (2016).

Article CAS Google Scholar

Kazemi, F., Mohammadpour, Z., Naghib, S. M., Zare, Y. & Rhee, K. Y. Percolation onset and electrical conductivity for a multiphase system containing carbon nanotubes and nanoclay. J. Market. Res. 15, 1777–1788 (2021).

CAS Google Scholar

Zare, Y. & Rhee, K. Y. Micromechanics simulation of electrical conductivity for carbon-nanotube-filled polymer system by adjusting Ouali model. Eur. Phys. J. Plus 136(8), 852 (2021).

Article CAS Google Scholar

Clingerman, M. L., King, J. A., Schulz, K. H. & Meyers, J. D. Evaluation of electrical conductivity models for conductive polymer composites. J. Appl. Polym. Sci. 83(6), 1341–1356 (2002).

Article CAS Google Scholar

Chang, L., Friedrich, K., Ye, L. & Toro, P. Evaluation and visualization of the percolating networks in multi-wall carbon nanotube/epoxy composites. J. Mater. Sci. 44(15), 4003–4012 (2009).

Article ADS CAS Google Scholar

Kara, S., Arda, E., Dolastir, F. & Pekcan, Ö. Electrical and optical percolations of polystyrene latex-multiwalled carbon nanotube composites. J. Colloid Interface Sci. 344(2), 395–401 (2010).

Article ADS CAS PubMed Google Scholar

Strugova, D., Ferreira Junior, J. C., David, É. & Demarquette, N. R. Ultra-low percolation threshold induced by thermal treatments in co-continuous blend-based PP/PS/MWCNTS nanocomposites. Nanomaterials 11(6), 1620 (2021).

Article CAS PubMed PubMed Central Google Scholar

Feng, C. & Jiang, L. Micromechanics modeling of the electrical conductivity of carbon nanotube (CNT)–polymer nanocomposites. Compos. A Appl. Sci. Manuf. 47, 143–149 (2013).

Article CAS Google Scholar

Messina, E. et al. Double-wall nanotubes and graphene nanoplatelets for hybrid conductive adhesives with enhanced thermal and electrical conductivity. ACS Appl. Mater. Interfaces. 8(35), 23244–23259 (2016).

Article CAS PubMed Google Scholar

Weber, M. & Kamal, M. R. Estimation of the volume resistivity of electrically conductive composites. Polym. Compos. 18(6), 711–725 (1997).

Article CAS Google Scholar

Li, J. et al. Correlations between percolation threshold, dispersion state, and aspect ratio of carbon nanotubes. Adv. Funct. Mater. 17(16), 3207–3215 (2007).

Article CAS Google Scholar

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).

Article CAS Google Scholar

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. A Appl. Sci. Manuf. 84, 472–481 (2016).

Article CAS Google Scholar

Stankovich, S. et al. Graphene-based composite materials. Nature 442(7100), 282–286 (2006).

Article ADS CAS PubMed Google Scholar

Zhang, H.-B. et al. Electrically conductive polyethylene terephthalate/graphene nanocomposites prepared by melt compounding. Polymer 51(5), 1191–1196 (2010).

Article CAS Google Scholar

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).

Article CAS Google Scholar

Zare, Y. & Rhee, K. Y. Effects of network, tunneling, and interphase properties on the operative tunneling resistance in polymer carbon nanotubes (CNTs) nanocomposites. Polym. Compos. 41(7), 2907–2916 (2020).

Article CAS Google Scholar

Download references

Biomaterials and Tissue Engineering Research Group, Department of Interdisciplinary Technologies, Breast Cancer Research Center, Motamed Cancer Institute, ACECR, Tehran, Iran

Yasser Zare

College of Engineering and Technology, American University of the Middle East, Egaila, 54200, Kuwait

Nima Gharib

Department of Materials Science and Chemical Engineering, BK21 FOUR ERICA-ACE Center, Hanyang University ERICA, Ansan, 15588, Korea

Dong-Hyun Nam & Young-Wook Chang

You can also search for this author in PubMed Google Scholar

You can also search for this author in PubMed Google Scholar

You can also search for this author in PubMed Google Scholar

You can also search for this author in PubMed Google Scholar

Y.Z. and N.G. wrote the main manuscript. D-H.N. and Y-W.C revised the paper.

Correspondence to Yasser Zare or Young-Wook Chang.

The authors declare no competing interests.

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

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, 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 changes were made. 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/4.0/.

Reprints and Permissions

Zare, Y., Gharib, N., Nam, DH. et al. Predicting of tunneling resistivity between adjacent nanosheets in graphene–polymer systems. Sci Rep 13, 12455 (2023). https://doi.org/10.1038/s41598-023-39414-w

Download citation

Received: 06 May 2023

Accepted: 25 July 2023

Published: 01 August 2023

DOI: https://doi.org/10.1038/s41598-023-39414-w

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.