Conduction transportation from graphene to an insulative polymer medium: A novel approach for the conductivity of nanocomposites

Yasser Zare; Muhammad Tajammal Munir; Kyong Yop Rhee; Soo-Jin Park

doi:10.1515/ntrev-2024-0131

Article Open Access

Conduction transportation from graphene to an insulative polymer medium: A novel approach for the conductivity of nanocomposites

Yasser Zare , Muhammad Tajammal Munir , Kyong Yop Rhee and Soo-Jin Park

Published/Copyright: December 23, 2024

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From the journal Nanotechnology Reviews Volume 13 Issue 1

Abstract

Some models have been proposed for the electrical conductivity of graphene-filled nanocomposites, but they have not reflected the characteristics of an imperfect interphase surrounding the graphene nanosheets. In this article, the size and conductivity of an imperfect interphase are used to develop a model for conductivity of the graphene/polymer system. Also, “Y,” the degree of conduction transfer through an imperfect interphase, is expressed as graphene dimensions and interphase conductivity to define the effective converse aspect ratio and effective filler portion in the samples. The developed model for nanocomposite conductivity is examined by the experimental data of some samples. Furthermore, the influences of various factors on “Y,” percolation onset, and nanocomposite conductivity are investigated. Thin and large nanosheets, poor filler conductivity, and high interphase conductivity produce a high “Y.” Likewise, “Y” and graphene volume portion ( φ f ) significantly govern the conductivity of samples. Y = 9 and φ f = 0.03 yield the highest nanocomposite conductivity of 16 S/m, while Y < 2 or φ f < 0.022 cannot improve the conductivity of samples.

Keywords: graphene; nanocomposite; conduction conveyance; interface/interphase; electrical conductivity; percolation threshold

Nomenclature

σ: nanocomposite conductivity
σ ¯: average normal conductivity
σ _f: graphene conductivity
D _c: minimum graphene diameter allowing the conduction transfer from conductive nanosheets to the polymer medium
t: graphene thickness
D: graphene diameter
ψ: interfacial conductivity
D _eff: effective diameter of nanosheets
α _eff: effective converse aspect ratio of graphene
φ eff: effective volume portion of graphene
Y: the degree of conduction transfer from graphene to the polymer medium
d: tunneling distance
φ f: filler volume portion
b: an exponent
α: converse aspect ratio of nanosheets
φ p: percolation threshold

1 Introduction

The electrical conductivity in polymer composites is attained once the filler amount gets an indispensable level as the percolation threshold, which constructs the conductive nets [1,2]. The large aspect ratio and vast surface area of graphene nanosheets produce a small percolation threshold in nanocomposites [3]. Thus, only a small quantity of graphene is enough to obtain significant stiffness [4,5], corrosion [6], damping properties [7], electrical conductivity [8,9], and microwave absorption [10] in nanocomposites providing biomedical [11] and electrical [12] applications. Moreover, the electrons can be transported via the tunnels among neighboring particles, and thus the nanocomposite conductivity does not require the attachment of nanosheets [13,14]. Accordingly, the tunneling effect governs the percolation threshold and conductivity of samples because the adjacent nanosheets can form the conductive nets and endorse the conductivity. However, only a few studies have investigated the tunneling effect in nanocomposites [15,16].

Some investigators have studied the strengthening effect of interphase on the nanocomposites containing carbon nanotubes (CNTs) [17,18], graphene [19,20], and clay [21,22]. Also, some simple models were proposed to show the effect of interphase properties such as its depth, modulus, and strength on the stiffness of nanocomposites [23,24,25]. Nevertheless, an inadequate compatibility between the matrix and the nanofiller causes poor interphase regions in nanocomposites, which cannot transport the significant features of nanofillers to the polymer medium deteriorating the effectiveness of nanofillers in the system [26,27]. Some researchers have analyzed the interfacial adhesion between the polymer and the nanofiller in the polymer nanocomposites [28,29]. The interphase is more conductive than the insulated polymer matrix in nanocomposites. So, the interphase can control the conductivity of nanocomposites. Also, the interphase surrounding the nanoparticles can be connected to produce the network [30,31]. Accordingly, the interphase has a significant effect on the percolation onset and nanocomposite conductivity concurrently.

A number of investigators have indicated the major interphase impacts on the network size and stiffness of samples [32], but these studies cannot demonstrate the interphase significance on the conductivity of nanocomposites. A predictive model for self-temperature-compensated piezoresistive properties of CNT/graphene polymer nanocomposites was suggested [33]. Also, the pressure/resistance sensitivity of the graphene elastomeric nanocomposite was investigated using a finite element percolation model [34]. The synergic effect of graphene nanosheets and CNTs on the electrical resistivity and percolation threshold of polymer nanocomposites was also studied [35]. Moreover, Sarikhani et al. [36] developed a generalized percolation framework that describes both the electrical and thermal conductivity in the polymer composites. However, the limited articles on the graphene samples have not reflected the impacts of interphase and tunneling characteristics on the nanocomposite conductivity. Additionally, the conduction shifting from graphene to the polymer matrix and its influences on the effective concentration of nanoparticles, percolation onset, and conductivity of nanocomposites were not studied.

In this work, a model for nanocomposite conductivity is developed by graphene size and interphase properties. Also, “Y” is defined by the diameter of nanosheets and the least nanosheet diameter transferring its conduction to the matrix. Moreover, the percolation threshold is expressed by the effective converse aspect ratio, interphase thickness, and tunneling distance. The conductivity of some samples from the literature is used to examine the new equations. Furthermore, the parameters’ roles in “Y,” percolation threshold, and nanocomposite conductivity are plotted and discussed.

2 Modeling analyses

“D _c” is the minimum diameter of graphene nanosheets providing the conductivity in the interphase. So, “D _c” is an important parameter that controls the conductivity of the graphene-filled system.

“D _c” is expressed as

(1) D c = σ f t 2 ψ ,

where “t” and “σ _f” show the thickness and conductivity of graphene nanosheets, respectively, and “ψ” is the interphase conductivity.

When the interphase is perfect, the average normal conductivity ( σ ¯ ) is equal to “σ _f,” but σ ¯ is slighter than “σ _f” in an imperfect interphase.

The effective diameter of nanosheets (D _eff) can be expressed as

(2) σ ¯ D = σ f D eff ,

where “D” is the graphene diameter.

So, the interfacial properties affect the effective converse aspect ratio (α _eff) and effective volume portion ( φ eff ) of graphene [37] in the nanocomposites as follows:

(3) α eff = α 8 D c 2 D 2 + 1 ,

(4) φ eff = 5 φ f 1 2 + 1 − 4 D c 2 D ( 1 − 4 D c ) 4 D ,

where “α” is the converse aspect ratio of nanosheets (t/D) and “ φ f ” is the filler volume portion.

The degree of conduction transfer via an imperfect interphase can be expressed as

(5) Y = D 4 D c .

When “D _c” is substituted from equation (1) into equation (5), “Y” is given by

(6) Y = ψ D 2 σ f t .

Also, substituting “Y” from equation (6) into equations (3) and (4) presents “α _eff” and “ φ eff ” as

(7) α eff = α 1 2 Y 2 + 1 ,

(8) φ eff = 5 φ f 1 2 + 1 − 1 2 Y 1 − 1 4 Y ,

expressing the roles of conduction transfer extent in the effective converse aspect ratio and effective filler portion in the nanocomposites.

Now, it is essential to define the percolation onset in a nanocomposite containing an imperfect interphase. The percolation threshold in polymer graphite nanocomposites assuming the tunneling distance (d) was given by [38]

(9) φ p = 27 π D 2 t 4 ( D + d ) 3 .

But D > > d condenses this equation to

(10) φ p = 27 π t 4 D .

As mentioned above, the interphase and tunneling regions about nanosheets change the percolation threshold. Actually, a thick interphase and a big tunnel facilitate the networking and reduce the percolation onset. Equation (10) can be developed to assume the effects of these terms as:

(11) φ p = 27 π t 2 4 t D + 2 ( D t i + D d ) ,

where “t _i” is the interphase thickness.

Assuming α = t/D in the above equation restructures “ φ p ” to

(12) φ p = 27 π t α 4 t + 2 t i + 2 d .

When the effective converse aspect ratio (α _eff) is substituted from equation (7) into equation (12), the percolation threshold can consider the role of “Y” as:

(13) φ p = 27 π t α 1 2 Y 2 + 1 4 t + 2 t i + 2 d .

Now, the effective converse aspect ratio and effective graphene portion due to the imperfect interphase as well as percolation onset are applied to develop a model for the conductivity of graphene-based nanocomposites.

The Hu group [39] advanced the power-law model for conductivity of polymer CNT products as:

(14) σ = ( φ f − φ p ) b σ f 10 0.85 [ log ( l / 2 R ) − 1 ] ,

where “l” and “R” represent the length and radius of CNTs, respectively, and “b” is an exponent. Hu et al. [39] demonstrated that equation (14) properly forecasts the electrical conductivity of polymer CNT nanocomposites. However, equation (14) neglects the effect of an imperfect interphase on the conductivity of nanocomposites. Actually, equation (14) disregards the effective amount and effective converse aspect ratio of filler depending on “Y” in the case of an imperfect interphase in a nanocomposite. The effective terms can be applied to progress the Hu model for the conductivity of the graphene polymer system. When graphene nanosheets are adopted into equation (14), the length and radius of CNTs are replaced by the diameter and thickness of graphene nanosheets, respectively.

When the size of graphene nanosheets as well as “α _eff” (equation (7)), “ φ eff ” (equation (8)) and “ φ p ” (equation (13)) are considered in equation (14), the Hu model for the conductivity of polymer graphene nanocomposites is advanced to:

(15) σ = σ f 10 − 0.85 [ log ( α eff ) + 1 ] ( φ eff − φ p ) b ,

which presents the nanocomposite conductivity by graphene conductivity, filler aspect ratio, percolation threshold, tunneling size, and interphase depth.

3 Results and discussion

3.1 Analysis of the developed model by experimental data

The developed model is utilized to guesstimate the conductivity in the examples from previous studies. Three samples and their characteristics are shown in Table 1. The measurements of percolation threshold are fitted to equation (13) for calculating the interphase size, tunneling size, and interfacial conductivity. The calculations are reported in Table 1.

Table 1

Studied samples and their properties

No.	Systems	t (nm)	D (μm)	φ p	t _i(nm)	d (nm)	ѱ (S/m)	D _c (nm)	Y	b
1	PI¹/graphene [40]	3	5	0.0015	30	12	700	214	5.8	5.25
2	Epoxy/graphene [41]	2	2	0.0050	5.0	9.0	500	200	2.5	11.0
3	PVDF²/graphene [42]	1	2	0.0030	3.0	3.0	200	250	2.0	8.70
4	PS³/graphene [43]	1	2	0.0010	9.0	10	300	167	3.0	3.42
5	ABS⁴/graphene [44]	1	4	0.0013	3.0	3.0	400	125	8.0	5.40

The values of t, D, and φ p were reported from original references, but other parameters were derived from the developed model.

1: polyimide; 2: poly(vinylidene fluoride); 3: polystyrene; 4: acrylonitrile butadiene styrene.

The thickest interphase, the largest tunnels, and the uppermost interfacial conductivity are detected in the polyimide (PI)/graphene sample because it shows the lowest percolation threshold among the examples. Accordingly, the interfacial/interphase parameters and tunneling size primarily affect the percolation threshold. Also, the calculations indicate that different levels of interfacial/interphase and tunneling regions are observed in the reported examples due to various characteristics of the polymer medium, nanosheets, and processing methods.

In the next step, the values of “D _c” (equation (1)) and “Y” (equation (6)) can be estimated for samples (σ _f = 10⁵ S/m). The lowest “D _c” and the highest “Y” are observed in the ABS/graphene system. On the other hand, the biggest “D _c” and the weakest “Y” are shown in the PVDF/graphene nanocomposite because it exhibits poor properties for interfacial/interphase regions. The calculations of various factors are applied in equation (15) to foresee the conductivity of systems. The tested and forecasted values of nanocomposite conductivity are displayed in Figure 1. It is publicized that the forecasts follow the tested results at whole filler amounts. Thus, the innovative model based on interfacial/interphase parameters such as “ψ,” “Y,” and “t _i” as well as the tunneling distance can properly forecast the conductivity of samples. It can be concluded that the proper levels of interfacial/interphase properties can obtain a high conductivity, but a poor interface insignificantly improves the nanocomposite conductivity. The values of “b” are also presented in Table 1. The least and the highest “b” are observed in PS/graphene and epoxy/graphene samples, respectively. Generally, the conductivity of nanocomposites weakens by higher “b,” but the nanocomposite conductivity is linked to numerous factors like the filler amount, interfacial/interphase parameters, and percolation threshold.

Figure 1

Tested and estimated ranks of conductivity from equation (15) for (a) PI/graphene [40], (b) epoxy/graphene [41], (c) PVDF/graphene [42], (d) PS/graphene [43], and (e) ABS/graphene [44] nanocomposites.

3.2 Parametric examinations

Figure 2 shows the impressions of “t” and “σ _f” on “Y” calculated using equation (6) at D = 2 μm and ψ = 400 S/m. The highest “Y” obtained is 7 at σ _f = 0.5 × 10⁵S/m and t = 1 nm, but “Y” decreases to 0.2 at σ _f > 1.5 × 10⁵ S/m and t > 3 nm. As a result, both “t” and “σ _f” inversely control the “Y.” In other words, thin and poor-conductive nanosheets can cause a high conduction transfer between nanoparticles and the polymer matrix.

Figure 2

Calculations of “Y” from “t” and “σ _f” using equation (6) at D = 2 μm and ψ = 400 S/m: (a) 3-D and (b) 2-D pictures.

Thin nanosheets significantly decrease the converse aspect ratio and “D _c.” So, thin nanosheets can promote the interfacial/interphase properties in nanocomposites. It is meaningful because thinner nanosheets produce a larger interphase region in nanocomposites compared to thicker ones [45]. Accordingly, thin nanosheets positively govern the interfacial/interphase levels, which increase the conduction transportation between the polymer matrix and nanoparticles. Besides, poor filler conductivity decreases the “D _c” and can be completely transferred to the polymer medium even by incomplete interfacial properties producing high “Y.” However, the transportation of high filler conductivity to insulated medium needs a robust interphase, so the incomplete interfacial adhesion commonly decreases the conduction transfer. Consequently, the advanced model appropriately demonstrates the role of “σ _f” in “Y.”

The variations of “Y” at unlike points of “D” and “ψ” with t = 2 nm and σ _f = 10⁵ S/m are presented in Figure 3. The maximum “Y” as 8 is obtained for D = 4 μm and ψ = 900 S/m, whereas the least Y = 0.1 is observed for D < 1.5 μm and ψ < 300 S/m. So, both “D” and “ψ” directly govern the “Y.” In fact, large nanosheets and high interfacial conductivity achieve a high “Y,” but short nanosheets and poor interfacial conductivity deteriorate the “Y.”

Figure 3

Dependencies of “Y” on “D” and “ψ,” as calculated using equation (6) at σ _f = 10⁵ S/m and t = 2 nm: (a) 3-D and (b) 2-D designs.

Large nanosheets can produce a low converse aspect ratio creating a big interfacial region. Therefore, the diameter of nanosheets positively controls the interfacial region in nanocomposites. Since a good conduction transfer requires a large interfacial region, it is rational to get a bigger “Y” by larger nanosheets. The role of interfacial conductivity in “Y” is also distinct because high interfacial conductivity can transport much conduction to the polymer matrix. It can be said that the high conductivity of the interfacial region facilitates the transferring of conduction. However, a low interfacial conductivity limits the conveyance of conduction between nanoparticles and the polymer background. Consequently, the effect of interfacial conductivity on the conduction transfer is properly expressed by equation (6).

The influences of “Y” and “t _i” on the percolation onset (equation (13)) at D = 2 μm, t = 1 nm, and d = 5 nm are displayed in Figure 4. A low percolation threshold of 0.0045 is observed at Y > 5 and t _i > 8 nm, while Y = 1 and t _i = 2 nm produce the highest percolation level of 0.011. Thus, the high levels of “Y” and interphase depth yield a desirable percolation threshold, but a poor “Y” and thin interphase increase the percolation level.

Figure 4

Roles of “Y” and “t _i” in the percolation threshold (equation (13)) at D = 2 μm, t = 1 nm, and d = 5 nm: (a) 3-D and (b) 2-D schemes.

A great “Y” indicates the large level of “D” and/or low “D _c” based on equation (5). Really, a big range of conduction transfer shows the large diameter of nanosheets and/or good interfacial/interphase parameters. A high diameter of nanosheets decreases the converse aspect ratio, which diminishes the percolation threshold because the percolation level is directly correlated with the converse aspect ratio (equation (12)). Furthermore, a low “D _c” and a thick interphase reveal the proper interfacial properties of nanocomposites. As stated, the interfacial/interphase can participate in the network, since they surround the graphene and produce the net before the linking of nanosheets. The formation of a big and strong interfacial/interphase region in nanocomposites facilitates the percolation of nanosheets, and thus the percolation threshold occurs by a small number of nanosheets. Accordingly, “Y” and “t _i” inversely control the percolation threshold considering equation (13).

Figure 5 displays the estimates of nanocomposite conductivity (summarized as conductivity here) calculated using equation (15) at unlike ranks of “Y” and “ φ f ” (D = 2 μm, σ _f = 10⁵ S/m, t = 2 nm, t _i = 5 nm, d = 5 nm, and b = 8). Y = 9 and φ f = 0.03 produce the highest conductivity as 16 S/m. However, Y < 2 or φ f < 0.022 cannot transfer the conductivity. Therefore, the high ranges of conduction transportation and filler volume portion result in desirable conductivity.

$Figure 5 Approximation of conductivity (equation (15)) at altered series of “Y” and “ φ f {\varphi }_{\text{f}} ” by (a) 3-D and (b) 2-D pictures.$

Figure 5

Approximation of conductivity (equation (15)) at altered series of “Y” and “ φ f ” by (a) 3-D and (b) 2-D pictures.

A high “Y” demonstrates the significant extent of conduction transfer. So, the superconductivity of nanoparticles can be assigned to the insulated matrix in this condition, which mainly enhances the conductivity. Alternatively, a poor transportation of conduction among particles and medium cannot promote the conductivity, since the polymers are normally insulated. Additionally, a big filler amount upsurges the content of conductive phase in the nanocomposites, which improves the conductivity. More “ φ f ” than the percolation threshold establishes the net, which can transfer the electrons through nanocomposites. Therefore, the model normally unveils the stimuli of “Y” and “ φ f ” on the conductivity. However, too high quantity of particles cannot increase the conductivity because they cannot further change the size and density of established nets [46,47].

The inspirations of “α” and “σ _f” on the conductivity are displayed in Figure 6a ( φ f = 0.01, Y = 5, t _i = 5 nm, d = 5 nm, and b = 8). Large “α” and poor “σ _f” cannot change the conductivity, although the most conductivity of 0.0025 S/m is obtained by σ _f = 2.5 × 10⁵ S/m and α = 0.001. As a result, high conductivity is obtained by a poor converse aspect ratio and a high filler conductivity, but only a large converse aspect ratio deteriorates the conductivity.

$Figure 6 Correlation of conductivity to (a) “α” and “σ f” and (b) “ φ p {\varphi }_{\text{p}} ” and “b” from equation (15).$

Figure 6

Correlation of conductivity to (a) “α” and “σ _f” and (b) “ φ p ” and “b” from equation (15).

A low converse aspect ratio increases the effectiveness of the filler phase in nanocomposites because it causes good interfacial/interphase properties. Consequently, the inverse relation between nanocomposite conductivity and converse aspect ratio is meaningful. Additionally, although a high “σ _f” negatively governs the “D _c” and “Y,” it positively affects the charge transferring in nanocomposites. So, the filler conductivity directly changes the conductivity.

Figure 6b also elucidates the impressions of “ φ p ” and “b” on the conductivity (σ _f = 10⁵ S/m, D = 2 μm, t = 2 nm, Y = 5 and b = 8). The greatest conductivity of 6 S/m is gained at φ p = 0.001 and b = 5, but the conductivity principally weakens at b > 7.5. Consequently, the percolation threshold and “b” oppositely affect the conductivity. Moreover, “b” is more important than “ φ p ” in the conductivity because a high “b” significantly reduces the conductivity.

A low percolation level causes considerable conductivity for nanocomposites by little concentration of nanoparticles. Also, little percolation threshold enhances the dimensions of nets in the nanocomposites [48,49,50]. Accordingly, the conductivity improves by the poor percolation level. Many models implicitly reported the improvement of conductivity by a low percolation threshold [51,52]. Also, “b” inversely manipulates the conductivity because its high level reduces the extent of “ φ eff - φ p ” term in the model. Hence, the novel model rightly shows the impressions of “ φ p ” and “b” on the conductivity.

4 Conclusions

The extent of conduction transference from graphene nanosheets to the polymer matrix is expressed by “Y.” Furthermore, a new model was developed for the nanocomposite conductivity based on the properties of interphase and tunnels. The interphase properties and tunneling distance mostly manage the percolation onset. The calculations of the advanced model agree with the tested results. Hence, the proposed model can properly estimate the conductivity of nanocomposites by the interphase features such as “ψ,” “Y,” and “t _i” as well as the tunneling distance. Thin and large nanosheets, poor filler conductivity, and high interfacial conductivity cause a high conduction transfer between nanoparticles and the polymer matrix (Y). The high levels of “Y” and interphase thickness also yield a low percolation threshold. Additionally, the high “Y,” high filler volume portion, poor converse aspect ratio, high filler conductivity, poor percolation threshold, and low “b” result in more conductivity in the nanocomposites. The “Y” and filler volume portion significantly change the nanocomposite conductivity, but the converse aspect ratio and filler conductivity negligibly affect it.

Funding information: This work was supported by the Korea government (MSIT) (2022M3J7A1062940).
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Received: 2024-06-26

Revised: 2024-10-19

Accepted: 2024-11-23

Published Online: 2024-12-23

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/ntrev-2024-0131

Keywords for this article

graphene; nanocomposite; conduction conveyance; interface/interphase; electrical conductivity; percolation threshold

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