Modification of Maxwell model for conductivity prediction of carbon nanotubes-filled polymer composites with tunneling effect

doi:10.1007/s10483-025-3210-9

Abstract

Abstract:

Carbon nanotubes (CNTs) have garnered great attention in recent years due to their outstanding electrical, thermal, and mechanical properties. The incorporation of small amounts of CNTs in polymers can substantially improve the sensitivity of the polymer's electrical conductivity. This paper presents a modified Maxwell model to evaluate the electrical conductivity of CNTs-filled polymer composites by introducing a transition zone to account for the tunneling effect. In this modified Maxwell model, the CNTs-filled polymer composite is modeled as a three-phase composite, consisting of a matrix (polymer), inclusions (CNTs), and a transition zone (tunneling zone). The effective electrical conductivity (EEC) of the composite is calculated based on the volume fractions and electrical conductivities of the matrix, inclusions, and transition zone. The model's validity is confirmed through the use of available test data, which demonstrates its capability to accurately capture the nonlinear conductivity behavior observed in CNTs-polymer composites. This study offers valuable insights into the design of high-performance conductive polymer nanocomposites, and enhances the understanding of electrical conduction mechanisms in CNT-dispersed polymer composites.

Key words: carbon nanotube (CNT), polymer, composite, electrical conductivity, tunneling, Maxwell model

2010 MSC Number:

O242.2

Jue ZHU, Longyuan LI, Ningtao ZHU. Modification of Maxwell model for conductivity prediction of carbon nanotubes-filled polymer composites with tunneling effect. Applied Mathematics and Mechanics (English Edition), 2025, 46(1): 25-36.

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URL: https://www.amm.shu.edu.cn/EN/10.1007/s10483-025-3210-9

https://www.amm.shu.edu.cn/EN/Y2025/V46/I1/25

Figures/Tables 9

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Table 1

Parametric values used in the prediction model"

Parameter	Validation (see Fig. 6)	Validation (see Fig. 7)	Validation (see Fig. 8)	Note
$ρ / r$	0.35	0.35	0.35	Assumed, based on experimental data^*
$λ$	50	50	50	Assumed, based on experimental data^**
$V p$	9.0%	0.15%	0.08%	Taken from test data
$E i / (S ⋅ m − 1)$	1 800 (CFs), 180 (CNTs)	650	2 500	Assumed, based on test data
$E VTZ / (S ⋅ m − 1)$	$E i$ /200	$E i$ /200	$E i$ /200	Assumed
$E mm / (S ⋅ m − 1)$	$2.5 × 10 − 7$	$5.0 × 10 − 13$	$2.0 × 10 − 13$	Taken from test data
$β$	1.25	1.25	1.25	Assumed
* According to Refs. [4] and [5], multi-walled CNTs have diameters $2 r < 30$ nm, which gives $ρ = 0.35 r < 5.25$ nm
** Although straight CNTs have the aspect ratio $> 50$ nm, the CNTs mixed in polymers are generally in highly curved shapes, and in some cases, they are even tied into knots, which could reduce their aspect ratios

Table 1

Fig. 7

Fig. 8

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