Contents lists available at ScienceDirect Physica B: Condensed Matter journal homepage: www.elsevier.com/locate/physb Considering the filler network as a third phase in polymer/CNT nanocomposites to predict the tensile modulus using Hashin-Hansen model Sanghoon Kim a , Navid Jamalzadeh b , Yasser Zare c , David Hui d , Kyong Yop Rhee a,∗ a Department of Mechanical Engineering, College of Engineering, Kyung Hee University, Yongin 446-701, Republic of Korea b Mechanical Properties Research Lab, Faculty of Mechanical Engineering, K.N. Toosi Univeristy of Technology, No. 17, Pardis St., Mollasadra Ave., Vanak Square, Tehran, Iran c Young Researchers and Elites Club, Science and Research Branch, Islamic Azad University, Tehran, Iran d Department of Mechanical Engineering, University of New Orleans, LA 70148, USA ARTICLE INFO Keywords: Polymer/CNT nanocomposites Hashin-Hansen model Tensile modulus Percolation threshold Network modulus ABSTRACT In this paper, a conventional Hashin-Hansen model is developed to analyze the tensile modulus of polymer/CNT nanocomposites above the percolation threshold. This model for composites containing dispersed particles utilizes the aspect ratio of the nanofiller (α), the number of nanotubes per unit area (N), the percolation threshold (φ p ) and the modulus of the filler network (E N ), assuming that the filler network constitutes a third phase in the nanocomposites. The experimental results and the predictions agree well, verifying the proposed relations between the modulus and the other parameters in the Hashin-Hansen model. Moreover, large values of “α”, “N” and “E N ” result in an improved modulus of the polymer/CNT nanocomposites, while a low percolation threshold results in a high modulus. 1. Introduction Much research has focused on the development of high-performance materials for advanced applications by the addition of nanoparticles into polymer matrices. Among the various types of nanoparticles, carbon nanotubes (CNTs) have attracted extensive attention as a novel reinforcement for polymer nanocomposites since 1991 [1–11]. CNTs consist of a single or several graphite layers with small diameters (1–100 nm) and large lengths (1–10 μm). Additionally, CNTs exhibit a Young's modulus of 1 TPa, a tensile strength in the range of 10–50 GPa, and an exceptional electrical conductivity [12–14]. These properties, along with their remarkable physical dimensions such as their high aspect ratio, large surface area, outstanding mechanical behavior and good conductivity properties, suggest that CNTs may be used as a promising reinforcement in advanced nanocomposites. However, the van der Waals attraction between CNTs causes agglomerates to form during the synthesis procedure [15,16]. Polymer/CNT nanocomposites exhibit a high electrical conductivity when the volume fraction of CNTs is higher than the percolation threshold [17–19]. That is, the percolation threshold is the minimum concentration of nanofiller of the filler network that results in an ac- ceptable conductivity. Many authors have studied the percolation threshold as an important parameter in polymer nanocomposites [20–22]. One main concern is determining whether a similar percola- tion effect also exists regarding the mechanical properties of nano- composites. This has been confirmed in the positive, where a similar abrupt change was reported for the tensile modulus of polymer nano- composites by the addition of filler concentration [23–26]. Researchers have identified the percolation threshold for mechanical properties through experimental and theoretical approaches. Although such an abrupt change in the mechanical behavior cannot be entirely attributed to the electrical percolation threshold, mechanical percolation is found to be similar to electrical percolation in polymer/CNT nanocomposites [27]. From the theoretical point of view, several models have been sug- gested that express the electrical conductivity above the percolation threshold, including power-law functions of various parameters [28,29]. Ouali et al. [30] considered both the percolation effect and an inverse rule for the mixtures to model the tensile modulus of conven- tional composites. Various researchers have estimated the tensile modulus of polymer nanocomposites above the percolation threshold using the Ouali model [31,32]. However, these conventional models cannot accurately predict the percolation threshold using only the tensile modulus because they do not consider the more unusual prop- erties of nanofillers, such as their high aspect ratio, big surface area and networking above a certain concentration. As a result, the percolation https://doi.org/10.1016/j.physb.2018.04.036 Received 27 February 2018; Received in revised form 22 April 2018; Accepted 24 April 2018 ∗ Corresponding author. Giheung, Yongin, Gyeonggi 449-701, Republic of Korea. E-mail address: rheeky@khu.ac.kr (K.Y. Rhee). Physica B: Condensed Matter 541 (2018) 69–74 Available online 25 April 2018 0921-4526/ © 2018 Elsevier B.V. All rights reserved. T