Increasing the electrical conductivity of polymer nanocomposites under the external field by tuning nanofiller shape

Accepted Manuscript Increasing the electrical conductivity of polymer nanocomposites under the external field by tuning nanofiller shape Yangyang Gao, Fan Qu, Wencai Wang, Fanzhu Li, Xiuying Zhao, Liqun Zhang PII:

S0266-3538(19)30437-3

DOI:

https://doi.org/10.1016/j.compscitech.2019.03.025

Reference:

CSTE 7608

To appear in:

Composites Science and Technology

Received Date: 14 February 2019 Revised Date:

23 March 2019

Accepted Date: 30 March 2019

Please cite this article as: Gao Y, Qu F, Wang W, Li F, Zhao X, Zhang L, Increasing the electrical conductivity of polymer nanocomposites under the external field by tuning nanofiller shape, Composites Science and Technology (2019), doi: https://doi.org/10.1016/j.compscitech.2019.03.025. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Increasing the electrical conductivity of polymer nanocomposites under the external field by tuning nanofiller shape Yangyang Gao1,2,3, Fan Qu1, Wencai Wang1,2,3, Fanzhu Li1, 2,3*, Xiuying Zhao1, 2,3*, Liqun Zhang1, 2,3*

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Abstract

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Key Laboratory of Beijing City on Preparation and Processing of Novel Polymer Materials, Beijing University of Chemical Technology, 10029, People’s Republic of China 2 Beijing Engineering Research Center of Advanced Elastomers, Beijing University of Chemical Technology, 10029, People’s Republic of China 3 State Key Laboratory of Organic-Inorganic Composites, Beijing University of Chemical Technology, 10029, People’s Republic of China

It is very important to improve the electrical conductivity of polymer nanocomposites (PNCs), which can be used under the external field. In this work, by employing a coarse-grained molecular dynamics simulation, the effect of the external

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fields (tensile field and shear field) on the conductive probability of PNCs has been investigated in details by tuning the nanofiller shape. By carefully analyzing the conductive network, compared with the rod filler and the sphere filler, the Y filler and

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the X filler can protect the conducive networks well from their breakage

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perpendicular to the external fields, which thus can significantly enhance the conductive probability under the external fields. In other words, compared with under the quiescent state, the decrease or the increase of the conductive probability under the external fields depends on the nanofiller shape. Meanwhile, it is interesting to find that both the directional conductive probabilities parallel to the shear field and perpendicular to the shear field increase with the shear rate for the Y fillers, which *

Corresponding

author:

[email protected]

[email protected] 1

or

[email protected]

or

ACCEPTED MANUSCRIPT further reflects their high conductive probability under the shear field. The reason can be attributed to two aspects: (1) the gradual aggregation of Y fillers under the external field; (2) the multi-arm structures of Y fillers. In summary, this work presents some

external fields by tuning the nanofiller shape.

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further understanding how to improve the electrical conductivity of PNCs under the

Keywords: Electrical conductivity, Nanofiller shape, External field, Molecular

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dynamics simulation

1. Introduction 2

ACCEPTED MANUSCRIPT Conductive polymer nanocomposites (PNCs) consisting of an insulating polymer matrix and a conductive filler (such as carbon black (CB), carbon nanotubes (CNTs) and graphene) have attracted the extensive attention [1-4]. As the volume fraction of

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the fillers rises above the percolation threshold, a continuous filler network is formed throughout the PNCs, leading to a transition from the non-conductivity to the conductivity [5]. Generally, the morphology of fillers plays a key role in determining

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the percolation threshold, which depends on their size, shape, concentration,

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alignment and so on [6, 7]. To improve the electrical conductivity of PNCs, the central question is to understand how to control the percolation network formation at low filler concentration.

Recently, the intensive studies have been devoted to address the relationship

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between the experimental parameters and the electrical conductivity, with the aim of realizing better performance of PNCs. For instance, it is reported that the percolation threshold is heavily dependent on the aspect ratio of CNTs, which can be described by

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a model [8]. Meanwhile, the polymer-filler interaction affects the dispersion state of

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fillers in the matrix, which eventually influences the conductive network and the electric conductivity of PNCs [9, 10]. By constructing the three-dimensional conductive network, the high electrical conductivity of PNCs can be obtained [11]. Because of the necessity of manufacturing PNCs, the tensile and shear fields exert a great influence on the electrical conductivity. Under the tensile field, the gradual decrease of the electrical conductivity is attributed to the decrease in the contact number [12, 13] and the increase in the contact distance between CNTs [13, 14]. 3

ACCEPTED MANUSCRIPT Under the shear field, the increase [15, 16] or the decrease [17, 18] of the electric conductivity depends on the initial dispersion state of fillers. Meanwhile, the shear induced orientation of fillers leads to the high conductive anisotropy of PNCs [19].

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Especially, by utilizing the shear flow and the electrostatic interaction, it is very convenient for the alignment and assembly of the nanofiller in the matrix [20]. In general, the change in the electrical conductivity upon the external fields is often

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attributed to the conductive network. However, the experimental approach can not

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analyze the conductive network and explain the mechanism of the conductive percolation on the microscopic scale. Computer simulation provides another good choice to study the electrical conductivity behavior of PNCs. By employing a three-dimensional Monte Carlo model, the dependence of the percolation threshold on

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the aspect ratio of nanofiller can be described by an exponential relationship [21]. By employing a molecular simulation, the high polymer-filler interaction can benefit the formation of the tightly conductive network, which exhibits the maximum

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conductivity of PNCs [22]. In addition, the wavy fiber reduces its effective aspect

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ratio, which increases the percolation threshold [23, 24]. The average conductive pathway density of CNTs is found to dominate the anisotropy of the electrical conductivity of the aligned CNT/polymer composites [25]. Interestingly, the partial alignment of CNTs can result in the maximum electrical conductivity of PNCs in contrast to the complete alignment [26, 27]. Under the tensile deformation, the decrease in the electrical conductivity is attributed to a subtle change in the topological structure of the conductive network for the CNTs filled PNCs [28]. Under 4

ACCEPTED MANUSCRIPT the shear field, the orientation of CNTs enhances the electrical conductivity along this direction and reduces it perpendicular to the shear direction [29]. Above a critical shear rate, the conductive network will be broken down because of the CNT

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aggregation [30]. Moreover, the formation and the destruction of the conductive aggregate depends on both the shear rate and the CNT aspect ratio, which further influences the electrical conductivity [31, 32]. In addition, Zare et al. [33] proposed a

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simple model for calculating the electrical conductivity of PNCs, which takes the

CNTs) into consideration.

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crucial parameters (such as the interphase thickness, tunneling distance between

Currently, molecular junctions of CNTs are obtained by using the electron beam welding method [34] and the chemical vapor deposition methods [35-37]. By

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carefully controlling the irradiation condition, nanofillers with various shapes (such as rod CNT, Y CNT, X CNT) have been created as shown in Fig. 1(a). In addition, Y CNT and X CNT have been investigated in many aspects, such as mechanical

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properties [38, 39], mechanical stability [40], thermal conductivity [41], electron

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transport properties [42]. However, there is no report for the application of Y CNT and X CNT in the electrical conductivity of PNC because of their limited production. Thus, here we mainly intended to understand it theoretically, which can extend its application in the electrical conductivity in the future. For the application of PNCs, the high electrical conductivity of PNCs is necessary, especially under the external fields (tensile field and shear field). Thus, in this work, in order to improve the conductive probability, we investigated the effect of the nanofiller shape on the 5

ACCEPTED MANUSCRIPT conductive probability of PNCs under the external fields by employing the molecular dynamics simulation, which has not been investigated to our knowledge. As shown in Fig. 1 (b), four kinds of the nanofiller shapes are considered: rod filler, Y filler and X

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filler, and sphere filler. We focused on the relationship between the conductive network and the conductive probability, which can analyze the mechanism of the high conductive probability of PNCs under the external fields.

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2. Simulation details of model

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A coarse-grained model of the nanofiller filled polymer nanocomposites has been adopted in this work. A standard bead-spring model [43] is adopted to simulate the polymer chains, which consist of thirty beads. The number of chains is 1400 for each system. For the rod filler, Y filler and X filler, each filler contains ten beads. Each

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sphere filler contains one bead. In addition, each bead in the polymer chain and filler has the size of σ and the mass of m. It has been shown that the chains with this length exhibit the static and dynamic characteristic behavior of long chains in the

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matrix [43]. Each bond in this model corresponds to three to six covalent bonds along

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the backbone of a real chemical chain when mapping the coarse-grained model to a real polymer.

The non-bonded interactions between all the beads are described by a truncated

and shifted Lennard-Jones (LJ) potential, given by   4ε U (r) =   

where ε is the energy parameter,

  σ 12  σ  6    −     r     r  0

r < rcutoff

(1)

r ≥ rcutoff

r is the distance between two interaction sites, 6

ACCEPTED MANUSCRIPT rcutoff stands for the distance at which the interaction is truncated and shifted so that

the energy are zero at r = rcutoff . The polymer-polymer interaction parameter and its 1/6 cutoff distance are ε pp = 1.0 and rpp = 2*2 , respectively. The nanofiller-nanofiller

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interaction parameter and its cutoff distance are set to be ε nn = 1.0 and rnn = 1.12 , which makes sure the uniform dispersion of nanofiller in the matrix. The polymer-nanofiller interaction parameter and its cutoff distance are ε pp = 1.0 and

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rpp = 2.5 , which simulates the weak attractive interaction.

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A stiff finite extensible nonlinear elastic (FENE) potential [43] is used to describe the bond energy between the connected beads in the polymer chain and nanofiller, which is given by

VFENE = − 0.5 kR02 ln[1 − (

(2)

ε and R 0 =1.5 σ , guaranteeing a certain stiffness of the bonds while σ2

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where k=30

r 2 ) ] R0

avoiding high-frequency modes and chain crossing.

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The angle energy of the nanofiller is enforced by a bending potential, given by

Uangle = K(θ − θ0 )2

(3)

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where θ is the bending angle formed by three consecutive rod beads, K is set to be 10000. θ0 are set to be 180 ° for rod filler, 120 ° or 180 ° for Y filler, and 90 ° or 180 ° for X filler, respectively. Since we did not focus on any specific polymer, the reduced LJ units σ and ε are set to unity, which means that all calculated quantities are dimensionless. Similar to our previous work [44-46], first the NPT ensemble is adopted to compress the system for 5000 τ

with a periodic boundary condition in three directions. The 7

ACCEPTED MANUSCRIPT temperature and pressure are fixed at T* =1.0 and P* = 0.0 respectively by using the Nose-Hoover temperature thermostat

and

pressure barostat.

Then,

further

equilibration is performed under NVT ensemble with T* =1.0 for 50000 τ . It is noted

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that each chain has moved at least 2 R g during the long simulation process, where

R g is the root mean square radius of gyration of chains. Meanwhile, we have checked that polymer chains have experienced fully relaxed, reaching their

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equilibrated states for all systems. The final number density of the polymer beads is

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nearly 0.85, which is corresponding to its bulk state. The equilibrated structures are then used as starting structures for production runs of the structural and dynamical analysis. The velocity-Verlet algorithm is applied to integrate the equations of motion with a timestep of δ t = 0.001 , where the time is reduced by LJ unit time ( τ ). All MD

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runs are carried out by using the large scale atomic/molecular massively parallel simulator (LAMMPS) [47].

First, to determine whether the conductive network is formed, a criterion is used

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to check whether any two nanofillers are connected. Two nanofillers are considered to

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be connected if their shortest distance is smaller than the tunneling distance (TD), for which an electron may be transferred between them via quantum mechanical tunneling. TD can roughly stand for the contact conductance between a pair of the nanofillers [48]. The value of TD has not been estimated accurately either theoretically or experimentally. Here, we chose TD to be 1.0 σ by considering the sandwiched polymer chain [49]. It is noted that the value affects the absolute conductivity, but does not affect the relativity. At the beginning of the computational 8

ACCEPTED MANUSCRIPT implementation, each nanofiller is assigned a site number and a cluster number. The site number is equal to the cluster number, ranging from 1 to N, where N is the total number of nanofillers. Next, each nanofiller is checked for connection with others. If

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two nanofillers are connected with each other, a common cluster number assigns for them. Finally, all nanofillers with the same cluster number are in the same cluster. Otherwise, they are not connected. If the one-dimensional network is formed

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continuously from one side to another, the system is conductive in this direction. If

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the three-dimensional network is formed throughout the system, the system is homogeneously conductive in three directions. Finally, more than 10000 equilibrated configurations are dumped to perform the data analysis. The time interval between two continuous frames is 10 τ . Finally, the number of the configurations is counted,

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which is conductive in the one-dimensional direction or the three-dimensional directions. The conductive probability Λ , directional conductive probability Λ || parallel to the external field direction, and directional conductive probability Λ⊥

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perpendicular to the external field direction are used to represent the probability of

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forming the filler network, which spans the systems in the three-dimensional directions, parallel to the external field direction, and perpendicular to the external field direction, respectively. 3. Results and discussion 3.1 Conductive probability of PNCs In this work, four kinds of nanofiller shapes are considered in Fig. 1(b): (I) rod filler, (II) Y filler, (III) X filler, and (IV) sphere filler. The effect of the nanofiller 9

ACCEPTED MANUSCRIPT shape on the conductive network formation of PNCs under the quiescent state has been investigated in our previous work [50]. The results show that the percolation threshold gradually increases from the rod filler, the Y filler, the X filler to the sphere

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filler, which is attributed to the decrease of their mean square radius of gyration. In other words, the rod filler is the best candidate for the high conductive probability under the quiescent state. In fact, two kinds of the common external fields (tensile

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field and shear field) are necessary during the manufacturing process of PNCs, which

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will significantly influence the conductive network [13-15, 18]. Under the external fields, the original conductive network will be broken down and a new one is formed. It is very important to improve the conductive probability of PNCs for the practical application. Thus, here we intended to investigate the effect of the nanofiller shape on

shear field).

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the conductive probability of PNCs under the external fields (the tensile field and the

3.1.1 Effect of the tensile field

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First, we investigated the effect of the tensile field on the conductive probability

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of PNCs by fixing the maximum tensile strain α =150%. The glass transition temperature of our system is about 0.4-0.5, which is below the simulated temperature

T * = 1.0 . According to the Polymer Data Handbook [51], the rubbery materials are incompressible, which leads to its Poisson’s ratio equal to 0.5. This means that the box volume keeps the same during the tensile process. Thus, when the length of the simulation box increases to L0α in one direction ( L0 is the length of the undeformed box), the length decreases to the L0α −1/ 2 in the other two directions, 10

ACCEPTED MANUSCRIPT which make sure that the volume remains constant. The strain rate is set to be dα = ( L(t ) − L0 ) / L0 = 0.05 / τ , which is similar to others [52]. It is noted that the conductive probability can be affected by the box shape, which should be avoided in

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the simulations. Thus, the initial simulation box is set as rectangle with the same volume of the cubic one, and the cell parameters are determined based on the tensile strain. After the tensile deformation, the rectangular box is stretched into the cubic

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one. As a result, the conductive probability is simulated in the boxes with the same shape. To better understanding it, Figure S1 gives an example.

nanofiller volume fractions

ϕ

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Figure 2 presents the variation of Λ , Λ || , and Λ⊥ as a function of the for different nanofiller shapes by fixing the maximum

tensile strain α =150%. The dependence of Λ on

ϕ

exhibits a typical percolation

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phenomenon. In the percolated region, the conductive network is formed but is not completely developed. Thus, tiny variation of the nanofiller concentration can obviously change the conductive network. Here, the percolation threshold is defined

ϕc

at Λ =0.5 in Fig. 1(a) [53]. Figure 3 shows

for different nanofiller shapes under the quiescent state and under the tensile field.

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ϕc

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as the nanofiller volume fraction

Under the quiescent state, ϕ c gradually increases from the rod filler, the Y filler, the X filler to the sphere filler. Attributed to the large mean square radius of gyration of the rod filler, the conductive network is much easy to be formed in the polymer matrix. However, under the tensile field,

ϕ c first decreases and then increases from the rod

filler, the Y filler, the X filler to the sphere filler. Here, the difference of

ϕ c under

the quiescent state and under the tensile field is used to denote the change of the 11

ACCEPTED MANUSCRIPT conductive probability of PNCs, which is -0.2%, 0.3%, 0.2% and 0.01%. The positive value denotes that the tensile field improves the conductive probability, while the negative value denotes the opposite effect. As a result, PNCs with Y filler exhibits the

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highest conductive probability, which is reflected by the high positive difference of

ϕ c . To explain it, we tuned to Λ || and Λ⊥ because Λ depends on them. It is noted that two directional conductive probabilities perpendicular to the tensile

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direction are averaged as the directional conductive probability perpendicular to the

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tensile direction Λ⊥ because they are very similar. From Fig. 2(b), Λ || shows a continuous decrease from the rod filler, the Y filler, the X filler to the sphere filler, which is generally attributed to the nanofiller orientation along the tensile direction. From Fig. 2(c),

Λ⊥

shows a similar trend with

Λ because the conductive network

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is easy to be formed parallel to the tensile direction than perpendicular to the tensile direction. To further understand the results, we adopted the second-order Legendre polynomials < P2 > to denote the nanofiller orientation degree for different

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nanofiller shapes: < P2 >= (3 < cos 2 θ > − 1) / 2 , where θ

denotes the angle

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between the arm vector of nanofiller and the tensile direction. It is noted that the rod filler has one arm, the Y filler has three arms and the X filler has four arms. < P2 > of the sphere filler is considered to be 0. Figure S2 presents that < P2 > gradually decreases from the rod filler, the Y filler, the X filler to the sphere filler at ϕ = 4.30% . The low < P2 > of the Y filler and the X filler is attributed to their multi-arm structure. Meanwhile, the high orientation degree and the large aspect ratio of the rod filler is responsible for the large Λ || . Then we characterized the nanofiller dispersion state 12

ACCEPTED MANUSCRIPT under the quiescent state and under the tensile field by calculating the inter-nanofiller radial distribution function (RDF) in Fig. S3 (a) and (b). Compared with under the quiescent state, the peaks at r = 1σ , and 2σ gradually increase under the tensile

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field, which reflects their direct aggregation structure. It is noted that the difference is small for the sphere filler. These results reflect that the tensile field helps in the formation of the conductive network along the tensile direction. However, the change

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of the inter-nanofiller connection perpendicular to the tensile direction depends on the

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nanofiller shape. For the rod filler and the sphere filler, the nanofiller orientation induces the breakage of the inter-nanofiller connection perpendicular to the tensile direction. As a result, the increase induced by the nanofiller orientation can not make up the decrease by the breakdown of the conductive network, which reduces the

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conductive probability in Fig. 2(a). However, for the Y filler and the X filler, both the direct aggregation structure and the multi-arm structure of nanofillers can help to protect the conductive network well from the breakage perpendicular to the tensile

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direction, which enhances the conductive probability in Fig. 2(a). To further

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characterize the conductive network, the main cluster size Cn (the number of the nanofiller beads within the main cluster) and the total number of clusters Nc are calculated to analyze the conductive probability [22]. In general, the conductive network tends to be formed for the system with higher Cn and smaller Nc . As shown in Fig. 4, Cn

first increases and then decreases from the rod filler, the Y filler,

the X filler to the sphere filler in the percolation region. Nc shows an opposite trend with Cn , which is consistent with the Λ . Moreover, to observe the conductive 13

ACCEPTED MANUSCRIPT network, the snapshots of some typical systems ( ϕ = 4.30% ) are presented for different nanofiller shapes under the quiescent state and under the tensile field in Fig. 5. Under the tensile field, the nanofillers gradually orientate along the tensile direction,

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especially for the rod filler. Meanwhile, for the Y filler and the X filler the main cluster size Cn becomes larger under the tensile field compared with under the quiescent state, which is reflected by the increase in the number of red beads in Fig. 4.

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In total, under the tensile field, the original conductive network is broken down and a

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new one is formed. Attributed to the direct aggregation structure and the multi-arm structure, the Y fillers filled PNCs exhibit the highest conductive probability under the tensile field. 3.1.2 Effect of the shear field

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In this section, we mainly investigated the effect of the shear field on the conductive probability of PNCs. Here, the SLLOD equations [54] is adopted to realize the shear simulation, which is one of the most widely used methods for studying the

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shearing systems. In addition, we used the special Lees-Edwards “sliding brick”

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boundary conditions [55] for the SLLOD method, which is effective for the shear simulation. The shear is applied to the simulation box by moving top xy plane of the simulation box along the x direction. First, we investigated the effect of the nanofiller shape on the conductive

probability of PNCs by fixing the shear rate γ& = 0.1 . Change of Λ , Λ || , and Λ⊥ with the nanofiller volume fraction

ϕ

is shown in Fig. 6 for different nanofiller

shapes. Then, ϕ c are obtained and shown in Fig. 3 for different nanofiller shapes 14

ACCEPTED MANUSCRIPT under the quiescent state and under the shear field. Similarly, under the shear field,

ϕ c first decreases and then increases from the rod filler, the Y filler, the X filler to the sphere filler. The difference of ϕ c under the quiescent state and under the shear

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field is -0.41%, 0.51%, 0.32% and 0.08% for the rod filler, the Y filler, the X filler and the sphere filler, respectively. This proves again that the Y filler endows the highest conductive probability of PNCs compared with the other fillers. Similar to the results

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under the tensile field, the high orientation degree and the large aspect ratio of the

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nanofiller in Fig. S2 is responsible for the continuous increase of Λ || in Fig. 6(b) from the sphere filler, the X filler, the Y filler to the rod filler. By calculating the inter-nanofiller radial distribution function (RDF) under the quiescent state and under the shear field in Fig. S3(a) and (c), the shear field induces the higher peaks at

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r = 1σ and 2σ , which reflects the direct aggregation structure of nanofillers. The conductive network of the rod filler is broken down under the shear field. However, for the Y filler and the X filler, similarly, both the direct aggregation structure and the

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multi-arm structure of nanofiller can help to form a new conductive network, which

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enhances the conductive probability in Fig. 6(a). Then, the main cluster size Cn and the total number of clusters Nc are calculated to analyze the conductive probability in Fig. S4. Similarly, Cn first increases and then decreases from the rod filler, the Y filler, the X filler to the sphere filler in the percolation region. Nc shows an opposite trend with Cn . In addition, to observe the conductive network, the snapshots of some typical systems ( ϕ = 4.30% ) are presented for different nanofiller shapes under the quiescent state and under the shear field in Fig. 7. Actually, for the Y filler and the X 15

ACCEPTED MANUSCRIPT filler the main cluster size Cn (red beads) becomes larger under the shear field compared with under the quiescent state. As a result, the conductive network spans the systems perpendicular to the shear direction, which is responsible for the high

probability of PNCs significantly under the shear field.

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conductive probability of PNCs. In total, the Y fillers can improve the conductive

Last, we intended to investigate the intensity of the external field on the

ϕ c under the shear

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conductive probability of PNCs. It is noted that the difference of

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field is larger than that under the tensile field. This indicates that the shear field induces a more significant change in the conductive network compared with the tensile field. Thus, here we adopted the shear field by tuning the shear rate γ& for the Y filler where PNCs exhibits the highest conductive probability. The dependence of

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Λ , Λ || , and Λ⊥ on ϕ and γ& are shown in Fig. 8. It is very interesting to find that Λ , Λ || , and Λ⊥ all first show a continuous increase and then keeps unchanged with γ& . As shown in Fig. S5, the Y fillers tend to aggregate directly with

γ& , which is reflected by the high peaks at

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the increase of

r = 1σ and 2σ . With

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the multi-arm structure of the Y fillers, the new conductive network can span the system in three directions, which enhances

Λ. As a result, ϕ c gradually decreases

from 4.27%, 3.93%, 3.83%, 3.76%, 3.69% to 3.69% with the increase of

γ& , which is

shown in Fig. S6. These results indicate that the conductive probability of PNCs gradually increases with the

γ&

for the Y filler, which is contrary to the rod filler.

Similarly, as shown in Fig. S7, the trend of the Cn and the Nc with

γ&

is consistent

with Λ for different nanofiller shapes. At last, Figure S8 presents the snapshots of 16

ACCEPTED MANUSCRIPT the conductive network for different

γ& ( ϕ = 4.30% ). The main cluster size Cn

beads) becomes larger with the increase of

(red

γ& , which benefits the formation of the

conductive network and enhances the conductive probability of PNCs. The Y fillers

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are actually the suitable candidate for fabricating the PNCs with the high conductive probability. 3.2 Discussion

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It is noted that when mapping the coarse-grained model to the real polymers, the interaction parameter ε is about 2.5-4.0 kJ mol-1 for different polymers [43]. The

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persistence length is about 0.676 σ for this coarse-grained model of polymer chains. The range of the persistence length is between 0.35nm and 0.76nm for real polymers [56]. Thus, σ can be roughly corresponding to about 1 nm. Each bead with its

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diameter σ equal to 1 nm roughly corresponds to 5 repeating units of polyethylene as an example for that the carbon-carbon bond length equal to 0.154 nm. This leads to the mass of each bead equal to ≈ 140g/mol. Then, the time scale τ = σ m ε is ≈

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10ps. In addition, the rotational diffusion coefficient (Dr) is about 1.42*10-4 τ −1 for the

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rod filler. The shear rate the Peclet number ( Pe =

γ&

varies from 0.02 τ −1 to 0.5 τ −1 in our simulation. Thus,

γ&

) varies from 140 to 3500, which is within the range of

Drot

the experimental Peclet numbers [57]. Thus, our simulation is roughly reasonable. In addition, in experiments the attractive interaction between fillers will induce the direct aggregation, which leads to the breakage of the conductive network. To break the aggregation structure of fillers, an effective strategy is to functionalize the filler surface by grafted polymer chains [58-61]. As a results, the filler surface is covered 17

ACCEPTED MANUSCRIPT with a layer of polymer, which leads to the similar interactions for ε pp , ε np and ε nn . Thus, in this work, ε pp , ε np , and ε nn are set to be 1.0, which roughly can stand for the system with a good dispersion state of grafted fillers in the matrix.

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According to the present works, the effects of the filler shape [32, 62], the filler flexibility [23, 24, 63, 64], the filler dispersion [32, 65, 66], the filler alignment [29, 30, 67], the filler aspect ratio [8, 68], the polymer-filler interactions [9, 10] and the

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fabrication process [69] on the electrical conductivity have been investigated. In this

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work, our main purpose is to understand the mechanism how the nanofiller shape influences the conductive probability of PNCs under the external fields. It is noted that in experiments [7], the percolation threshold in the CNT filled PNCs ranges from 0.01 vol% to 4.75 vol% for different systems, which is a little smaller than our results

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(from 3.76% to 4.82%). To explain the difference, first the aspect ratio of CNTs and the length of polymer chains in experiments are much greater than those in this simulation because of the limitation of computational power. Secondly, the formation

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probability of the conductive network is adopted to denote the conductive probability,

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which depends on the tunneling distance. In addition, the conductive probability just varies from 0.0 to 1.0 (one order of magnitude). However, the electrical conductivity is adopted in experiments [7], which can change over more than 10 orders of magnitude near the percolation threshold. Thirdly, a loose conductive network is formed in the simulation, which leads to the relatively small difference of percolation threshold under the quiescent state and under the tensile field. However, the closely connected conductive network is formed in experiments. Thus, the sudden change in 18

ACCEPTED MANUSCRIPT the orientation degree of nanofillers [70-72] or the electrical conductivity [6, 73, 74] with the shear rate is observable at high shear rates in experiments. Last, the CNT or graphene filled PNCs [75] have a higher electrical conductivity than the CB filled

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PNCs [76] in experiments. Thus, in general the volume fraction of CB is larger than that of CNT or graphene. Meanwhile, both of their electrical conductivity and shape are not same. For example, CB has a much lower electrical conductivity than CNT

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and graphene. In our simulation, the rod filler filled PNCs also show a higher

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conductive probability than the sphere filler filled PNCs. However, only the nanofiller shape is not same in the simulation. In addition, the application of Y CNT and X CNT in the electrical conductivity of PNC has not been reported because of their limited production. Thus, further works should be done for better quantitative comparison

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between the experiments and the simulation in the future. In total, our simulation can roughly reflect the experimental systems. The main new phenomena revealed here are summarized below: (1) compared with in the quiescent state, the decrease or the

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increase of the conductive probability under the external fields depends on the

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nanofiller shape. (2) Compared with the rod filler and the sphere filler, the Y filler and the X filler can significantly protect the conducive network well from breakage and enhance the conductive probability of PNCs under the external fields. (3) It is interesting to find that both the conductive probabilities parallel to the shear field and perpendicular to the shear field increase with the shear rate for the Y filler, which reflects the high conductive probability of PNCs under the shear field. 4. Conclusions 19

ACCEPTED MANUSCRIPT In this work, we adopted a coarse-grained molecular dynamics simulation to investigate the effect of the nanofiller shape on the conductive probability of the nanofiller filled polymer nanocomposites (PNCs) under the external fields (tensile

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field and shear field). Four kinds of nanofiller shapes are considered: rod filler, Y filler, X filler, and sphere filler. Interestingly, the conductive network of the rod filler is broken down under the external fields. On the contrary, the Y filler, the X filler and

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the sphere filler can help to form the new percolation network, which can significantly

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enhance the conductive probability of PNCs under the external fields, especially the former two. As a result, the decrease or the increase of the conductive probability of PNCs under the external fields depends on the nanofiller shape. In addition, for the Y filler, it is interesting to find that both the conductive probabilities parallel to the shear

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field and perpendicular to the shear field increase with the increase of the shear rate, which is responsible for their high conductive probability. To explain it, we found that Y fillers form a direct aggregate structure with the increase of the shear rate.

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Meanwhile, the multi-arm structures of Y fillers can help to form the conductive

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network perpendicular to the external field direction. In summary, this work can provide a potential and simple method to improve the conductive probability of PNCs under the external fields by tuning the nanofiller shape. Acknowledgements

The authors acknowledge financial supports from the National Natural Science Foundation of China (21704003, 51673013), National 973 Basic Research Program of China 2015CB654700(2015CB654704), the Foundation for Innovative Research Groups of the NSF of China (51221002), the start-up funding of Beijing University of 20

ACCEPTED MANUSCRIPT Chemical Technology for excellent introduced talents (buctrc201710), and supported by CHEMCLOUDCOMPUTING of Beijing University of Chemical Technology are both greatly appreciated.

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Fig. 1 (a) Snapshots of nanofillers with four kinds of shapes: (I) rod carbon nanotube (CNT), (II) Y CNT, (III) X CNT, and (IV) sphere filler. (b) The coarse-grained model of nanofillers with four kinds of shapes. ( T ∗ =1.0)

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Table of Content Graphics Yangyang Gao1,2,3, Fan Qu1, Wencai Wang1,2,3, Fanzhu Li1, 2,3*, Xiuying Zhao1, 2,3*, Liqun Zhang2, 2,3*

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*

Corresponding

author:

[email protected]

[email protected] 34

or

[email protected]

or