A comprehensive evaluation of piezoresistive response and percolation behavior of multiscale polymer-based nanocomposites

Composites Part A 130 (2020) 105735

Contents lists available at ScienceDirect

Composites Part A journal homepage: www.elsevier.com/locate/compositesa

A comprehensive evaluation of piezoresistive response and percolation behavior of multiscale polymer-based nanocomposites ⁎

M. Haghgooa, M.K. Hassanzadeh-Aghdamb, , R. Ansaria, a b

T

⁎

Faculty of Mechanical Engineering, University of Guilan, Rasht, Iran Department of Engineering Sciences, Faculty of Technology and Engineering, East of Guilan, University of Guilan, Rudsar-Vajargah, Iran

A R T I C LE I N FO

A B S T R A C T

Keywords: Electrical conductivity Percolation threshold Piezoresistivity Multiscale nanocomposite (MSNC)

This article investigates electrical conductivity and piezoresistivity of carbon nanotube (CNT)-polymer nanocomposites using an efficient analytical model. The effects of chopped carbon fibers on the electrical conductivity and percolation behavior of multiscale polymer-based nanocomposites containing CNTs are examined at various maximum angular orientations and different polymer matrix barrier heights. The multiscale nanocomposite (MSNC) electrical conductivity and percolation onset are found to be dependent on the carbon fiber and CNT geometry and dispersion. The tunneling effect is discussed as an important mechanism to explain the low percolation threshold and nonlinear electric behavior of MSNC. A comparison between nanocomposites filled with CNTs and MSNC reinforced with CNTs and chopped carbon fibers demonstrates different percolation behaviors. Moreover, the influences of CNT position and orientation changes on the piezoresistivity of polymer nanocomposites are studied. Resistance change ratio as a function of applied strain demonstrates a non-linear behavior due to tunneling resistance change between CNTs.

1. Introduction Potential applications of nanocomposites as electrically conductive materials include electronic devices, photovoltaic cells and highly sensitive strain sensors [1–3]. The electrical conductivity of polymers can be increased by the addition of conductive fillers such as carbon nanotube (CNT) and carbon fiber for insulating polymers [4,5]. This is largely due to the high electrical conductivity and high aspect ratio of the carbon nanofillers, and the important involvement of electron tunneling [6,7]. Also, comparing with metal materials which have great electric properties, a polymer prepared with carbon nanofillers has good environmental adaptability and good compatibility with the matrix [8]. Furthermore, the discovery of CNTs with good electrically conductive behavior at low filler loadings has led to wide interest on the electrical response of CNT-reinforced nanocomposites [9,10]. Lisunova et al. [11] demonstrated that the method of composite preparation affects the values of percolation threshold. They studied the percolation behavior of multi-walled carbon nanotube (MWCNT)polymer nanocomposite prepared by mechanical mixing in a powder and in a hot-pressed compacted state. Their data showed the presence of percolation threshold at a very small volume fraction (0.0004–0.0007) of the MWCNTs. They explained this phenomenon by high aspect ratio of the MWCNTs and their segregated distribution ⁎

inside the polymer matrix. High aspect ratio conductive nanofillers like CNTs are relatively expensive because of the necessity for extra treatments and purification. Therefore, nonofiller content that is demanded for a conductive nanocomposite should be minimized as much as possible. Jang et al. [12] exhibited the weak electron transport through the composite when CNTs were sparsely dispersed in the matrix which fabricated MWCNT/PDMS sensing composite using the solution casting method. They also incorporated the eight-chain model and Simmons′ equation for contact resistance [13] into a Gaussian statistics-based formulation to calculate the effective length of a single CNT well dispersed in the matrix. They found that considering waviness and good desperation for CNTs led to a great agreement with their experiments. Due to the large ratio between the conductivities of CNTs and matrix, there is a concentration threshold above which the nanocomposite is said to be conductive [14]. The conductive fillers need contact each other to construct connecting paths in the matrix. The process of forming connecting paths can be well explained by the percolation theory, and the critical point is defined as percolation threshold [4]. Percolation threshold is the transition point from insulator to conductor when a certain volume fraction of conducting fillers is reached. The introduction of CNTs into polymers leads to lower percolation threshold which is originated from high aspect ratio and conductivity of CNTs. In particular, any parameter that can change the inter-filler distance will

Corresponding authors. E-mail addresses: [email protected] (M.K. Hassanzadeh-Aghdam), [email protected] (R. Ansari).

https://doi.org/10.1016/j.compositesa.2019.105735 Received 29 September 2019; Received in revised form 25 November 2019; Accepted 17 December 2019 Available online 18 December 2019 1359-835X/ © 2019 Elsevier Ltd. All rights reserved.

Composites Part A 130 (2020) 105735

M. Haghgoo, et al.

included the CNTs′ piezoresistivity influence in a multiscale way. They calculated the strain of an embedded CNT by applying a mechanical tensile strain. Strain and CNT transformation provided the necessary information at the atomic scale to incorporate the internal network change of CNTs. They found that the tunneling of conductive network plays a major role in the nanocomposite piezoresistivity. In this work, electrical conductivity and piezoresistivity of the CNTpolymer nanocomposites as well as electrical conductivity of the carbon fiber/CNT-polymer MSNC are comprehensively studied. Combination of two types of conductive fillers, chopped carbon fiber and CNT forms the conductive network. This combination is used to maximize the conductivity of the MSNC. The objectives in the present study are to investigate the effects of the carbon fiber electrical conductivity and barrier height of matrix influences on the electrical and percolation conductive behavior of multi-inclusion nanocomposites.

influence the percolation threshold [15]. After percolation threshold, the electrical properties of the material exhibit a non-linear behavior and electron tunneling is the dominant mechanism for transferring charge within the matrix. The tunneling distance and surface area of conductive fillers influence the percolation threshold value [16,17]. Increasing demands of the electronic industry cannot be met by high filler loading in polymer. Developing hybrid fillers with considerable advantages are a necessity. The selection of conductive fillers is important to produce a highly conductive nanocomposite since different conductive fillers have their own unique properties [18]. Incorporation of CNT and carbon fiber into the polymer materials results in increasing the electrical conductivity of multiscale polymer-based nanocomposites by generating more conductive paths. The combination of CNT and carbon fiber presents uniquely different aspect ratios and dispersion characteristics. Thus, incorporation of two conductive fillers into polymer enhances the conductivity and improves the material percolation behavior due to a double percolation phenomenon. Although the percolation threshold of CNTs is very low due to their high intrinsic conductivity and high aspect ratio, the high cost obstructs their large scale utilization. Another reason that limits employing CNTs as conducting fillers is difficulties related with the dispersion of MWCNTs during proceeding; particularly when the CNT volume fraction is higher than 0.005 [17]. Large aspect ratio CNTs stand as ideal candidates for fillers in piezoresistive materials. Aligning them in a preferential direction may increase their piezoresistive sensitivities [19]. Electrically conductive CNT-polymer nanocomposites can be applied to various fields, such as piezoresistive or resistance-type strain sensors for human motion, soft skins, smart wearable sensors and robotics, whose main requisite is large deformation [20]. Piezoresistive is a passive phenomenon that is defined as the coupling phenomenon between a mechanical stimulus and a change in electrical resistance and goes in the mechanical to electrical direction [21].The piezoresistivity of CNT-polymer nanocomposites is linked to three major mechanisms: (I) the inherent piezoresistivity of CNTs (II) the deformation of conductive paths that they form within the polymer (III) the change of inter-filler distances, influencing the electron tunneling between adjacent CNTs [22]. Aspect ratio of CNT plays a very important role in the piezoresitive response of nanocomposite. High aspect ratio CNTs provide better stress transfer within the composite and yields very low percolation threshold. This is related to the increase of contact points for higher aspect ratio CNTs [19]. Several experimental studies were conducted to understand the effects of material properties on piezoresistivity of composites. Hu et al. [3] fabricated a strain sensor from a polymer nanocomposite with MWCNTs. They reported that much higher sensitivity can be obtained in the nanocomposite sensors when the volume fraction of CNT is close to the percolation threshold. They postulated that the tunneling effect is the principal mechanism of the sensor under small strain. Aviles et al. [23] investigated the electromechanical properties of vinyl ester polymer nanocomposites comprising three carbon nanostructures of different topologies (MWCNT, few-layer graphene oxide flakes and CVD synthesized three dimensional graphene shells). They found the highest mechanical properties for composites containing MWCNT and graphene oxide flakes. This combination yielded the highest piezoresistive sensitivity. They reported better electrical conductivity by combination of MWCNT and cubic shells. They concluded that the volume of the filler within the composite played an important role in the conductive network formation. Several analytical models have been worked to model the effects of nanoscale fillers on the piezoresistive behaviors of nanocomposite materials. Park et al. [24] suggested a resistance change model to describe the combination of linear and non-linear modes of electrical resistance change as a function of strain. These linear and nonlinear regions were the overall pattern of electrical resistance change versus strain for specimens of their experimental study. Hu et al. [25] proposed a comprehensive multiscale three-dimensional model to predict the piezoresistivity behavior of a nanocomposite material. They

2. Theoretical approach Due to the vast difference between the spatial scales and electrical conduction mechanisms associated with these two sets of fillers, the strategy to determine the effective electrical conductivity of carbon fiber/CNT-polymer MSNC is considered into two steps. In the first step, the nanocomposite is composed of two phases, the polymer matrix and the CNT. CNTs are randomly distributed in a representative volume cuboid with dimensions of L x × L y × Lz . The location of the first point and azimuthal and polar angles (φi , θi ) of the i th CNT are generated randomly as [26]

(x i0 , yi0 , z i0) = (rand × L x , rand × L y , rand × Lz )

(1)

φi = 2π × rand

(2)

cosθi = (1 − cosθf ) × rand + cosθf

(3)

where i is the index of i -th CNT and ‘rand’ denotes uniformly distributed random numbers in the interval {0: 1} and θf is the maximum CNT alignment angle. Due to the highly anisotropic properties of CNTs, the electrical properties of the CNT-polymer composites strongly depend on the orientation of the CNTs. Fibers overlap each other by increasing the average fiber orientation [27]. To evaluate orientation effects, anisotropy can be introduced in this model by restricting the orientation angle of any filler between zero and θf . If θf = 0 , all fillers are aligned and this represents a fully anisotropic case. Increase in the degree of anisotropy of CNTs results in a lower electrical conductivity, especially when θf < 30° . Whereas, θf = 90 indicates an isotropic case where percolation threshold gets stabilized at a comparatively lower value. This result can be explained by considering the average number of contacts each CNT makes with others. In a fully anisotropic configuration θf < 10° an end-to-end connection is demanded [28]. With increasing degree of random orientation, the probability of CNTs overlapping each other becomes larger. Since the desired electrical conduction is assumed to be along X axis, the CNT orientation is measured in terms of the angle with respect to X axis. The ending point of the CNT can be determined accordingly [29].

(x i1, yi1 , z i1) = (x i0 + li sinθi cosφi , yi0 + li sinθi sinφi , z i0 + li cosθi )

(4)

where li is the length of the i -th CNT. After determining the position and orientation of CNTs, the shortest distance between each pair of CNTs is determined. If the shortest distance between CNTs is less than the effective distance of tunneling (cutoff distance), the pair is in contact. The electrical resistance of CNT network mainly consists of the tunneling resistance of electron tunneling at CNT joints while CNT intrinsic resistance is very low comparing tunneling resistance. The intrinsic resistance of a filler is given by Drude model [30]

Rf = 2

4l πD 2σf

(5)

Composites Part A 130 (2020) 105735

M. Haghgoo, et al.

where l , D and σf are the filler length, diameter and electrical conductivity, respectively. At this stage, contact resistance is introduced by assuming tunneling resistance at CNT (conductor) joints. Electron density related to a single state in a conductor of length L is e / L . Thus, the current carried by the k state can be introduced by [31]

I=

Rc =

σ = σ0 +

∑ ℏ ∂k f (E )

(6)

k

πD Rtun

L

+∞

∫ f (E ) M (E )dE

(7)

−∞

ξ=

2e 2 (μ1 − μ 2 ) I= M h e

fp =

(

(

8me φ ℏ

)

da2 h a 2e 2

exp

(

4πda 2me φ h

M

(17)

⎟

(19)

Substituting Eqs. (5) and (14) into Eq. (19) leads to the expression for the electrical conductivity of nanocomposite as follows

⎡ ξ f ⎢ σ =1+ σ0 3 ς 2σ0 ⎢ 4l + ⎣

(10)

4l πh2d

a

e 2 2mφ

exp

(

4πda h

⎤ ⎥ 2mφ ⎥ ⎦

)

(20)

3. Multiscale carbon fiber/CNT-polymer nanocomposites In the second step, carbon fibers are randomly distributed in the CNT/polymer nanocomposite matrix and the effective electrical conductivity of carbon fiber/CNT-polymer MSNC is estimated. The effective electrical conductivity of carbon fiber/CNT-polymer MSNC considering the carbon fibers as microscopic inhomogeneities is presented. The RVE of carbon fiber/CNT-polymer MSNC with randomly distributed carbon fibers in the polymer matrix was shown in Fig. 1(a). After calculating the electrical conductivity of the CNT-polymer nanocomposite and considering it as matrix, the electrical conductivity of carbon fiber/CNT-reinforced polymer composite system can be calculated. As depicted in Fig. 2(b) and (c), two electrodes are separated by an insulating film, the potential barrier height of insulating matrix obstructs the current flow between the electrodes. Penetrations by the electric tunnel effect outflow the current through the insulating region between the two electrodes if the electrons could mount the potential barrier. By reducing the thickness of the insulating film, the image potential increases the flow of current between electrodes. The image potential Vi can be expressed by using mirror image method as [34]

(12)

As shown in Fig. 2(d), da = 2t = d − D is the distance between the surfaces of neighboring CNTs and t is the thickness of interphase [35]. Assuming the contact area after percolation threshold as a = da2 , R c is obtained by

Rc =

⎠⎦

(18)

⎜

(11)

h 1 2e 2 M . exp − da

⎝2

ξ f ⎛ σ 4l ⎞ σf + =1+ σ0 3 ς 2σ0 ⎝ πD 2R c ⎠

here m (9.10938291 × 10−31kg) is the mass of the electron, and φ (eV) is the potential barrier height originated from the insulating polymer between conductive fillers. From Eqs. (9)–(11), contact resistance can be determined as:

Rc =

)

2 3 ⎛ l/ς ⎞ ⎤ 32 ⎜ D + t ⎟ ⎥

when waviness of CNT is considered, the model is expressed by Eq. (19)

where dt is tunneling distance given by [29]

dt = ℏ/ 8mφ

3⎡ 3⎛ l/ς ⎞ + t ⎢1 + 4 ⎜ D ⎟ + ⎝ 2 +t⎠ ⎣

ς = l/ leff

)

)

(

D 2

π

+ 6 D3

here ς is the CNT waviness. The effectiveness of CNT decreases with waviness which is caused by CNT large aspect ratio. The distance between two ends is much less than the CNT′s length. The waviness factor is evaluated as the actual length divided by effective length. Effective length is expressed as the minimum distance between two ends of each CNT as depicted in Fig. 2(d) [39]

The factor τ is the transmission probability of one electron from one end of the conductor to the other end. Considering dvdW as the Van der Waals distance between the surfaces of neighboring CNTs and d as the shortest distance between the axis of neighboring CNTs, transmission probability can be evaluated as [26]:

(

π 2 Dl 4 32 π 3

(9)

d ⎧ exp − vdW 0 ≤ d ≤ D + d vdW ⎪ dt ( ) − d − D ⎨ exp D + d vdW < d ≤ D + d cutoff ⎪ dt ⎩

(16)

Percolation threshold can be expressed by [38]

(8)

h 1 2e 2 Mτ

1 3

1 − fp

where μ1 and μ 2 are the electrochemical potentials of high and low states which are plotted in Fig. 2(a). The contact resistance (tunneling resistance) at CNT joints relates to the distance between neighboring CNTs. Assuming that the number of conduction channels is constant over the energy range μ 2 < E < μ1 and incorporating the Landauer-Buttiker formula with a rectangular potential barrier, the contact resistance can be expressed as [34]:

τ=

1

1

f 3 − f p3

here the function M is related to the number of channels at energy level E depending on CNT wall types. The current carried per channel 2e number is equal to h , (h = 2π ℏ) . The current flowing at any point is given by [31]

Rc =

(15)

conductivity of extended filler. The symbol ξ relates to the percentage of filler belonging to network phase after percolation and f is the CNT volume fraction and will change from 0 to 1 while CNT volume fraction increases from percolation threshold. After percolation threshold, a certain number of fillers incorporate in the conductive networks represented by Eq. (16)

the sum over k into an integral ∑k , →2 × 2π ∫ dk and extended the result to multi-channel M conduction, the current can be given as [33]

2e h

ξfσe 3

where σ0 accounts for the matrix electrical conductivity. The small 4l value of σ0 can be ignored. The factor σe ≡ σf + 2 is the electrical

where e (−1.602176565 × 10−19 °C) is the electron charge and ℏ(6.626068 × 10−34m2kg/s) is the Planck′s constant [32]. In addition, 1 ∂E is electron velocity in the x-direction over the barrier. Converting ℏ ∂k

I=

(14)

Deng and Zheng suggested a simple model for electrical conductivity of polymer composite as [37].

1 ∂E

e L

da h2 4πda exp ⎛ (2mφ)1/2⎞ ae 2 (2mφ)1/2 ⎠ ⎝ h

) (13)

From Eq. (13) and assuming contacted CNTs are separated at their tunneling distance, the contact resistance between two neighboring CNTs can be evaluated as [36] 3

Composites Part A 130 (2020) 105735

M. Haghgoo, et al.

Fig. 1. (a) Carbon fiber/CNT-polymer multiscale nanocomposite and (b) strain effect on the RVE.

−e 2 ⎞ ⎡ x Vi = ⎛ ⎢ + 4 ⎝ πKX ⎠ ⎣ 2 ⎜

⎟

∞

nX

∑ ⎛ (nX )2 − x 2 ⎜

n=1

−

⎝

1 ⎞⎤ 1.15e 2X 2 ln2 ≈− nX ⎠ ⎥ 8πKxX (X − x ) ⎦ ⎟

Fig. 2. (a) Difference of potential in two electrodes, (b) CNT dispersion inside electrode plates with minimum orientation angle, (c) maximum orientation angle, (d) conductive network and (e) shortest distance between two neighboring CNTs.

(21) where K is the dielectric permittivity of the matrix, x is the distance from the surface of the electrode, X is the insulating layer thickness (in angstrom). Considering the net flow of electrons from lower potential electrode to higher potential electrode, the potential barrier has to be modified considering the effects of image potential [13] as follows

φ¯ =

1 Δs

s2

x φ0 + (Δφ − eV ) + Vi dx X

∫{ s1

}

s1 =

46 ⎤+ 6 s2 = X ⎡1 − ⎥ ⎢ 3 φ KX 20 2 VKX Kφ0 + − 0 ⎦ ⎣ (22)

(24)

(25)

where V is the applied voltage across the thin film [41] given by

here Δs stands for the difference between the limits of the barrier at Fermi level. Integrating yields [40]

V 5.75 ⎤ ln ⎡ s2 (X − s1 ) ⎤ φ¯ = φ0 − ⎛ ⎞ (s1 + s2) − ⎡ ⎢ ⎝ 2X ⎠ ⎣ K (s1 − s2 ) ⎥ ⎦ ⎦ ⎢ ⎣ s1 (X − s2 ) ⎥

6 Kφ0

V=

eX Ac K

(26)

The contact area at the overlapping position Ac is approximated by Ac = D 2 . The effective resistance of a carbon fiber originates from the intrinsic resistance of the conductive fillers and the tunneling resistance as follows

(23)

where φ0 is the height of the rectangular potential barrier which can be approximated as the potential barrier height of polymer, s1 and s2 stand for the limits of the barrier at Fermi level [40]

R e = Rf + Rtun 4

(27)

Composites Part A 130 (2020) 105735

M. Haghgoo, et al.

the main conductive process [46] given by

The tunneling resistance can be defined as

Rtun

4ρdt = πD 2

R (ε ) =

(28)

here D and ρ are the carbon fiber diameter and CNT-polymer matrix

4(l + ρdtun σf ) πD 2σf

(29)

The electrical conductivity of an extended carbon fiber is estimated 4l by replacing R e into σe = 2 , as

ΔR R − R0 1−v d 1−v ⎞ ε ⎞ exp ⎛ 0 ε −1 = = ⎛1 + R0 R0 2 ⎠ ⎝ ⎝ dt 2 ⎠

πD R e

σe =

⎜

lσf (l + ρdtun σf )

(30)

To validate the analytical model, the predictions are compared with the experimental data [10] for MSNC reinforced with CNTs and chopped carbon fiber (CF). The MWCNTs′ diameter and length are 10 nm and 2 μm, respectively. The diameter and length of carbon fiber were 0.11 μm and 4 μm, respectively. Height of barrier is approximately taken as 5eV . For CNTs with a mutual distance between 0.47 and 1.4 nm, the tunneling effect is considered. Fig. 3 shows a comparison between the two sets of results. It is observed that the analytical prediction agrees well with the experimental result [10]. It should be noted that for the current theoretical modeling, CNTs are assumed to be uniformly dispersed in the polymer. With individual CNT considered as a conductor, charge conduction between CNTs may be considered by the quantum tunneling of electrons. Fig. 4 shows the effect of matrix height of barrier on the electrical conductivity of CNT-polymer nanocomposites. As the volume fraction of CNT is increased, the electrical conductivity of the composite increases. From the figure, nanocomposite exhibits a percolation threshold at about 0.002 vol fraction of the MWCNTs. When the CNT volume fraction is close to this value, the CNT-CNT conduction joints begin to create. At this moment, an only increase of small volume of CNT will result forming conduction channels, leading to produce the sudden change of electrical conductivity. The impact of barrier height on the electrical conductivity of CNT-polymer nanocomposites is investigated by changing the barrier height from 5eV to1.5eV . After the

(31)

4. Nanocomposite volume expansion For most MSNC, mechanical response arises from the properties of its different phases in addition to the interaction between them [42]. As a consequent, different multiscale modeling techniques have been developed to predict the continuum properties of MSNC in two stages including the determination of orthotropic elastic properties of CNT and polymer combination and considering this phase as reinforcement and the polymer as the matrix phase to calculate the bulk effective properties of scaled up nanocomposite [43,44]. Once the percolation threshold has been recognized, a proper formulation to study the relationship between resistivity and mechanical deformation is advisable. CNT-polymer nanocomposite experiences a uniaxial strain as shown in Fig. 1(b). CNT positions are changed in accordance with the total strain. The response of CNTs to the applied strain is modeled by the widely accepted fiber re-orientation model. The re-orientation model assumes that the midpoint (x ic , yic , z ic ) of i -th CNT is transferred with the magnitude of total strain and the Poisson′s ratio of the composite [45].

(x ic , yic , z ic ) = (x ic + x ic Δε , yic − νyic Δε , z ic − νz ic Δε ).

(39)

5. Results and discussion

ξflσf 3(l + ρdtun σf )

⎟

where v is the Poisson′s ratio.

Substituting the electrical conductivity into Eq. (15) leads to a model for conductivity of carbon fiber-CNT-reinforced polymer-based nanocomposites, considering the tunneling regions as follows

σ=

(38)

where R (ε ) is the resistance of the nanocomposite, n is the number of CNTs in a single conductive path and N is the number of conductive path in conductive network. The average distances between two neighboring CNTs change from d 0 to d with the applied strain ε . The relative change of resistivity will change with applied strain due to the demolition and construction of conductive paths. The relative change of resistivity caused by the change of the distance between neighboring CNTs can be written as [24]

2mφ¯

tunneling resistivity, ρ = e2M (Ω. m) , respectively. Substituting Rf and Rtun into Eq. (27) gives the total resistance of an extended carbon fiber as

Re =

4n dh exp(dt d ) 3N D 2e 2dt

(32)

Then, the starting and ending points of i -th CNT after strain can be expressed as

(x i0 , yi0 , z i0) = (x ic , yic , z ic ) − [(x ic , yic , z ic ) − (x¯i0 , y¯i0 , z¯i0)] × (li / li )

(33)

(x i1, yi1 , z i1) = (x ic , yic , z ic ) + [(x i1, yi1 , z i1) − (x ic , yic , z ic )] × (li / li )

(34)

where

(x¯i0 , y¯i0 , z¯i0) = (x i0 + x i0 Δε , yi0 − yi0 ν Δε , z i0 − z i0 ν Δε )

(35)

(x¯i1, y¯i1 ,

(36)

li =

z¯i1)

=

(x i1

+

x i1 Δε , yi1

−

yi1 ν Δε ,

z i0

−

z i1 ν Δε )

(x i1 − x i0 )2 (1 + Δε )2 + [(yi1 − yi0 )2 + (z i1 − z i0 )2](1 − ν Δε )2 . (37)

The positions of CNT junctions will be changed due to the applied strain. After calculating the new position and dimension of the representative CNT to the RVE, the new distance between CNT pair is measured. If minimum distance between the pair is higher than d vdw + D and smaller than dcutoff + D , then the pair are in contact and tunneling resistance is present among them. A simple tunneling model is used to compute the piezoresistive response of conductive nanocomposite. The model considers the electron tunneling mechanism as

Fig. 3. Comparison between analytical study and experimental data for chopped carbon fiber-CNT-reinforced polymer multiscale nanocomposite electrical conductivity. 5

Composites Part A 130 (2020) 105735

M. Haghgoo, et al.

Fig. 4. Electrical conductivity versus CNT volume fraction of CNT-polymer nanocomposite under different values of height of barrier.

Fig. 6. Electrical conductivity versus carbon fiber volume fraction curves of carbon fiber/CNT-polymer MSNC under different values of CNT volume fraction.

percolation threshold, the addition of a few percent of CNT, the electrical conductivity of composite approaches a limiting value and remains within the same order of magnitude with further addition of CNT into the matrix. It is known that the conduction in a tunnel junction decreases with the barrier width [47] as a result of energy barrier of insulting matrix that impedes electrons to move from one electrode (nanofiller) to another. However, when the barrier is removed by tunneling, there is a driving force for the electrons to move across the barrier. Fig. 5 demonstrates the CNT level of alignment influence on the overall electrical conductivity of CNT polymer nanocomposites. Anisotropy is introduced by restricting the alignment of the CNTs to a range of angular orientation θ . A small value of θ indicates a high degree of anisotropy while isotropic distribution corresponds to θ = 90° . It is found that the CNT level of alignment has a significant effect on the nanocomposite electrical conductivity. As can be seen from the figure, increasing CNT level of alignment would lead to a decrease in the electrical conductivity of the nanocomposite. Such a phenomenon can be explained by the reduction of conductive paths due to aligned CNT arrangement and large gap between the CNTs. In an anisotropic configuration (θ < 10°) , most CNTs will be aligned. End to end contacts will be required to provide demanded conduction and the probability of electron tunneling is very small [28]. These results suggest that the

electrical conductivity of CNT-polymer nanocomposites can be increased by dispersing CNTs in randomly oriented configuration. Whereas, θ < 90° indicates an isotropic case where the percolation threshold gets stabilized at a comparatively lower value. Similar observations of higher conductivity measured in non-aligned CNT-reinforced nanocomposites have been reported by Du et al. [14]. The electrical conductivity of MSNC is investigated in Figs. 6–10. CNT volume fraction is held at 0.005. Within a polymer matrix, a continuous channel may get formed by both CNT and carbon fiber paths. In calculations, the electrical conductivity of carbon fiber and CNT are taken to be 625S/cm and 11,300 S/cm, respectively. Also carbon fiber critical electron tunneling (cutoff) distance is taken 0.01μm [48]. We previously observed that the percolation of CNT nanocomposite happened around 0.002 vol fraction. In Figs. 6-10, a drastic second boost in electrical conductivity can be noted when carbon fiber volume fraction reaches around 0.02 vol fraction. The second increase results from the formation of large number of percolation channels that can be easily formed by the thin CNT connecting carbon fibers. Fig. 6 shows the predicted electrical conductivities of MSNC versus carbon fiber volume fraction for different CNT volume fractions. Because the CNT diameter is in the nanometer scale and carbon fiber

Fig. 5. Electrical conductivity versus CNT volume fraction curves of CNTpolymer nanocomposite under different values of CNT maximum off-alignment angle.

Fig. 7. Electrical conductivity versus carbon fiber volume fraction curves of carbon fiber/CNT-polymer MSNC under different values of carbon fiber aspect ratios. 6

Composites Part A 130 (2020) 105735

M. Haghgoo, et al.

diameter is in the micrometer scale, the specific surface area of the CNT is much greater than that of carbon fibers. This is beneficial for CNT to transform the electrical conductivity even in low volume fraction. As a result, for low carbon fiber volume fraction, the electrical conductivity of MSNC is almost the same as that of the matrix. Whereas carbon fiber volume fraction increases, the average distance between them decreases. When the average distance between the carbon fibers become small enough to create the electron tunneling mechanism, the electrical conductivity of MSNC starts to enhance. In this case, the values of the electrical conductivity of the MSNC increase rapidly with increasing the carbon fiber volume fraction. After the carbon fiber volume fraction exceeds the percolation threshold value, the variation of the electrical conductivity of the MSNC tends to be gentle. Whereas the carbon fiber volume fraction is 0.05, the value of the electrical conductivity of the MSNC with higher CNT volume fraction is somewhat lower comparing MSNC with lower CNT volume fraction. It can be explained by the percolation phenomenon in a MSNC with different percolation thresholds for different fillers. Also, Fig. 6 shows that the value of percolation threshold of the MSNC increases slightly with increasing CNT volume fraction. This happening is due to the vast difference between the spatial scales and electrical conduction mechanism associated with two sets of fillers. A percolation like behavior is due to the large disparity of electrical conductivity between the polymer matrix and CNTs. In this case, CNT is dispersed into the non-conductive polymer and provided conductivity to matrix and decreased the primary large disparity of electrical conductivity between matrix and filler which was the main cause of percolation behavior [32,49]. Comparing with Fig. 3 demonstrates that with the same amount of CNT, it is clear that the case whereas CNT and carbon fiber are mixed into the MSNC is greater to the case where only CNT is used. The incorporation of carbon fiber serves to prevent insulating matrix to have a dominant negative effect on the electrical conductivity mechanism. Fig. 7 shows the electrical conductivity change versus carbon fiber volume fraction for different copped carbon fiber aspect ratios. The influences of the aspect ratio of the carbon fibers on the electric conductivity behavior of the MSNC are significant. There are fewer numbers of overlaps between chopped carbon fibers due to lower aspect ratio. Large aspect ratio helps carbon fibers to intervene in their neighborhood and the electrical conductive channel can be easily formed. The smaller length of carbon fibers makes it difficult to be contacted along their lengths and their bigger size obstacles their surfaces to be connected. These factors are destructive to form new conductive networks and abolish percolation threshold probability. Fig. 8 is a plot of predicted electrical conductivity of the carbon fiber-CNT-reinforced polymer MSNC as a function of the carbon fiber volume fraction for different matrix barrier heights. When the carbon fiber volume fraction is low, the distance between carbon fiber is too large; the probability of contact between them is very small and the electrical conductivity of MSNC is approximately the same as that of the matrix, thus the variation of the electrical conductivity with increasing the carbon fiber volume fraction is not considerable. When the carbon fiber volume fraction increases, the average distance between carbon fibers becomes small. The percolation threshold value of the MSNC with low height of barrier is lower than that of the MSNC with higher height of barrier. Whereas height of barrier decreases, the electron flow between carbon fibers increases. The smaller height of barrier provides sufficient electron movement inside the insulating matrix between carbon fibers and makes the carbon fibers more easily to be connected. These factors provide beneficial conditions for smaller percolation threshold. Comparing with Fig. 4 shows that there is a more positive response when carbon fiber and CNT are used together than with CNT alone in barrier height case. At carbon fiber volume fraction about 0.035, these three graphs reach a similar value and matrix height of barrier does not affect the MSNC electrical conductivity at this level of carbon fiber volume fraction. Fig. 9 shows the electrical conductivity of MSNC versus carbon fiber

Fig. 8. Electrical conductivity versus carbon fiber volume fraction curves of carbon fiber/CNT-reinforced polymer MSNC under different values of polymer height of barrier.

Fig. 9. Electrical conductivity versus carbon fiber volume fraction curves of carbon fiber/CNT-reinforced polymer MSNC under different values of carbon fiber electrical conductivity.

Fig. 10. Piezoresistive change ratio versus nanocomposite strain, comparison between present study results and experimental data [3]

7

Composites Part A 130 (2020) 105735

M. Haghgoo, et al.

volume fraction for different carbon fiber electrical conductivities. The electrical conductivity increases with increasing intrinsic carbon fiber electrical conductivity. It can be concluded that the higher the carbon fibers′ electrical conductivity, the higher the overlap ratio between the carbon fibers, the longer is the conduction network. It is proved that the electrical conductivity of the filler is an important driving force to overcome barrier height. Fig. 9 shows clearly that the intrinsic electrical conductivity of carbon fibers becomes dominant as the carbon fiber volume fraction increases over the percolation threshold. When carbon fiber electrical conductivity increases, the percolation threshold does not change. This is because the percolation threshold still relies on the large disparity between filler and matrix. The percolation threshold is strongly influenced by the aspect ratio and the degree of dispersion and as it is depicted in Fig. 7 much lower percolation threshold would be obtained with fillers with larger aspect ratio. The most relevant parameter in the formation of percolated networks is the volume around filler inscrutable by other fillers [19]. Although carbon fiber possibly forms the micro-scale structures and facilities the formation of the percolated network, much higher carbon fiber intrinsic electrical conductivity is required to shift the percolation threshold span [9]. It is interesting to see that conductance increase trend takes place at 0.02 of carbon fiber volume fraction. In order to compare the analytical predictions with the experimental data, the relative resistance change of the CNT-reinforced polymer nanocomposite is expressed in terms of the stretching. Polymer barrier height is hold at 1.5eV . In the comparison case, MWCNTs volume fraction is above the percolation threshold and increases with increasing the level of resistivity. Fig. 10 includes analytical and experimental results. The experiment data is in good agreement with the predicted response, confirming the effectiveness of the proposed model to predict the piezoresistivity of CNT-polymer nanocomposites. After being subjected to strain, CNT junctions will move apart, although the Poisson′s ratio influence will push some together in transverse directions. This will cause a remarkable increase in tunneling resistance when CNT shortest distance is lower than the cutoff distance. Simmons theory declares that the average CNT distance increases during the extension process. Thus, some conductive paths are damaged and the resistance increases with applied strain. The physical interpretation can be elaborated by the enhancement of the total number of joints due to more CNTs in the composite. Enhancement of the number of CNTs increases the probability to form CNT joints. The variation in intrinsic resistance is due to the movement of CNT joints which should be proportional to the applied strain. The higher CNT volume fraction will result in smaller changes in electrical resistance because the contact number for CNTs increases with increased filler volume fraction. The number of filler-to-filler contact joints within the conductive network yields a strong effect on the piezoresistivity of nanocomposite. As a result, piezoresistivity of CNT-polymer composites with higher CNT volume fraction is more sensitive to the applied strain. This means that the number of conductive paths is small and a small change in the network configuration produces large changes in electrical resistance. The tunneling resistance is the main mechanism of the piezoresistive effect of the nanocomposites. The resistance of CNT itself is much lower than the tunneling resistance. The piezoresistivity should be independent of the sample volume, such as the size of the unit cell. There is a sufficient number of CNTs in the RVE and a stable conductive network is formed. To reduce the computation cost, the numerical simulation is conducted in a unit cell with dimensions of 40μm(length) × 40μm(width) × 40μm(thickness) containing CNTs with a length of 10μm and diameter of 40nm . Carbon fibers are absent in the RVE because of the long length of carbon fiber that demands very large RVE with enormous numbers of CNTs. Considering the relative strain of such a huge number of CNTs is very time consuming. Relative resistance change as a function of strain for different matrix barrier heights is shown in Fig. 11. Tunneling resistance would not play a role until larger levels of strain. Still, the effect of the polymer barrier height

Fig. 11. Piezoresistive change ratio versus nanocomposite strain for different polymer heights of barrier.

on the resistance change ratio of nanocomposite is very dominant even at low strain. It can be easily found from the figure that the shortest distance between neighboring CNTs needs to be lower than the critical tunneling distance to keep the fine and stable state of conductive network at higher barrier height. The polymer matrix is considered as an insulating phase. At lower strain, the average CNT distance remains under the tunneling distance and the conductive network could perform properly. The expansion of the matrix can destroy the conductive network lapped with the CNTs. That is, the piezoresistivity of the former material is much higher than that of the latter. The effect of CNT volume fraction on the piezoresistive behavior is evaluated as a function of applied strain in Fig. 12. The strain causes the breakage of MWCNTs′ conductive networks by enhancing the gap between fillers. As shown in Fig. 12, the relative change of resistivity increases rapidly when CNT volume fraction is low. The increase of CNT volume fraction reduces the electrical resistance ratio. This is because when CNT volume fraction increases, the distance between CNTs decreases and some deteriorated conductive paths are rebuilt. As can be seen from the figure, CNT/polymer nanocomposite usually shows a linear response under low strains and followed by non-linear behavior at larger strains. This non-linear behavior is apparent for low CNT volume fraction even in low strains. The non-linear manner in material

Fig. 12. Piezoresistive change ratio versus nanocomposite strain for different CNT volume fractions. 8

Composites Part A 130 (2020) 105735

M. Haghgoo, et al.

response happens when material subjects to conversion from homogeneous form to nonhomogeneous form during stretching. The polymer nanocomposite with low CNT volume fraction is found to undergo greater non-linearity as the main conclusion from the figure.

[12]

[13]

6. Conclusion

[14]

In this paper, the electrical conductivity and piezoresistivity of the CNT-polymer nanocomposites were studied. Also, the effect of carbon fibers on the electrical conductivity of polymer-based MSNC containing CNTs was evaluated. The model embodied the most essential features of electrical process including tunneling resistance between neighboring fillers. The outcome of the proposed method is a widely useful model that has the necessary connection with the physics of electrical process in MSNC. The influence of CNT volume fraction and matrix barrier height on the CNT-polymer nanocomposite electrical conductivity and piezoresistivity was investigated. Through the consideration of several data sets available in the literature, the calculated results could successfully capture the behavior of different experimental measurements. Simulation results indicated that the off alignment in the CNTs in conductive networks plays a dominant role in the variation of electrical conductivity. The analytical results showed significant non-linearity of piezoresistivity and dependence of it on CNT volume fraction. To some extent, the developed model in this work in envisaged to help understand the trend of the composite system factors on the overall electrical conductivity and piezoresistivity of polymer-based nanocomposite. The proposed analytical method can capture the sharp increase of conductivity even at a low volume fraction of fillers.

[15] [16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

CRediT authorship contribution statement [25]

M. Haghgoo: Data curation, Formal analysis, Project administration, Writing - original draft, Writing - review & editing. M.K. Hassanzadeh-Aghdam: Conceptualization, Formal analysis, Methodology, Resources, Software, Supervision, Writing - original draft, Writing - review & editing. R. Ansari: Project administration, Resources, Software, Supervision, Writing - original draft.

[26] [27]

[28] [29]

Declaration of Competing Interest

[30] [31]

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

[32]

[33]

References

[34]

[1] Aguilar J, Bautista-Quijano J, Avilés F. Influence of carbon nanotube clustering on the electrical conductivity of polymer composite films. Express Polym Lett 2010;4(5):292–9. [2] Hu N, et al. The electrical properties of polymer nanocomposites with carbon nanotube fillers. Nanotechnology 2008;19(21):215701. [3] Hu N, et al. Tunneling effect in a polymer/carbon nanotube nanocomposite strain sensor. Acta Mater 2008;56(13):2929–36. [4] Bauhofer W, Kovacs JZ. A review and analysis of electrical percolation in carbon nanotube polymer composites. Compos Sci Technol 2009;69(10):1486–98. [5] Inam F, Reece MJ, Peijs T. Shortened carbon nanotubes and their influence on the electrical properties of polymer nanocomposites. J Compos Mater 2012;46(11):1313–22. [6] Castellino M, et al. Conductivity in carbon nanotube polymer composites: A comparison between model and experiment. Compos A Appl Sci Manuf 2016;87:237–42. [7] Ambrosetti G, et al. Solution of the tunneling-percolation problem in the nanocomposite regime. Phys Rev B 2010;81(15):155434. [8] Liang J-Z, Yang Q-Q. Effects of carbon fiber content and size on electric conductive properties of reinforced high density polyethylene composites. Compos B Eng 2017;114:457–66. [9] Chen Y, et al. Theoretical estimation on the percolation threshold for polymer matrix composites with hybrid fillers. Compos Struct 2015;124:292–9. [10] Al-Saleh MH, Saadeh WH. Hybrids of conductive polymer nanocomposites. Materials Design (1980–2015) 2013;52:1071–6. [11] Lisunova M, et al. Percolation behaviour of ultrahigh molecular weight

[35] [36] [37] [38]

[39] [40]

[41] [42] [43]

[44]

9

polyethylene/multi-walled carbon nanotubes composites. Eur Polym J 2007;43(3):949–58. Jang S-H, Yin H. Effective electrical conductivity of carbon nanotube-polymer composites: a simplified model and its validation. Mater Res Express 2015;2(4):045602. Simmons JG. Electric tunnel effect between dissimilar electrodes separated by a thin insulating film. J Appl Phys 1963;34(9):2581–90. Du F, Fischer JE, Winey KI. Effect of nanotube alignment on percolation conductivity in carbon nanotube/polymer composites. Phys Rev B 2005;72(12):121404. Matos MA, et al. Predictions of the electro-mechanical response of conductive CNTpolymer composites. J Mech Phys Solids 2018;114:84–96. Mutlay İ, Tudoran LB. Percolation behavior of electrically conductive graphene nanoplatelets/polymer nanocomposites: theory and experiment. Fuller Nanotubes Carbon Nanostruct 2014;22(5):413–33. Ma P-C, et al. Enhanced electrical conductivity of nanocomposites containing hybrid fillers of carbon nanotubes and carbon black. ACS Appl Mater Interfaces 2009;1(5):1090–6. Radzuan NAM, et al. The effect of milled carbon fibre filler on electrical conductivity in highly conductive polymer composites. Compos B Eng 2017;110:153–60. Avilés F, Oliva-Avilés AI, Cen-Puc M. Piezoresistivity, strain, and damage self-sensing of polymer composites filled with carbon nanostructures. Adv Eng Mater 2018;20(7):1701159. Cob J, et al. Influence of concentration, length and orientation of multiwall carbon nanotubes on the electromechanical response of polymer nanocomposites. Mater Res Express 2019;6(11):115024. Oliva-Avilés A, et al. On the contribution of carbon nanotube deformation to piezoresistivity of carbon nanotube/polymer composites. Compos B Eng 2013;47:200–6. Can-Ortiz A, Abot JL, Avilés F. Electrical characterization of carbon-based fibers and their application for sensing relaxation-induced piezoresistivity in polymer composites. Carbon 2019;145:119–30. Avilés F, et al. A comparative study on the mechanical, electrical and piezoresistive properties of polymer composites using carbon nanostructures of different topology. Eur Polym J 2018;99:394–402. Park M, Kim H, Youngblood JP. Strain-dependent electrical resistance of multiwalled carbon nanotube/polymer composite films. Nanotechnology 2008;19(5):055705. Hu N, et al. Investigation on sensitivity of a polymer/carbon nanotube composite strain sensor. Carbon 2010;48(3):680–7. Bao W, et al. A novel approach to predict the electrical conductivity of multifunctional nanocomposites. Mech Mater 2012;46:129–38. Feng C, Jiang L. Investigation of uniaxial stretching effects on the electrical conductivity of CNT–polymer nanocomposites. J Phys D Appl Phys 2014;47(40):405103. Zeng X, et al. Characteristics of the electrical percolation in carbon nanotubes/ polymer nanocomposites. J Phys Chem C 2011;115(44):21685–90. Bao W, et al. Tunneling resistance and its effect on the electrical conductivity of carbon nanotube nanocomposites. J Appl Phys 2012;111(9):093726. Bushman A, Fortov VE. Model equations of state. Sov Phys Uspekhi 1983;26(6):465. Datta, S., Electronic Transport in Mesoscopic Systems. Cambridge Studies in Semiconductor Physics and Microelectronic Engineering. 1995, Cambridge: Cambridge University Press. Feng C, Jiang L. Micromechanics modeling of the electrical conductivity of carbon nanotube (CNT)–polymer nanocomposites. Compos A Appl Sci Manuf 2013;47:143–9. Littlejohn SD. Electrical properties of graphite nanoparticles in silicone: flexible oscillators and electromechanical sensing. Springer Science & Business Media; 2013. Zabihi Z, Araghi H. Monte Carlo simulations of effective electrical conductivity of graphene/poly(methyl methacrylate) nanocomposite: Landauer-Buttiker approach. Synth Met 2016;217:87–93. Takeda T, et al. Modeling and characterization of the electrical conductivity of carbon nanotube-based polymer composites. Polymer 2011;52(17):3852–6. Gong S, Zhu Z, Meguid S. Carbon nanotube agglomeration effect on piezoresistivity of polymer nanocomposites. Polymer 2014;55(21):5488–99. Deng F, Zheng Q-S. An analytical model of effective electrical conductivity of carbon nanotube composites. Appl Phys Lett 2008;92(7):071902. Zare Y, Rhee KY. A simple methodology to predict the tunneling conductivity of polymer/CNT nanocomposites by the roles of tunneling distance, interphase and CNT waviness. RSC Adv 2017;7(55):34912–21. Berhan L, Sastry A. Modeling percolation in high-aspect-ratio fiber systems. II. The effect of waviness on the percolation onset. Phys Rev E 2007;75(4):041121. Li C, Thostenson ET, Chou T-W. Dominant role of tunneling resistance in the electrical conductivity of carbon nanotube–based composites. Appl Phys Lett 2007;91(22):223114. Yu S, et al. Superior piezoresistive strain sensing behaviors of carbon nanotubes in one-dimensional polymer fiber structure. Carbon 2018;140:1–9. Alian AR, Kundalwal SI, Meguid SA. Multiscale modeling of carbon nanotube epoxy composites. Polymer 2015;70:149–60. Kundalwal SI, Kumar S. Multiscale modeling of stress transfer in continuous microscale fiber reinforced composites with nano-engineered interphase. Mech Mater 2016;102:117–31. Kundalwal SI, Meguid SA. Multiscale modeling of regularly staggered carbon fibers embedded in nano-reinforced composites. Eur J Mech A Solids 2017;64:69–84.

Composites Part A 130 (2020) 105735

M. Haghgoo, et al.

terephthalate). J Appl Polym Sci 2013;130(1):543–53. [48] Motaghi A, Hrymak A, Motlagh GH. Electrical conductivity and percolation threshold of hybrid carbon/polymer composites. J Appl Polym Sci 2015;132(13). [49] Zhang Q, et al. Low percolation threshold in single-walled carbon nanotube/high density polyethylene composites prepared by melt processing technique. Carbon 2006;44(4):778–85.

[45] Gong S, Zhu ZH. On the mechanism of piezoresistivity of carbon nanotube polymer composites. Polymer 2014;55(16):4136–49. [46] Zhang P, et al. The effects of agglomerate on the piezoresistivity of conductive carbon nanotube/polyvinylidene fluoride composites. Sens Actuat A 2018;281:176–84. [47] Maiti S, et al. Low percolation threshold in polycarbonate/multiwalled carbon nanotubes nanocomposites through melt blending with poly (butylene

10