Modeling the dielectric breakdown strength and energy storage density of graphite-polymer composites with dielectric damage process

Journal Pre-proof Modeling the dielectric breakdown strength and energy storage density of graphite-polymer composites with dielectric damage process

Xiaodong Xia, Bai-Xiang Xu, Xiazi Xiao, George J. Weng PII:

S0264-1275(20)30064-2

DOI:

https://doi.org/10.1016/j.matdes.2020.108531

Reference:

JMADE 108531

To appear in:

Materials & Design

Received date:

18 December 2019

Revised date:

26 January 2020

Accepted date:

28 January 2020

Please cite this article as: X. Xia, B.-X. Xu, X. Xiao, et al., Modeling the dielectric breakdown strength and energy storage density of graphite-polymer composites with dielectric damage process, Materials & Design(2018), https://doi.org/10.1016/ j.matdes.2020.108531

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© 2018 Published by Elsevier.

Journal Pre-proof

Modeling the dielectric breakdown strength and energy storage density of graphite-polymer composites with dielectric damage process Xiaodong Xia1, Bai-Xiang Xu2, Xiazi Xiao1,* and George J. Weng3,* 1

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School of Civil Engineering, Central South University, Changsha 410083, PR China 2 Mechanics of Functional Materials Division, Institute of Material Science, Technische Universität Darmstadt, Darmstadt 64287, Germany 3 Department of Mechanical and Aerospace Engineering, Rutgers University, New Brunswick, NJ 08903, USA

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Abstract

Recent experimental data has demonstrated significant influences of graphite volume

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concentration on the dielectric breakdown and energy storage behavior of graphite-polymer composites, but no existing homogenization theory has been established to illustrate such

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dependence in the context of alternating current (AC) loading. In this paper we develop a novel homogenization scheme to connect the microstructural parameters of constituent phases and the

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AC frequency to the dielectric breakdown strength and energy storage density of the overall composite. The major microstructural features covered are the graphite volume concentration,

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imperfect bonding effect, graphite aspect ratio, percolation threshold, filler-dependent electron tunneling, Maxwell-Wagner-Sillars polarization, and frequency-dependent electron hopping and

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dielectric relaxation. A thermodynamic framework is developed to describe the evolution of a dielectric damage parameter with respect to the electric field. We highlight the developed theory with validation through the experimental data of graphite/PVDF composite over a wide range of graphite volume concentration and AC frequency. The results indicate that the dielectric breakdown strength of the graphite-polymer composite decreases with respect to the graphite volume concentration, while the energy storage density increases with it. Keywords: energy storage density, dielectric breakdown strength, dielectric damage process, graphite-polymer composite, interface effects. ______________________ * Corresponding author. E-mail: [email protected] (X. Xiao), [email protected] (G.J. Weng). 1

Journal Pre-proof 1. Introduction High dielectric (high-k) materials, especially the carbon-based composites, have attracted significant applications in the modern energy and electronics industry [1, 2], such as the energy storage systems [3-5], high power density batteries [6] and electromagnetic interference shielding devices [7-9]. Typical carbon fillers include graphite, carbon nanotube (CNT) and graphene. Graphite is a crystalline form of the element carbon with its atoms arranged in a hexagonal structure [10]. However, single-phase graphite filler cannot achieve the high energy

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storage density while maintaining sufficient dielectric and mechanical breakdown strength. Therefore, a small amount of graphite fillers are usually added into the polymer matrix to

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constitute the graphite-polymer composites [11, 12]. In this way excellent energy storage density and dielectric breakdown strength can be attained simultaneously [13].

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While studies on the fundamental electrical properties, such as dielectric permittivity and

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electrical conductivity, of carbon-based composites are very active under direct current (DC) and alternating current (AC) electrical loading [14], few research concentrate on the related topics of

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dielectric breakdown strength and energy storage density [15, 16]. Along this line one may refer to the recent reviews of Uyor et al. [17] and Huang et al. [18]. Indeed there is a dire need to

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illustrate the influences of graphite volume concentrations and AC frequency on the dielectric breakdown and energy storage behaviors of graphite-polymer composites [10]. At present no

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existing homogenization theory is available to connect the microstructural parameters of the constituent phases and the AC frequency to the effective dielectric breakdown strength and

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energy storage density of the composites [19]. Existing experimental data has revealed that a small amount of carbon fibers can remarkably enhance the dielectric energy storage density while drastically cutting down the dielectric breakdown strength [12]. The reasons for such opposing trends have not been quantitatively explained in the literature. This has motivated the present study. In undertaking this investigation, we immediately encountered the issues that dielectric breakdown and energy storage behaviors are both nonlinear topics; they all involve the dielectric damage process [20-22]. It is with this view that we intend to resolve the stated issues. To establish a suitable homogenization scheme for the present problem, several essential issues need to be confronted. The first one is the determination of overall fundamental electrical properties of the composite under a perfect interface condition. Note that this is not a common problem of effective properties of graphite-polymer composite due to the extra low aspect ratio 2

Journal Pre-proof of the graphite fillers and the existence of a percolation threshold. The effective permittivity and conductivity obtained by many homogenization schemes may not provide a percolation threshold or may go outside of the Hashin-Shtrikman (HS) bounds [23, 24]. Percolation threshold of effective electric properties is indeed a key feature for the graphite-polymer composite that cannot be ignored. In addition, frequency-dependent behavior should also be accounted for in the homogenization scheme. Towards this end, the concept of complex dielectric permittivity for a two-phase composite will be adopted in the homogenization scheme. In this way, the theory will possess the frequency dependence, percolation threshold, and the calculated electrical properties

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will always lie on or within the HS bounds.

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The second issue is the imperfect interface conditions between the graphite and polymer matrix phases. Several categories of interface effects need to be included and they can all be

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addressed with the introduction of an ultrathin interlayer with specific properties between the graphite and polymer phases. The first category is the imperfect bonding between the constituent

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phases. This one has previously been considered in Dunn and Taya [25], Nan et al. [26] and

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Duan et al. [27]. The second category is unique for the dielectric and conduction problems. In the context of dielectric phenomena, it involves the Maxwell-Wagner-Sillars polarization [28-30]

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under DC loading and Debye dielectric relaxation [31] under AC loadings, while in the context of electrical conduction it involves the electron tunneling under DC [32] and electron hopping under AC conditions [33, 34]. All of these interfacial features will be considered in the

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development of the theory. It turns out that the second category of the interface effects is

composites.

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responsible for the extremely high permittivity and conductivity of the carbon-based polymer

The third one is the thermodynamics-based evolution equation to study the dielectric breakdown and energy storage behavior of the graphite-polymer composite. In this work, a thermodynamic framework will be developed to describe the evolution of a damage parameter based on the dielectric damage mechanics (DDM). Two key features are involved here: (i) the thermodynamic driving force for the dielectric damage process, and (ii) the corresponding dielectric dissipation potential. The second one needs to be determined at a given electric field and AC frequency. The thermodynamic driving force comes from the reduction of the dielectric energy density of the composite as the dielectric damage process proceeds. It is the conjugate to the dielectric damage parameter. Then the dielectric dissipation potential provides the evolution 3

Journal Pre-proof of the dielectric damage parameter via the normality structure. The dissipation potential is a function of the thermodynamic driving force, electric field, and dielectric damage parameter. This part needs to be integrated into the homogenization scheme with all interface effects to capture the dielectric breakdown strength and energy storage density of the graphite-polymer composites under AC electrical loading. In what follows we will develop such a novel homogenization scheme to address the frequency-dependent dielectric breakdown strength and energy storage density of graphitepolymer composite. The developed homogenization theory will be validated with the

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experimental data of graphite/PVDF composite at various graphite volume concentrations and

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AC frequencies. Briefly, the theoretical development will consist of the four parts: (i) the basic features for the frequency-dependent complex dielectrics of graphite-polymer composite; (ii) the

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homogenization method of graphite-polymer composite under the perfect interface condition; (iii) the imperfect interface conditions between the graphite and polymer phases; (iv) the dielectric

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damage process of graphite-polymer composite.

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We now turn to the development of the theory for dielectric breakdown strength and energy storage density.

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2. Basic features for the complex dielectric permittivity in graphite-polymer composite

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For the study of frequency dependence, we first introduce the basic concept of complex dielectric permittivity. When the graphite-based composite is subjected to a cyclic AC electric

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field in the infinity as

E(t )  Εeiωt ,

(1)

the steady-state electric displacement of the composite will also take the cyclic form as

J(t )  Jeiωt ,

(2)

where i is the imaginary constant,   2 f (in rad/s) denotes the angular frequency of the electrical loading, and f (in Hz) represents the AC frequency of the external electrical loading. Ε and J are the amplitudes of the electric field and electric displacement, respectively.

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Journal Pre-proof To pave the way for the frequency-dependent analysis of energy storage properties for graphite-polymer composite, the complex dielectric permittivity, ε* , is introduced to connect the amplitudes of the electrical field and electric displacement J =ε* Ε,

with ε*  ε  i

σ , ω

(3)

where the real part of complex permittivity is the original dielectric permittivity, and the negative imaginary part is related to the original electrical conductivity divided by the angular frequency. In addition, the relationship between the complex dielectric permittivity, ε* , and the complex

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electrical conductivity [35], σ * , can be illustrated as

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ε*  iσ* / ω, with σ*  σ  iωε.

(4)

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Note that the complex dielectric permittivity will be adopted as the homogenization parameter in

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the homogenization scheme for the graphite-polymer composite.

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3. The homogenization scheme of graphite-polymer composite under the perfect interface condition

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An SEM image of the graphite/PVDF polymer is shown in Fig. 1 [12]. Graphite fillers are randomly distributed in the polymer matrix. A typical morphology is sketched in Figs. 2(a) and

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(b) to reveal the microstructure of graphite-polymer composite before and after the dielectric

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damage, respectively. Under the condition of perfect graphite-polymer interface, there are several homogenization schemes that could be called upon to determine the effective complex permittivity of graphite-polymer composite without dielectric damage. The most common ones include Bruggeman’s effective-medium approximation (EMA) [36], the Mori-Tanaka (MT) method [37] and Ponte Castaneda-Wills (PCW) model [38]. As both MT and PCW methods require each inclusion to be completely surrounded by the matrix, they do not permit the formation of percolation path as depicted in Fig. 2(a). As a consequence the percolation threshold of the graphite-polymer composite cannot be calculated by either method. PCW approach has another limitation that the calculated effective permittivity can go outside the Hashin-Shtrikman (HS) bounds [39] due to the extreme small aspect ratio of the fillers [32]. Among these three, EMA is the most suitable one for the present problem. The calculated 5

Journal Pre-proof complex permittivity always lies on or within the HS bounds and it can also provide the percolation threshold [32]. This approach will be adopted to serve as the backbone of the homogenization scheme under the perfect interface condition. Then various interfacial phenomena can be built upon it to establish a more general homogenization procedure. With EMA, the effective complex permittivity of two-phase graphite-polymer composite, ε*e , at the undamaged state can be derived by means of zero sum of individual scatterings

through Maxwell’s far-field matching as [40] 1

1

 0,

(5)

of

c0 (ε*0  ε*e )1  S0ε*e1   c1 (ε1*  ε*e )1  S1ε*e1 

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where the curly brackets < > signify the orientational average of the said quantity inside it, c0 and c1 are the volume concentrations of polymer and graphite phases, respectively; S 0 and S1

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are the corresponding Eshelby S-tensors [41] or depolarization tensor in electrostatics [42]; ε*i

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represents the complex permittivity tensor of phase i. The subscripts of “0”, “1” and “e” stand for the properties of matrix phase, inclusion phase and effective medium, respectively. It is to be

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noted that, to qualify as EMA, the properties of the reference medium in the scattering analysis must be taken as those of the effective medium, i.e., the composite.

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As the graphite inclusions are transversely isotropic with the direction 3 (i.e. axis- x3 of the graphite filler) to be the rotation symmetry axis, Eq. (5) degenerates to the scalar equation for the

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oriented

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effective complex permittivity,  e* , of the overall composite when inclusions are randomly

c0

  0*   e*  3*   e* 1  2(1*   e* )  c   0, 1 * * * * * * * * *   e  (1/ 3)( 0   e ) 3   e  S11 (1   e )  e  S33 ( 3   e ) 

(6)

where ε0* and ε1* (or ε3* ) are the complex dielectric permittivity for the matrix phase and inclusion filler, respectively. Note that these complex quantities can be given in terms of the corresponding real and imaginary parts: ε0*  ε0  iσ0 / ω , ε1*  ε1  iσ1 / ω (or ε3*  ε3  iσ3 / ω ) and

εe*  εe  iσe / ω . In addition, S11 and S33 are components of the S-tensor for the graphite inclusion, with the expressions as [42] S11  S22 

 arccos    (1   2 )1/2  , S33  1  2S11 ,   1, 2 3/2  2(1   ) 6

(7)

Journal Pre-proof where  denotes the aspect ratio (the thickness-to-diameter ratio) of the disc-type graphite filler. However, micro imperfections inevitably can be created in the composite during manufacturing and service. As the external AC electric loading increases, the micro-regions that lose its permittivity begin to emerge, as shown in Fig. 2(b). This will lower the dielectric permittivity of the overall composite. The effective electric displacement of the graphite-polymer composite, J e , at the dielectric damage state can be written as J e  1  γ(c1 ) D  εe E, 0  D  1,

(8)

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where εe is the effective permittivity of the composite at the undamaged state, which has been determined by the homogenization scheme in Eq. (6). D is the dielectric damage parameter of

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the composite, with D  0 and D  1 denoting the undamaged and dielectric breakdown states of

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the overall composite, respectively. γ(c1 ) represents the maximum influence of dielectric

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damaged micro-regions on the effective permittivity of the composite. This quantity is c1 dependent, as higher graphite volume concentration will introduce more dielectric damaged

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micro-regions during manufacturing and service. At low filler concentrations we take γ(c1 ) to be a linear function as

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γ(c1 )  γ0  c1 η,

(9)

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where γ0 is the value of γ(c1 ) for the pure polymer matrix, and η reflects the influence of

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graphite fillers on the generation of damaged micro-region during the dielectric damage process.

4. The imperfect interface condition between the graphite and polymer phases The effective complex permittivity in Eq. (6) is given under the perfect interface condition. However, the interface condition is usually imperfect in practice. Three categories of interface effects will be considered in this section: (i) imperfect bonding effect between the constituent phases, (ii) the Maxwell-Wagner-Sillars (MWS) polarization [28-30] and Debye dielectric relaxation [31] effects for the interfacial permittivity, and (iii) electron tunneling [43] and electron hopping [44] effects for the interfacial conductivity. To account for the above interface effects, the graphite inclusion is assumed to be surrounded by a thin interlayer to form the coated

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Journal Pre-proof inclusion phase. The effective permittivity and conductivity of the coated graphite inclusion can be derived by the Mori-Tanaka method [37]





(c) i

(c) i

(int) freq

  (1  cint )( i   (int) freq ) , 1  (int) (int)   cint Sii ( i   freq )   freq 

(10)

(int) freq

  (1  cint )( i   (int) freq ) , 1  (int) (int)   cint Sii ( i   freq )   freq 

(11)





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(int) (int) where  freq and  freq are the frequency-dependent permittivity and conductivity of the interlayer

with consideration of the mentioned interface effects. They will be illustrated in the Sect. 4.1 and

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4.2, respectively. Sii is the ii  component of the Eshelby S-tensor for the graphite filler. cint is

     2  2 

     h .   h  2  2 

2

2

(12)

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cint  1 

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the volume concentration of the interlayer in the coated graphite configuration

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Here,  and h represent the thickness of the graphite filler and interlayer, respectively. 4.1. Imperfect interfacial bonding, Maxwell-Wagner-Sillars polarization and Debye dielectric

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relaxation effects for the interfacial permittivity

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Due to the imperfect interfacial bonding effect, the original permittivity of the interlayer will be lower than that of the constituent phases. Then, the filler-dependent MWS polarization

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effect and frequency-dependent Debye relaxation effect need to be considered for the interfacial permittivity. Electrons cannot move across the interface between the graphite and polymer phases, because of the large disparity of electrical conductivity between these two adjacent phases. It results in the formation of nanocapacitors on the interface, as depicted in Fig. 3. In addition, AC frequency also reveals remarkable influences on the MWS polarization for the interfacial permittivity. More electrons flow across the interface between the graphite and polymer phases as the AC frequency increases. As a consequence, it will decrease the amount of the electrons accumulated on the interface. The final permittivity of the interlayer with consideration of the above imperfect interfacial bonding, MWS polarization and Debye relaxation interface effects can be derived as [31]

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Journal Pre-proof (int)  (int) freq ( )   inf 

(int) (int)  static   inf , 1   2t2

(13)

(int) (int) where t is the characteristic time for MWS polarization.  static and  inf are dielectric

permittivity of the interlayer with MWS polarization effect at the static and infinite frequency states, respectively (14)

(int) (int)   inf   0(inf) /  (c1 , c1* ,  inf ),

(15)

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(int)   static   0(int) /  (c1 , c1* ,  static ),

(int) in which  0(int) and  0(inf) are the original permittivity of the interlayer at the static and infinite

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  frequency states, respectively.  static and  inf are the corresponding characteristic parameters for

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the MWS polarization effect. c1* is the percolation threshold for the complex permittivity of the two-phase composite

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9 S33 (1  S33 ) . 2 9 S33  15S33  2

(16)

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c1* 

which is identical to percolation threshold of the electrical conductivity or complex conductivity

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of two-phase composite [32, 35]. In addition, function  (c1 , c1* ,  ) serves to characterize the change of the dielectric permittivity after the graphite volume concentration reaches the

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percolation threshold (cf. [35])

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F (1, c1* ,  )  F (c1 , c1* ,  )  (c1 , c ,  )  , F (1, c1* ,  )  F (0, c1* ,  ) * 1

with F (c1 , c ,  )  * 1

1



arctan(

c1  c1*



1 ) . 2

(17)

4.2. Imperfect interfacial bonding, electron tunneling and electron hopping effects for the interfacial conductivity Due to the imperfect interfacial bonding effect, the original conductivity of the interlayer will be lower than that of the constituent phases. Next, the filler-dependent electron tunneling [43] and frequency-dependent electron hopping effects [44] need to be included for the interfacial conductivity. As the graphite volume concentration increases, the average distances between the adjacent graphite fillers continuously decreases. The graphite connective networks begin to build 9

Journal Pre-proof up as the graphite volume concentration passes through the percolation threshold, c1* . It remarkably enhances the probability of electron tunneling. Moreover, AC frequency also has significant influences on the electron tunneling effects for interfacial conductivity. It can be attributed to the fact that extra electrons will hop from one graphite filler to another as the AC frequency increases. After including the imperfect interfacial bonding, electron tunneling and electron hopping effects, the final conductivity of the interlayer can be derived as [34] (int)  (int) freq ( )   static p ( ),

(18)

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(int) where  static is the electrical conductivity of the interlayer with electron tunneling effect at the

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static state

(int)   static   0(int) /  (c1 , c1* ,  static ),

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and p( ) is the Dyre’s hopping function [34]

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t (arctan t )

p( ) 

(19)

2

,

(20)

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ln(1   2t2 )1/2   (arctan t )2

 with  0(int) is the original electrical conductivity of the interlayer at the static state;  static is the

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characteristic parameter for electron tunneling effect at the static state; t is the characteristic time of electron hopping effect. It is noted that the hopping function p  1 as the frequency

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  0, recovering the static interfacial conductivity.

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Substituting Eqs. (13) and (18) into Eqs. (10) and (11), we obtain the final permittivity and conductivity for the coated graphite filler, respectively, at a given level of graphite volume concentration and AC frequency. With the material constants listed in Table 1 for later comparison with the experimental data in [12], the coated graphite permittivity and conductivity are shown in Figs. 4(a) and (b), respectively. It is seen that both quantities increase with the graphite volume concentration. In addition, the coated graphite permittivity decreases with respect to the AC frequency, while the coated graphite conductivity displays an opposite trend. The final complex permittivity of the coated graphite phase,  i( c )*   i( c )  i i( c ) /  , will be utilized to replace the original  i* of the inclusion phase in Eq. (6) for the evaluation of the effective complex permittivity of the overall composite,  e*   e  i ie /  , with the consideration 10

Journal Pre-proof of all categories of interface effects. Then, the effective dielectric permittivity,  e , and electrical conductivity,  e , of the composite can be obtained from the real and imaginary parts of  e* , respectively. The obtained  e will be utilized to evaluate the dielectric energy storage density in Sect. 5. 5. Dielectric damage process of graphite-polymer composite We assume that the micro-regions that lose its dielectrics begin to emerge in the graphitepolymer composite as the external electric field reaches a critical level. The dielectric damage

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mechanism (DDM) will be introduced to characterize the generation of the damaged micro-

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regions during the dielectric damage process. The concept of the DDM method was enlightened by the continuum damage mechanics (CDM) approach in the mechanical field [45, 46].

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In the DDM approach, the dielectric energy density of the overall composite can be derived

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as

(21)

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1 U E  1  γ(c1 ) D  εe E 2 . 2

As the dielectric damage process is an irreversible dissipation process, the principle of non-

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equilibrium thermodynamics must be satisfied under a given electric field U E |E dD  0, D

(22)

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

where the equality holds only for the reversible process. As the thermodynamic system evolves

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towards equilibrium, the change of U E continues to decrease. The thermodynamic driving force for the dielectric damage process, which is the conjugate to the dielectric damage parameter, is given by f driv  

U E 1 |E  γ(c1 )εe E 2 , D 2

(23)

under a given electric field. The driving force for dielectric damage is enlightened by the analogous concept for domain switch in ferroelectric ceramics [47, 48]. Similar thought has also been adopted in the electric creep evolution of ferroelectric domains [49] as well as the mechanical damage process of graphene-metal nanocomposites [50].

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Journal Pre-proof Then, a dielectric dissipation potential is required to characterize the evolution of the dielectric damage parameter. Here we suggest that the dissipation potential for the dielectric damage process,  D , takes the form D 

2 (1  D)m 1 1 f driv , 2 S0 E n

(24)

where m and n are the dielectric damage exponents of the composite, and S0 is the energy strength for the dielectric damage process. This formation of dielectric dissipation potential is

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enlightened by the dissipation potential in the mechanical damage potential [45, 50]. Next, the evolution equation for the dielectric damage parameter is derived by

(25)

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  D , E  Ecr ,  E (1  D) f driv D 0, E  Ecr , 

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where Ecr denotes the threshold electric field for the dielectric damage process. Similar

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evolution equations have been adopted by Bonora [51] and Xia et al. [50] in the mechanical damage process.

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In order to transfer the rate form of the evolution equation in Eq. (25) into an incremental form, the derivative of the dielectric damage parameter with respect to the electric field can be given as

(26)

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(1  D) m dD  P ( D, E )  f driv , E  Ecr  , with P( D, E )  . dE 0, S0 E n E  Ecr

Note that the increase of the dielectric damage parameter with respect to the electric field increment is proportional to the thermodynamic driving force for the dielectric damage process,

fdriv . The growth coefficient P( D, E ) is isotropic and dependent on the electric field. This evolution equation is similar to the time-dependent Ginzburg-Landau (TDGL) kinetic equation [52, 53] dτi δG   Lij , dt δτ j

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(27)

Journal Pre-proof where τ i denotes the order parameter, Lij is anisotropic growth coefficients for the TDGL kinetic equation, G is Gibbs free energy of the overall system, δG / δτ j represents the thermodynamic driving force for the considered nonlinear process. As the external electric load exceeds the threshold electric field, Ecr , the dielectric damage process proceeds, which is reflected by the evolution of dielectric damage parameter. The dielectric breakdown phenomenon occurs when the dielectric damage parameter reaches the critical value, D  1 . The corresponding electric field is denoted as the dielectric breakdown

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strength, Ebd , and the dielectric energy storage density of graphite-polymer composite is defined

U bd  U E |E =E

1 = 1  γ(c1 )  εe Ebd2 . 2

(28)

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bd , D 1

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via U E at the dielectric breakdown strength

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This completes the development of the whole frequency-dependent homogenization theory

6. Numerical results

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6.1. Computational procedure

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for the dielectric breakdown strength and energy storage density of graphite-polymer composite.

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To place the present homogenization scheme for energy storage properties in proper prospective, the theoretical prediction will be validated by the experimental data of

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graphite/PVDF composite [12]. First, the weight percentage ( wt % ) in the experiment needs to be converted to the volume percentage ( vol % ) of graphite fillers for the theory by

  1  vol %  m  m   g   1  ,  wt %   

(29)

where m  1780 kg/m3 and  g  2260 kg/m3 are the mass densities of the PVDF polymer and graphite filler, respectively. The in-plane dielectric permittivity and electrical conductivity of the graphite filler are given as 1  15 vac ,  1  6.32 104 S/m,

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(30)

Journal Pre-proof where  vac =8.85 1012 F/m denotes the permittivity in the vacuum. Then the out-of-plane permittivity and electrical conductivity of the graphite filler are taken as  3  p1 ,  3  q 1 ,

(31)

with p  0.667 and q  0.001 . In addition, other physical and microstructural features utilized in the numerical calculation are listed in Table 1 based on the experimental conditions. The initial condition of dielectric damage parameter for graphite-polymer composite is set at D  0 in the numerical calculation. The overall composite is subjected to an AC electric field

of

with increasing amplitude in the infinity. Once the threshold electric field has been reached ( E  Ecr ) , the thermodynamic driving force can be obtained from Eq. (23) with an increment of

ro

the electric field, E . The increment of the dielectric damage parameter, D , with respect to

-p

the electric field can be obtained by solving the Eq. (26) via the Runge-Kutta method. Specific iteration procedure for the evolution of the dielectric damage parameter can be expressed as

re

E  g1 ( D, E)  2g2 ( D, E)  2g3 ( D, E)  g4 ( D, E), 6

(32)

lP

D 

where

g1 ( D, E )  P( D, E )  f driv |E  E ( x ), D  D ( x ) ,

na

g 2 ( D, E )  P( D, E )  f driv |E  E ( x ) E /2, D  D ( x )  g1 ( D , E ) E /2 , g3 ( D, E )  P( D, E )  f driv |E  E ( x ) E /2, D  D ( x )  g2 ( D , E ) E /2 ,

(33)

ur

g 4 ( D, E )  P( D, E )  f driv |E  E ( x ) E , D  D ( x ) g3 ( D , E ) E .

Jo

Then, the final electrical displacement, J e , and dielectric energy storage density, U E , for this iteration can be calculated by Eqs. (8) and (21), respectively, under E  E ( x )  E and D  D ( x )  D . Next, another increment of the electric field is implemented until the dielectric

damage parameter has reached 1, at which the dielectric breakdown phenomenon of the overall composite occurs. 6.2. The effective dielectric permittivity and electrical conductivity of graphite-polymer composites As a first application, we explore the effective dielectric permittivity and electrical conductivity of graphite-polymer composites under the AC electrical loading. Influences of 14

Journal Pre-proof dielectric damage process are neglected temporarily in this subsection. Figs. 5(a) and (b) depict the effective permittivity and conductivity with respect to the graphite volume concentration at a given AC frequency (100 Hz). Three categories of interface conditions are considered in the numerical calculation. To start with, the effective conductivity and permittivity of the graphitepolymer composite are achieved under the perfect interface connection, as depicted in the green lines of Figs. 5(a) and (b). The calculated conductivity is seen to be higher than the experimental data. The interfacial connection between graphite filler and polymer matrix cannot be perfect. Drawbacks inevitably exist in the interface phase during the manufacture and service. The

of

imperfect connection will lower the electrical properties of overall composites. The existence of the imperfect interfacial connection has been confirmed by the experiment [54]. However, the

ro

effective permittivity and conductivity under constant  int

and  int are lower than the

-p

experimental data, as revealed by the yellow lines of Figs. 5(a) and (b). Finally, the theoretical prediction with all categories of interface effects was obtained and found to agree with the

re

experimental data [12], as shown by the blue lines of Figs. 5(a) and (b).

lP

Figs. 6(a) and (b) depict the effective permittivity and conductivity with respect to the AC frequency at three different graphite filler loadings. Three graphite volume concentrations are

na

selected for investigation, i.e. c1  2.38% , 3.98% and 4.79%, in accordance with the experimental data. The first case corresponds to that below the percolation threshold, while the

ur

last two refer to those above the percolation threshold. The calculation is conducted with all categories of interface effects. The effective permittivity and conductivity, respectively,

Jo

decreases and increases with respect to the AC frequency. It is found that the graphite volume concentration has significant influence on the effective permittivity and conductivity at a low AC frequency, but the significance reduces markedly at high AC frequency range. The predicted permittivity and conductivity agree with the experimental data within the frequency range from 102 to 107 Hz [12]. The present frequency-dependent homogenization scheme is thus validated for the effective permittivity and conductivity of graphite/PVDF composites under AC electrical loading.

6.3. The dielectric damage process of graphite-polymer composite

15

Journal Pre-proof Then we investigate the influence of graphite volume concentration on the dielectric damage and breakdown process of graphite-polymer composite under AC electrical loading. The selected graphite volume concentrations are identical to those in Sect. 6.2. Fig. 7(a) shows that the calculated electric displacement of the graphite-polymer composite with respect to the electric field at 100 Hz. At the first stage ( E  Ec ), no dielectric damage process happens. The effective electric displacement increases linearly with respect to the electric field, and the growth rate of the electric displacement enhances with respect to the graphite volume concentration, which can be attributed to the larger effective permittivity at higher graphite filler loading. Then,

of

when the external electric loading exceeds the threshold electric field ( E  Ec ), the dielectric

ro

damage process occurs. The electrical displacement continues to increase with the electric field before the dielectric breakdown phenomenon occurs. The dielectric energy storage density also

-p

increases nonlinearly with respect to electric field, as revealed by the U E  E curves of the

re

graphite-polymer composite in Fig. 7(b). This trend also agrees with the experimental data of energy storage density for BaTiO3/PVDF composites [55]. Next, when the external electric

lP

loading reaches the dielectric breakdown strength, the effective electrical displacement and energy storage density of the composite both rapidly decrease to zero. At this stage the dielectric

na

breakdown phenomenon occurs for the graphite-polymer composite. When looking into this phenomenon further, we can see that the dielectric breakdown can

ur

be predicted by the evolution of dielectric damage process. The damage parameter is equal to

Jo

zero when E  Ec , as depicted in Fig. 8(a). Then the dielectric damage process proceeds when the external electric loading reaches the threshold electric field, Ec . When the electric field increases, the dielectric damage parameter, D , increases rapidly from 0 to 1. Fig. 8(b) displays the dielectric damage rate with respect to the electric field. It is revealed that the dielectric damage rate initially remains zero as E  Ec . Afterward, the dielectric damage rate increases as the dielectric damage process proceeds. In addition, the maximum damage rate increases with respect to the graphite volume concentration, which results in the lower dielectric breakdown strength at a higher graphite volume concentration. The underlying physics can be attributed to the increasing number of micro regions that lost its dielectric permittivity at higher graphite volume concentration.

16

Journal Pre-proof 6.4. The dielectric breakdown strength and energy storage density of graphite-polymer composite Finally, we investigate the dielectric breakdown strength and energy storage density of the graphite-polymer composite. As depicted in Figs. 9(a) and (b), the experimental data appear to be in good agreement with the theoretical predictions after all categories of interface effects are considered [12]. The comparison confirms that the filler- and frequency-dependent interface effects are both essential in order to obtain an accurate dielectric breakdown strength and energy

of

storage density of the graphite-polymer composite. The dielectric breakdown strength decreases with the increase of graphite volume concentration, as shown by the blue line of Fig. 9(a).

ro

However, the energy storage density increases with it, as shown by the blue line of Fig. 9(b). The graphite inclusions have positive influence on the energy storage of graphite-polymer composites.

-p

The opposing trends between the dielectric breakdown strength and energy storage density can

re

be explained by the evolution of the dielectric damage parameter with respect to the electric field at different graphite volume concentrations, as illustrated in Fig. 8(a). When the graphite volume

lP

concentration increases, it leads to an increasing volume percentage for the dielectric damaged micro-regions, which in turn lower the dielectric breakdown strength of the composite. In the

na

meantime, an enhanced effective permittivity of the composite is achieved at a higher graphite volume concentration.

ur

In addition, Figs. 10(a) and (b) reveal the influence of AC frequency on the dielectric breakdown strength and energy storage density of graphite-polymer composite. Fig. 10(a)

Jo

indicates that the dielectric breakdown strength of the composite increases with respect to the AC frequency. The underlying physics is ascribed to the slower dielectric damage process at a higher AC frequency. However, the energy storage density reveals a decreasing trend, as shown in Fig. 10(b), which can be attributed to the lower dielectric permittivity at the high frequency range.

6.5. Further discussions of the model: (i) effect of aspect ratio; (ii) variation of driving force with respect to the dielectric damage process (i) Effect of aspect ratio Now, we investigate the influence of the aspect ratio of graphite fillers on the overall behavior of the graphite/PVDF composite. It can be concluded from Fig. 11 that, when the aspect ratio of graphite filler increases, the dielectric breakdown strength will increase, while the energy 17

Journal Pre-proof storage density will decrease. In addition, as the aspect ratio increases continuously, the graphite volume concentration will be lower than the percolation threshold. Then, the energy storage density decreases nearly to zero and the dielectric breakdown strength enhances significantly. Other microstructure parameters can also be investigated via the similar procedure. (ii) Variation of driving force with respect to the dielectric damage process Next, we illustrate the variation of driving force with respect to the dielectric damage process. Fig. 12 shows that the driving force of dielectric damage increases with the dielectric damage parameter. This trend agrees with that of the mechanical damage process for

ro

of

graphene/metal nanocomposites [50].

6.6. Additional tests of the theory with graphene/PVDF and graphite/epoxy composites

-p

As a further test of the present theory, we now consider two more material systems, i.e. graphene/PVDF composite and graphite/epoxy composite. These two composites have been

re

tested by Uyor et al. [56] and Patsidis et al. [57], respectively. The material parameters utilized

lP

in the first one are listed in Table 2, and those in the second one in Table 3. For graphene/PVDF composite, the predicted results of electrical properties agree well with

na

the experimental data [56]. The effective permittivity increases with respect to the graphene volume concentration at 100 Hz after considering all categories of interface effects, as depicted in Fig. 13(a). In Fig. 13(b), the dielectric breakdown strength and energy storage density of the

ur

composite decrease and increase with respect to the graphene volume concentration, respectively.

Jo

The overall trends are very similar to those of graphite/PVDF composite in Figs. 9(a) and (b). In addition, Figs. 14(a) and 14(b) show that the calculated results of graphite/epoxy composite again agree well with the corresponding experimental data [57].

7. Conclusions In the present work, we have developed a homogenization scheme for the dielectric breakdown strength and energy storage density of graphite-polymer composites with dielectric damage process. First, we introduce an effective-medium approximation for the complex dielectric permittivity in the context of AC electrical loading. Then, three categories of interface effects are considered, including the imperfect bonding between the constituent phases, the Maxwell-Wagner-Sillars polarization and Debye dielectric relaxation for the interfacial 18

Journal Pre-proof permittivity, and the electron tunneling and electron hopping for the interfacial conductivity. Next, the thermodynamic driving force and dielectric damage potential are derived for the dielectric damage process. We establish the evolution equation of the dielectric damage parameter based on the DDM. These interface effects and the dielectric damage process are integrated into the original effective-medium approximation to constitute the whole frequencydependent homogenization scheme for the dielectric breakdown strength and energy storage density of graphite-polymer composites. After the homogenization scheme is developed, we have validated it with the experimental

of

data of graphite/PVDF composites over a wide range of graphite volume concentration under AC

ro

electrical loading. It is demonstrated that the dielectric breakdown strength of the graphitepolymer composite decreases with respect to the graphite volume concentration, but increases

-p

with the AC frequency. However, the corresponding dielectric energy storage density reveals an opposite trend. It increases with respect to the graphite volume concentration, and decreases with

re

the AC frequency.

lP

We conclude by saying that the present theory is capable of quantitatively predicting the effective dielectric breakdown strength and energy storage density for the graphite-polymer

na

composite at a given graphite filler loading and AC frequency. The present homogenization theory could be applied to provide guidance for the design and application of carbon-based

Jo

Competing interests

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composites in energy storage devices.

We declare we have no competing interests. Acknowledgements X.D. Xia thanks the support from National Natural Science Foundation of China (Grant No. 11902365) and the 'Future Talents' short-term scholarship from TU Darmstadt. G. J. Weng thanks the support from NSF Mechanics of Materials and Structures Program under CMMI1162431. X.Z. Xiao thanks the support from National Natural Science Foundation of China (Grant No. 11802344) and Natural Science Foundation of Hunan Province, China (Grant No. 2019JJ50809).

19

Journal Pre-proof Data Availability This work was not based on original data, except the experimental curves published in Ref. [12] https://journals.sagepub.com/doi/pdf/10.1177/0021998317693675 for comparison with our theory. References [1] J. Zhu, D. Yang, Z. Yin, Q. Yan, H. Zhang, Graphene and graphene-based materials for

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Journal Pre-proof Table 1 Physical and microstructural features utilized in the numerical calculation of the dielectric breakdown strength and energy storage density of graphite/PVDF composite for validation with Weng et al. [12]. Physical and microstructural features of the constituent phases

Values

* Percolation threshold of the two-phase composite, c1

0.0249

Aspect ratio of the graphite filler, 

0.0148

5.0  10 6

Thickness of the interlayer phase, h (m)

6.0  10 7

ro

Electrical conductivity of PVDF phase,  0 (S/m)

of

Thickness of the graphite filler,  (m)

-p

Relative dielectric permittivity of PVDF phase,  0 /  vac

re

(int) Electrical conductivity of the interlayer phase at the static state,  0 (S/m) (int) Relative dielectric permittivity of the interlayer phase at the static state,  0 /  vac

lP

 Scale parameter for the electronic tunneling at the static state,  static

na

 Scale parameter for MWS polarization at the static state,  static  Scale parameter for MWS polarization at the infinite frequency,  inf

3.2 1011 6

1.0 104 3 0.0105

9.3 1011 5.5 1010 6.0  10 6

Relaxation time of dielectric relaxation for interfacial permittivity, t (s)

2.0 104

Jo

ur

Characteristic time of electron hopping for interfacial conductivity, t (s)

Dielectric damage exponent of the two-phase composite, m

-2

Dielectric damage exponent of the two-phase composite, n

-1

Threshold electric field for dielectric damage process, Ecr (V/m) Maximum volume concentration of micro-region losing permittivity for PVDF,  0

5.0 104 0.01

Parameter reflecting c1 -influence on the generation of micro-region, 

1.0110 6

Energy strength for the dielectric damage process, S 0

1.39 1015

25

Journal Pre-proof Table 2 Physical and microstructural features utilized in the numerical calculation of the dielectric breakdown strength and energy storage density of graphene/PVDF composite for validation with Uyor et al. [56]. Physical and microstructural features of the constituent phases

Values 0.0078

* Percolation threshold of the two-phase composite, c1

0.00448

Thickness of the graphene filler,  (m)

5.0  10 8

Thickness of the interlayer phase, h (m)

6.0  10 9

ro

Electrical conductivity of PVDF phase,  0 (S/m)

of

Aspect ratio of the graphene filler, 

-p

Relative dielectric permittivity of PVDF phase,  0 /  vac

re

(int) Electrical conductivity of the interlayer phase at the static state,  0 (S/m) (int) Relative dielectric permittivity of the interlayer phase at the static state,  0 /  vac

lP

 Scale parameter for the electronic tunneling at the static state,  static

na

 Scale parameter for MWS polarization at the static state,  static

3.2  10 10 10

1.0 104 2

1.05 101 5.3 10 9 5.5  10 7

Characteristic time of electron hopping for interfacial conductivity, t (s)

6.0  10 2

Relaxation time of dielectric relaxation for interfacial permittivity, t (s)

1.2 103

Jo

ur

 Scale parameter for MWS polarization at the infinite frequency,  inf

Dielectric damage exponent of the two-phase composite, m

-2

Dielectric damage exponent of the two-phase composite, n

-1

Threshold electric field for dielectric damage process, Ecr (V/m) Maximum volume concentration of micro-region losing permittivity for PVDF,  0

5.0 104 0.185

Parameter reflecting c1 -influence on the generation of micro-region, 

1.0110 6

Energy strength for the dielectric damage process, S 0

2.86 1017

26

Journal Pre-proof Table 3 Physical and microstructural features utilized in the numerical calculation of the dielectric breakdown strength and energy storage density of graphite/epoxy composite for validation with Patsidis et al. [57]. Physical and microstructural features of the constituent phases

Values 0.0050

* Percolation threshold of the two-phase composite, c1

0.00286

Thickness of the graphite filler,  (m)

5.0  10 6

Thickness of the interlayer phase, h (m)

6.0  10 7

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Electrical conductivity of epoxy phase,  0 (S/m)

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Aspect ratio of the graphite filler, 

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Relative dielectric permittivity of epoxy phase,  0 /  vac

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(int) Electrical conductivity of the interlayer phase at the static state,  0 (S/m) (int) Relative dielectric permittivity of the interlayer phase at the static state,  0 /  vac

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 Scale parameter for the electronic tunneling at the static state,  static

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 Scale parameter for MWS polarization at the static state,  static

3.2  10 10 2

1.0 109 1

1.05 101 1.3 105 5.5  10 2

Characteristic time of electron hopping for interfacial conductivity, t (s)

6.0  10 2

Relaxation time of dielectric relaxation for interfacial permittivity, t (s)

1.2 103

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 Scale parameter for MWS polarization at the infinite frequency,  inf

Dielectric damage exponent of the two-phase composite, m

-2

Dielectric damage exponent of the two-phase composite, n

-1

Threshold electric field for dielectric damage process, Ecr (V/m) Maximum volume concentration of micro-region losing permittivity for epoxy,  0

5.0 104 0.01

Parameter reflecting c1 -influence on the generation of micro-region, 

1.0110 6

Energy strength for the dielectric damage process, S 0

3.44 1021

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Figure Captions Fig. 1. The surface SEM image of graphite-PVDF composites. (Reprinted with permission from Weng et al., J. Compos. Mater. 51, 3769-3778 (2017). Copyright 2017 SAGE Publications.) Fig. 2. Schematic of the graphene-polymer nanocomposites subjected to AC electrical loading: (a) before dielectric damage process, (b) with micro-region that loses its dielectric permittivity. Fig. 3. Schematic of Maxwell-Wagner-Sillars effects due to the formation of numerous nanocapacitors in the graphite-polymer composites.

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Fig. 4. The effective (a) dielectric permittivity and (b) electrical conductivity of the coated graphite filler with respect to the AC frequency under different graphite volume concentrations.

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Fig. 5. The effective (a) dielectric permittivity and (b) electrical conductivity of graphite-polymer composite with respect to the graphite volume concentration under the perfect and imperfect interface conditions.

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Fig. 6. The effective (a) dielectric permittivity and (b) electrical conductivity of graphite-polymer composite with respect to the AC frequency under different graphite volume concentrations.

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Fig. 7. The effective (a) electric displacement (b) dielectric energy density of graphite-polymer composite with respect to the electric field under different graphite volume concentrations.

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Fig. 8. The (a) dielectric damage parameter and (b) dielectric damage rate of graphite-polymer composite with respect to the electric field under different graphite volume concentrations. Fig. 9. The (a) dielectric breakdown strength (b) energy storage density of graphite-polymer composite with respect to the graphite volume concentration under perfect and imperfect interface conditions.

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Fig. 10. The (a) dielectric breakdown strength and (b) energy storage density of graphite-polymer composite with respect to the AC frequency under different graphite volume concentrations.

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Fig. 11. The dielectric breakdown strength and energy storage density of graphite-polymer composite with respect to the graphite aspect ratio.

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Fig. 12. The driving force of dielectric damage with respect to the dielectric damage parameter for graphite-polymer composite. Fig. 13. The effective (a) dielectric permittivity, (b) dielectric breakdown strength and energy storage density of graphene/PVDF composite with respect to the graphite volume concentrations at 100 Hz. Fig. 14. The effective (a) dielectric permittivity, (b) dielectric breakdown strength and energy storage density of graphite/epoxy composite with respect to the graphite volume concentrations at 100 Hz.

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Fig. 1. The surface SEM image of graphite-PVDF composites. (Reprinted with permission from Weng et al., J. Compos. Mater. 51, 3769-3778 (2017). Copyright 2017 SAGE Publications.)

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AC electrical

Graphite fillers as the inclusions

loading

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Percolation path

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Polymer as the matrix

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(a)

AC electrical

Graphite fillers as the inclusions

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loading

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Percolation path

Polymer as the matrix Dielectric damaged micro-regions

Filler and frequency dependent interface (b) Fig. 2. Schematic of the graphene-polymer nanocomposites subjected to AC electrical loading: (a) before dielectric damage process, (b) with micro-region that loses its dielectric permittivity. 30

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+

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+ + + + + +

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AC electrical loading

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Fig. 3. Schematic of Maxwell-Wagner-Sillars effects due to the formation of numerous nanocapacitors in the graphite-polymer composites.

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(a)

(b) Fig. 4. The effective (a) dielectric permittivity and (b) electrical conductivity of the coated graphite filler with respect to the AC frequency under different graphite volume concentrations.

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(a)

(b) Fig. 5. The effective (a) dielectric permittivity and (b) electrical conductivity of graphite-polymer composite with respect to the graphite volume concentration under the perfect and imperfect interface conditions.

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(a)

(b) Fig. 6. The effective (a) dielectric permittivity and (b) electrical conductivity of graphite-polymer composite with respect to the AC frequency under different graphite volume concentrations.

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(a)

(b) Fig. 7. The effective (a) electric displacement (b) dielectric energy density of graphite-polymer composite with respect to the electric field under different graphite volume concentrations.

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(b) Fig. 8. The (a) dielectric damage parameter and (b) dielectric damage rate of graphite-polymer composite with respect to the electric field under different graphite volume concentrations.

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(b) Fig. 9. The (a) dielectric breakdown strength (b) energy storage density of graphite-polymer composite with respect to the graphite volume concentration under perfect and imperfect interface conditions.

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(a)

(b) Fig. 10. The (a) dielectric breakdown strength and (b) energy storage density of graphite-polymer composite with respect to the AC frequency under different graphite volume concentrations.

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Fig. 11. The dielectric breakdown strength and energy storage density of graphite-polymer composite with respect to the graphite aspect ratio.

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Fig. 12. The driving force of dielectric damage with respect to the dielectric damage parameter for graphite-polymer composite.

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(a)

(b) Fig. 13. The effective (a) dielectric permittivity, (b) dielectric breakdown strength and energy storage density of graphene/PVDF composite with respect to the graphite volume concentrations at 100 Hz.

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(a)

(b) Fig. 14. The effective (a) dielectric permittivity, (b) dielectric breakdown strength and energy storage density of graphite/epoxy composite with respect to the graphite volume concentrations at 100 Hz.

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Journal Pre-proof Credit Author Statement Xia developed the original idea, made computations; Xu and Xiao contributed to discussions; Weng developed

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the theory, built the irreversible thermodynamics with Xia, and finalized the manuscript.

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Declaration of Interest Statement The authors declare no potential conflicts of interest with respect to the research, authorship, and

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publication of this article.

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Graphical abstract

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Highlights 1. Dielectric breakdown strength and energy storage density are predicted by the homogenization scheme and validated by the experimental data.

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2. Energy storage properties of overall composite are connected to the microstructural features

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of constituent phases as well as the filler and frequency dependent interface effects.

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3. Thermodynamic evolution equations with dissipation potential and driving force are derived

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for the dielectric damage process.

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