Effective thermal conductivity of polymer composites

International Journal of Heat and Mass Transfer 117 (2018) 358–374

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Review

Effective thermal conductivity of polymer composites: Theoretical models and simulation models Siping Zhai a, Ping Zhang a,⇑, Yaoqi Xian a, Jianhua Zeng a, Bo Shi b a Guangxi’s Key Laboratory of Manufacturing Systems and Advanced Manufacturing Technology, School of Mechanical and Electrical Engineering, Guilin University of Electronic Technology, No. 1 Jinji Road, Guilin, Guangxi 541004, China b College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Jiangsu 210016, China

a r t i c l e

i n f o

Article history: Received 21 June 2017 Received in revised form 28 August 2017 Accepted 17 September 2017

Keywords: Thermal conductive polymer materials Polymer composite Effective thermal conductivity Theoretical model Simulation model

a b s t r a c t Polymer composites of highly effective thermal conductivity (ETC) are commonly used in various industries, for renewable energy systems and Electronic Systems. Owing to the low thermal conductivity (TC) of polymers, inserting particles with ultra-high TC into the polymer matrix makes it possible for polymer composites to possess high ETC. The ETC of polymer composites is determined by several factors, including the particle and matrix properties and microscopic structures. Modelling methods are powerful tools to understand how these factors influence the ETC of polymer composites. Modelling methods can be combined with experimental data and can be used to qualitatively and quantitatively analyse the impact of various factors on the ETC. Moreover, modelling methods can be used as a guide for the choice and design of particle-filled composites for engineering applications. Herein, we review the recent research on ETC models of polymer composites. First, the classical theoretical models of ETC for polymer composites are introduced. Then, novel theoretical and simulation models are described. We focus on the influence of the theoretical models and the simulation models of polymer composites at multiple scales. Finally, we conclude and give an outlook regarding the ETC models of polymer composites. Ó 2017 Elsevier Ltd. All rights reserved.

Contents 1. 2. 3.

4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classical theoretical models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical models for composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Effect of particles on ETC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1. Complex shape particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2. Folded and crooked particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3. Hybrid particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Effect of microstructures on ETC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Aggregation structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. Connection mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3. Pore structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Effect of volume fraction on ETC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation models of composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Simulation models for interface resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Undressed interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Modified interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Simulation models for bulk composites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Microscale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2. Mesoscale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3. Macroscale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

⇑ Corresponding author. E-mail address: [email protected] (P. Zhang). https://doi.org/10.1016/j.ijheatmasstransfer.2017.09.067 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.

360 361 361 361 361 363 363 364 364 365 367 368 369 369 369 369 370 370 370 371

S. Zhai et al. / International Journal of Heat and Mass Transfer 117 (2018) 358–374

359

Nomenclature Symbols f volume fraction of particle (vol%) RI interfacial thermal resistance between particle and matrix (Km2/W) keff effective thermal conductivity of composites (W/mK) km thermal conductivity of matrix (W/mK) thermal conductivity of particle (W/mK) kp eff

kp

r j Li keff ;i keff

effective thermal conductivity of particle adding the effect of RI radius of particle phase of B–L model or EMT depolarization factor effective thermal conductivity of composites along the i-axis (i = x, y and z) effective thermal conductivity tenser of composites

km thermal conductivity tenser of matrix kp thermal conductivity tenser of particle sp Eshelby tensor of particle I identity tensor p aspect ratio of particle B0 , B1 , B2 experimental coefficients transformational coordinates n, f, g a orientation angle of h-BN c c kg , knf effective thermal conductivities of the unit cell along the g direction and in the n f plane L, W, H length, width, height of unit cell a, b size of h-BN sðiÞ second-rank Eshelby tensor common to the ith heterogeneity and all of its layers f ðiÞa volume fraction of the ai th layer of the ith heterogeneity i

ða Þ AðiÞi

h n hm C

m

Keff K D keff ðnÞ kP ðnÞ c ki ðnÞ RI ðnÞ F Le e

kp LCNT LGNP HðpÞ ya k k11 , k12 , s km;equ aK hopt kp;i

global strain concentration tensor thickness of particle number of layers of Multilayer graphene thickness of monolayer graphene specific heat phonon velocity effective phonon mean free path phonon mean-free path lateral sizes of multilayer graphene layer-dependent effective thermal conductivity of composites layer-dependent thermal conductivity of graphene equivalent thermal conductivity of composites along the i-axis (i = x, y and z) layer-dependent interface resistance flatness ratio Equivalent length of graphene nanoplate or carbon nanotube equivalent thermal conductivity of graphene nanoplate or carbon nanotube length of carbon nanotube length of graphene nanoplate geometrical factor amplitude of the wavy penny-shaped ellipsoidal heterogeneity length of the wavy penny-shaped ellipsoidal heterogeneity k22 components of the effective compliance tensor synergistic effect parameter equivalent thermal conductivity of unit cell Kapitza radius, km RI optimal particle ratio thermal conductivity of particle i

fi fm fc

volume fraction of particle i mass fraction of particle (wt%) critical volume fraction of particle or percolation threshold knc effective thermal conductivity of the dead-end particle f nc volume fraction in the dead-end particle effective thermal conductivity the single aggregate ka fb volume fraction of the backbone particles fa volume fraction of the aggregates f int volume fraction of the particles in an aggregate effective thermal conductivity in parallel with the direck1 tion of the heat flow ko , kr thermal conductivity of the layer with the oriented and random graphite flakes q, qo , qr density of the composite and the oriented and randomly graphite flakes kp;e thermal conductivity of the element 0

f v oid f v oid

effective fraction of pores volume fraction of the pore

km

effective thermal conductivity of the effective matrix

eff 0

keff pv oid f matrix kv oid f max qp , qm ks fs kv l c

c

kx , kz

effective thermal conductivity of the porous composite shape factor of pores volume fraction of the matrix thermal conductivity of the pore maximum volume fraction densities of the particle and matrix thermal conductivity of the solid phase volume fraction of the solid phase thermal conductivity of the gas and liquid phases effective thermal conductivity of the cell in-plane and through-thickness

Abbreviations TC thermal conductivity ETC effective thermal conductivity EMA effective medium approximation M-G Maxwell–Garnett H–J Hasselman–Johnson B–H Bruggeman–Hanai IE integral embedding ITR interfacial thermal resistance B–L Bruggeman–Landauer EMT effective medium theory M-T Mori–Tanaka MWCNT multi-walled carbon nanotube MD molecular dynamics GNP graphene nanoplate CNT carbon nanotube SWCNT single-walled carbon nanotube CTR contact thermal resistance SEM scanning electron microscopy MC Monte Carlo FEM finite element method EMD equilibrium molecular dynamics NEMD non-equilibrium molecular dynamics RNEMD reverse non-equilibrium molecular dynamics DPD dissipative particle dynamics LBM lattice Boltzmann method LGA lattice gas automata SPH smoothed particle hydrodynamics TIM thermal interface materials FDM finite difference method EFG element-free Galerkin

360

5.


4.2.4. Multiscale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conflict of interest. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Supplementary material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction Polymers are common in industrial production because they are lightweight and show excellent corrosion resistance and electrical insulation, as well as many other advantages. Nevertheless, their thermal application is limited by their low thermal conductivity (TC). Hence, particles with ultra-high TC are frequently added to the polymers to form particle-filled polymer composites in order to enhance the thermal conductive performance. Polymer composites of high TC, needed for effective thermal dissipation, are widely used in renewable energy systems and Electronic Systems [1–3], such as light-emitting diodes [4–6], Li-ion batteries [7,8], solar cells

371 372 372 372 372 372

[7,9–11] and microelectronic packaging [12–14]. The local overheating and heat accumulation that occurs due to increasing power density may lead to the degradation and failure of functional systems. For example, the solar cell is a renewable energy system [15,16] and has important research significance. However, the conversion efficiency of solar cells is severely decreased with increasing operating temperatures, which influences the efficient and stable conversion of solar energy [17]. Therefore, rapid heat transfer from these systems to ensure a safe and reliable operation is essential. Using polymer composites of high TC can effectively enhance the heat transfer of the energy systems. The TC of polymer composites, as well as the effective thermal conductivity (ETC), is

Fig. 1. The relationship diagram of the classical theoretical models.

Table 1 List of equations and remarks of frequently used classical theoretical models. Model M-G model [21–23]

Equation keff ¼ km

Fricke model [24] keff ¼ kp H–J model or Benvensite model [25,34] B–H model [26,27] Every model [28] B–L model or EMT [23,30]

keff ¼ km 1f ¼

Remarks 2km þkp þ2f ðkp km Þ 2km þkp f ðkp km Þ

h i

eff 2km þkeff p þ2f ðkp km Þ

2km þkp f ðkp km Þ eff

kp keff km kp

kp keff

eff

1=3

ð1þ2mÞ=ð1mÞ h i keff kp ð1mÞ 3=ð1mÞ 3 ð1 f Þ ¼ kkm km kp ð1mÞ eff P kj keff j f j k þðd1Þk ¼ 0, when j ¼ 2 and d ¼ 3, B-L model is EMT model. eff

keff ;x ¼ keff ;y ¼ km keff ;z ¼ km

M-T model [33]

S is related to the kp , km and the shape of particle

1

1þf ðS1Þ

j

Nan model [31]

km kp

1þf S

keff ¼ km

2þf ½bx ð1Lx Þð1þcos2 hÞþbz ð1Lz Þð1cos2 hÞ 2f ½bx Lx ð1þcos2 hÞþbz Lz ð1cos2 hÞ

1þf ½bx ð1Lx Þð1cos hÞþbz ð1Lz Þ cos h 1f ½bx Lx ð1cos2 hÞþbz Lz cos2 h 2

m ¼ kmrRI j is the phase and d ¼ L1i bi ¼ k

kp;i km , ðkp;i km Þ

m þLi

Li and cos2 h are related to the geometry

2

1 1 1 I þ f sp I Ap sp I þ f sp A p sp

1 Ap ¼ km kp km


361

Fig. 2. Distribution of the h-BN composites and structure parameters of unit cell.

determined by many factors, including the properties of the particle, the properties of the matrix and the microstructures of the composite [18,19]. To understand the effect of these factors on the ETC of polymer composites, modelling methods – both theoretical models and simulation models – are powerful tools [20]. Starting from Maxwell’s [21] work more than a hundred years ago, scholars have proposed many models to study the ETC of particle-filled composites. For experimental research, modelling methods can be used not only to define the scope of the parameter settings but also to analyse the experimental results. When experiments are expensive to conduct or when it is hard to reproduce true working conditions, simulation models can address these challenges to describe the physical processes. Furthermore, qualitative and quantitative analyses of modelling methods can guide the choice and design of polymer composites for actual engineering applications. Modelling methods can be classified into two main categories: theoretical models and simulation models. The theoretical modelling method can describe the heat-transfer mechanisms while the simulation modelling method can accurately describe the characteristics of microstructures. This review describes the recent analyses and investigations regarding the modelling methods of ETC of particle-filled polymer composites. These modelling methods include theoretical models and simulation models. We focus on the effects of various factors in theoretical models of the ETC of particle-filled polymer composites. In addition, we describe the different scales of simulation models of particle-filled polymer composite ETC. 2. Classical theoretical models The classical theoretical model of the ETC for composites is divided into two major categories: effective medium approximation (EMA) and the micromechanics method. The fundamental model of the EMA is the Maxwell–Garnett (M-G) model [21–23], which is suitable for the lower volume fraction of particles (f ) or particles that are far apart from each other. The Fricke’s model [24] extended the M-G model with ellipsoidal particles. Many scholars have expanded the M-G model, such as Hasselman–Johnson (H–J) [25], who considered the effect of the interface and particle size. The Bruggeman–Hanai (B–H) [26,27] model improved the M-G model by using the integral embedding (IE) principle and adding the effect of thermal interaction, thus it is applicable

for higher f . Then, on the basis of the B–H model, Every’s [28] model added the influence of interfacial thermal resistance (ITR) between the particles and matrix, for which the symbol is RI . The Bruggeman–Landauer (B–L) [26,29] model is more suitable for complex composites. When the composites are two-phase, as well as a three-dimensional block structure, the B–L model is the wellknown effective medium theory (EMT) model [23,30]. The Nan [31] model is a more general EMA model that considers the effect of f , RI , particle size and orientation distribution. The major micromechanics methods are the variational principle and the mean field approximation, including Hashin–Shtrikman bounding [32], the Mori–Tanaka (M-T) model [33] and Benvensite’s model [34]. Meanwhile, many researchers have proposed semi-empirical models [35–37]. The relationship diagram of the classical theoretical models is shown in Fig. 1. The equations of frequently used classical theoretical models are summarised in Table 1. 3. Theoretical models for composites 3.1. Effect of particles on ETC 3.1.1. Complex shape particles The shape of the particles is one of the most significant factors that affect the ETC of composites. Many classical theoretical models of ETC for composites have considered the particle shape but all these models assume that the shape of particles was a fundamental shape, such as a sphere, ellipsoid, fibre or sheet. Based on this assumption, these models consider the shape of a particle using a simple aspect ratio p. The definition of p is not same under different particle shapes. In fact, some particles have a complex shape that cannot be simply described with p. Wang et al. [38] paid attention to the roles of artificially designed complex particle shapes (Y shaped, double-Y shaped and quad-Y shaped, etc.) in the ETC of composites. Based on the correlations, they proposed a semiempirical model of keff with particles of different shapes:

keff ¼ B0 km þ B1 km f þ B2 kp

ð1Þ

where km and kp are the intrinsic TC of the matrix and particle, respectively, and B0 , B1 and B2 are coefficients that vary with the shape of the particles according to the particular experiment. By the unit cell method, Chen and associates [39] proposed a model

362


Fig. 3. Sketch of (a) the folded, wrinkled and equivalent GNPs and (b) the crooked and equivalent CNTs.

to calculate the ETC of aligned hexagonal boron nitride (h-BN) platelet polymer composites. The sizes of the unit cell and h-BN are shown in Fig. 2. The coordinate transformation method – i.e. transforming the n f g coordinate into the x y z coordinates – was used to manage the h-BN orientation angle a with the x axis (Fig. 2). Further, the anisotropic ETC of composites is equal to that in the unit cell and can be expressed as: c

c

2

keff ;x ¼ knf cos2 a þ kg sin a c

2

c

c knf

ð2aÞ

c

keff ;z ¼ knf sin a þ kg cos2 a

ð2bÞ

where kg and are the ETCs of the unit cell along the g direction and in the n f plane, which are defined as

H

c

kg ¼ 2d

H km

c knf

þ 2LWk

mþ

2 14 ¼ L 2 2d þ k m

ð2cÞ

p2LWb ffiffi 2 3a ðkp;g km Þ

3 L ffiffi p2LWa 2HWkm þ 3abðkp;nf km Þ

þ 2d

W

W

km

þ pffiffi3HLk

5

2LHa pffiffi m þ0:75 3abðkp;g km Þ

ð2dÞ Here, kp;g and kp;nf are the TC of h-BN along the g direction and in the n f plane, respectively. This model uses the equivalent thickness of the interfacial thermal barrier to contain RI. Proceeding from the perspective of micromechanics [40], the effect of geometry and p of the heterogeneity were contained in sðiÞ (second-rank Eshelby tensor common to the ith heterogeneity and all of its layers). Kim et al. supposed that the matrix contains l distinct types of ellipsoidal heterogeneities (i ¼ 1; 2; . . . ; l), each consisting of ni layers (ai ¼ 1; 2; . . . ; ni , i ¼ 1; 2; . . . ; l). Each type of heterogeneity has distinct thermal properties, shapes, and orientation distributions. The modified M-T model for the keff of composites can be expressed as:

(

keff ¼ km (

Iþ

" n #) l 1 i X X ðai Þ Iþ f ðiÞa sðiÞ I AðiÞ sðiÞ i¼1

" n l i X X i¼1

ai ¼1

i

ai

f ðiÞa i

ða Þ sðiÞ AðiÞi

sðiÞ

1

#)1 ð3Þ

ða Þ

where f ðiÞa and AðiÞi are the f and the global strain concentration i

tensor of the ai th layer of the ith heterogeneity (ai ¼ 1; 2; . . . ; ni , i ¼ 1; 2; . . . ; l), and the middle dot ‘‘” is used to denote the tensor single-dot product. The experimental data about the ETC of the composites with diverse length multi-walled carbon nanotubes (MWCNTs) match the model values although the model slightly over-estimates the values, which is mainly because this model assumes particles are well dispersed in the matrix and ignores the RI . Recently, Huang et al. [41] considered the effect of particle size distribution on the ETC and used the lognormal distribution function to approximate the distribution and improved the Nan model by using the mean particle size, which was obtained by the lognormal distribution function instead of the intrinsic particle size. It is noteworthy that this model assumed a spherical particle. The shape of graphene, including the radial size and the lateral size, often show a significant impact on the ETC of graphene/polymer composites [42,43]. Shahil and co-workers [44,45] treated the layer as the thickness of the multilayer graphene to improve and perfect Nan’s model. The thickness of multilayer graphene, h, is defined as the product of n and hm , where n and hm are the layers of the multilayer graphene and the thickness of monolayer graphene, respectively. Meanwhile, the size of the graphene and RI are also added to this model. This is given by:

keff ¼ kp

3km þ 2f kp km ð3 f Þkp þ km f þ

RI km kp f nhm

ð4Þ

where n is the number of layers in the multilayer graphene. According to the kinetic theory that Shahil and co-workers used kp ¼ ð1=3ÞC mKeff , where 1=Keff ¼ 1=K þ 1=D, to account for the effect of the particle size. Here, Keff is the effective phonon mean free path. It is noteworthy that this model only considers the lateral size effect without substrate scattering. Subsequently, some scholars considered the layers of graphene and the impact of the layers on the TC of particles and RI . Shen et al. [46] built a threedimensional model that contained a graphene sheet embedded within an epoxy system to calculate the microscopic coefficients and improved Nan’s model by using the number of graphene layers and the lateral size to calculate the macroscopic coefficients. Shen

363


Fig. 4. The schematic diagram of (a) a unit cell with a half GNP/SWCNTs and half GNP sandwich structure and (b) the composite.

et al. assumed all the graphite nanoplatelets of different shapes were identical. The ETC of composites, keff ðnÞ, is expressed as:

keff ðnÞ ¼

3 þ f 2k

kcz ðnÞkm

ð 3 f 2k

C m þLz kz ðpÞkm

Þ

ð1 Lz Þ þ k

kðnÞkm Ly c m þLz ðkz ðnÞkm Þ

þk

kcx ðnÞkm ð1 c m þLx ðkx ðnÞkm Þ

kCx ðnÞkm Lx c m þLx ðkx ðnÞkm Þ

Lx Þ

keff 2 Ff h iþ1 ¼ km 3 HðFpÞ þ 1= F 2 kp LGNP =2RI kp =km 1 ð5aÞ

where kP ðnÞ is the TC of the embedded graphene sheets, and Li (i = x, y and z) is the depolarization factor related to the particle shape. The equivalent TC of composites along the i-axis (i = x, y and z), c ki ðnÞ, is given by: c ki ðnÞ

¼

kP ðnÞ 1 þ ð1þ2pÞRIhðnÞLi kP ðnÞ

ð5bÞ

where RI ðnÞ is the layer-dependent interface resistance, which is obtained from the molecular dynamics (MD) simulation. 3.1.2. Folded and crooked particles Graphene nanoplates (GNPs) are usually folded and wrinkled and the carbon nanotubes are usually crooked in the matrix; this geometry may lead to a decrease in the ETC of composites [47] as well as an enhancement of the interface connection [48]. The concept of the flatness ratio F for GNPs was proposed by Chu et al. [47]. The flatness ratio F is represented using a simple e fraction: Le =LGNP ¼ kp =kp , where LGNP is the length of the GNP, pðLGNP Þ and gðLGNP Þ are the distribution function of the length and e flatness of GNPs; Le and kp are the equivalent length and equivalent TC of GNPs. Fig. 3(a) shows the structure and the equivalent length of GNPs. Supposing that isotropic composites include randomly embedded GNPs, the ETC of composites on account of Nan’s model is given by:

3þ keff ¼ km

km

2F 2 f

ffiffi

p 2RI þ13:4 h LGNP

3 Ff

forward another model with F and ignored the interaction between the two particles. This model uses the well-known geometrical factor HðpÞ for h and LGNP and is given by:

ð6Þ

pffiffiffi The term 13:4 h is the kp that is only influenced by the thickness of GNPs. This model is suitable for isotropic composites containing randomly embedded GNPs. Chu et al. [48] also put

ð7Þ

The straightness ratio of the carbon nanotubes (CNTs) is identical to F. Fig. 3(b) provides a diagram of both non-straight and straight CNTs. In order to analyse the effect of the nonstraightness of CNTs on the ETC of composites, a simple model by Deng et al. [49,50] is given in the following form:

keff Ff =3

¼1þ km km =F kp = 1 þ 2RI kp =LCNT þ HðFpÞ

ð8Þ

where LCNT is the length of the straightness of the CNTs and kp is the axial TC of CNTs. It is remarkable that RI represents the tube-end interfacial thermal resistance. A different way to deal with both non-flat GNPs [51] and non-straight CNTs [52] was proposed by Kim et al. The geometry of non-flatness and non-straightness is represented using a simple cosine function: y ¼ ya cos ð2px=kÞ, where ya and k represent the amplitude and the length of the wavy penny-shaped ellipsoidal heterogeneity (Fig. 3(a)). Here, the waviness ratio in the lateral direction that resembles the flatness ratio and straightness ratio is defined as ya =k. By modifying the M-T model, the transverse effective thermal conductivity tensor, keff , can be expressed as:

keff ¼

h

1 þ c2

3=2

3=2

k11 ð1 þ c2 =2Þ þ 2k12 ð1 þ c2 Þ

i 1 3c2 =2 þ k22 ðc2 =2Þ ð9Þ

where c ¼ 2pya =k and k11 , k12 and k22 are components of the effective compliance tensor acquired from the M-T model. In this model, Kim et al. assumed that the particles were randomly dispersed in the composites. 3.1.3. Hybrid particles At present, with the consummation of the models of singleparticle composites, researchers have turned their attention to hybrid-particle composites. When composites are filled with sev-

364


Table 2 List of details of theoretical models in Section 3.1. Researchers

Theoretical basis

Range

Considerations

Wang et al. [38]

Empirical formula

f ; complex particle shape

Chen et al. [39]

Infinitesimal method and Fourier law

Kim et al. [40]

Micromechanics

Application range of f is between 1 vol% and 20 vol% and kp =km is less than 100 Test content range of f up to 60 vol% and D from 0.6 lm to 18 lm Test content range of f m was between 0 and 2 wt %

Shahil et al. [44] Shen et al. [46] Chu et al. [47]

EMA and kinetic theory EMA EMA

Chu et al. [48] Deng et al. [49,50]

Micromechanics Micromechanics

Kim et al. [51,52] Yu et al. [55]

Micromechanics EMA

Chen et al. [56] Agrawal et al. [58]

Thermal resistance method and EMA Law of minimal thermal resistance and equal law of specific equivalent thermal conductivity Micromechanics

Yu et al. [59]

Test content range of f 6 10 vol.% Test content range of f was 1–6 vol% for multilayers GNP and 0–25 vol% for few-layers GNP Test content range of f was 0–10 vol% Application range was low T and test content range of f was 0–1 vol% Test content range of f m was 0–20 wt% Test content range of f was 0–65 vol% Application range was near the hopt Application range was less than f c Test content range of f m was 0–2 wt%

eral kinds of particles, a synergistic effect that increases the ETC may occur [53,54]. Several research groups have reported theoretical models to reflect this synergistic phenomenon. Yu et al. [55] made a novel silicone grease that was reinforced with hybrid particles consisting of graphene and aluminium oxide. They found that the synergistic effect of hybrid particles can decrease the RI and increase the ETC of this grease and proposed a correction model by modifying the Every model as follows:

keff s ¼ km ð1 f Þ3ð1RI km =rÞ=ð1þ2RI km =rÞ

ð10Þ

where s is the synergistic effect of graphene and aluminium oxide particles. Furthermore, the experimental data are in reasonable agreement with the predictions of this model. Chen et al. [56] analysed the synergistic effect on the ETC of composites with hybrid particles that consisted of single-walled carbon nanotubes (SWCNTs) and GNPs using a two-step model. In the first step, they calculated the equivalent TC of unit cell km;equ based on the thermal resistance method. The unit cell is conceived as a half GNP/ SWCNTs and half GNP sandwich structure in which the SWCNTs are bridges that connect two GNPs (Fig. 4(a)). This step interprets the synergistic effects, i.e. the heat-transfer network formed by the hybrid particles. Then the composites are treated as an equivalent uniform matrix that consists of unit cells and non-bridging SWCNTs (Fig. 4(b)). Based on the Nan model, the ETC of the composites is given as: kp

keff f p km;equ ¼1þ 3 p þ aK kp km;equ r km;equ

ð11Þ

where f , kp , p and r are the parameters of non-bridging SWCNTs. The experimental results [57] are well matched with this model and the optimal particle ratio hopt is reported. Meanwhile, a simple series model of hybrid-particle-reinforced polymer composites was reported by Agrawal et al. [58] as: 131

2

keff

0 1=3 6XB 1 B 1 12f i ¼ 26 þ 4 @ km k m p i¼1;2

km

1=3 2p 3f i

þ

2 1=3 4f i

9p

p kp;i km

C7 C7 A5

ð12Þ

p; f ; RI ; particle orientation; complex particle shape p; f m ; complex particle shape; heterogeneity orientation distribution p; f ; RI ; complex particle shape p; f ; RI ; complex particle shape h; f ; RI ; F p; f ; RI ; F p; f ; RI ; F p; f m ; F; void p; f ; RI ; hybrid particle synergistic effect p; f ; RI ; synergistic effect r; f ; hybrid particle p; f ; F; hybrid particle synergistic effect; direction distribution

where kp;i and f i are the TC and f of the particle i. However, this model assumes that the particles keep away from each other and ignores the synergistic effect of hybrid particles and RI . Subsequently, based on the micromechanics theory, Yu et al. [59] modified the M-T model to study the synergistic effect of hybrid GNPs and MWCNTs-filled polymer composites on the ETC. This model has the same equation as Kim’s model [40] and they took GNPs and MWCNTs as the first and second heterogeneities to consider the synergistic effect of hybrid particles. The flatness ratio of GNPs and the straightness ratio of MWCNTs are also considered in this model and it is consistent with the experiment results. It is noteworthy that this model assumed the particles were ideally dispersed. Table 2 shows a summary of the details of theoretical models of particle effects, including the theoretical basis, application range, test content range and considerations of these models. 3.2. Effect of microstructures on ETC 3.2.1. Aggregation structure Composites with high particle loading are prone to forming particle aggregates. Aggregation structure can result in an increase in the ETC of composites due to the fast heat-transfer path for neighbouring particles [60]. Prasher et al. [61] and Evans et al. [60] explicitly considered the linear chain aggregate and presented a three-level homogenisation theory to determine the ETC of welldispersed nanocomposites. A single aggregate is assumed to be a dead-end particle (consisting of particles and matrixes) embedded in a backbone. The whole nanocomposite is comprised of the aggregates and matrixes (Fig. 5(a)). The first level calculates the ETC of the dead-end particle knc using the B–L model. This model is given by:

ð1 f nc Þðkm knc Þ f nc kp knc ¼0 þ ðkm þ 2knc Þ kp þ 2knc

ð13aÞ

where f nc is the volume fraction of particles without backbone particles in a dead-end particle. According to Nan’s model, the ETC of the single aggregate, ka , is given by:

ka ¼ knc

3 þ f b ½2bx ð1 Lx Þ þ bz ð1 Lz Þ 3 f b ½2bx Lx þ bz Lz

ð13bÞ


365

Fig. 5. Schematic of (a) the three-level homogenisation model and (b) the two-level homogenisation model.

ka 3 ¼ ð1 f int Þ km

ð14aÞ

where f int is the volume fraction of the particles in an aggregate. The effect of RI is also considered in the first-level model. The second level uses Prasher’s model [61] to demonstrate the ETC of the anisotropic composite:

k1 f f 3 þ ¼ 1 ð1 f int Þ f int f int km

ð14bÞ

where k1 is the ETC in parallel with the direction of the heat flow.

Fig. 6. Sketch map of Zhou’s model.

where f b is the volume fraction of the backbone particles in an aggregate. The effect of RI is embodied in the second-level model. In the third level, the ETC of the whole system using the M-G model is given by:

keff ðka þ 2km Þ þ 2f a ðka km Þ ¼ km ðka þ 2km Þ f a ðka km Þ

ð13cÞ

where f a is the volume fraction of the aggregates in the composites. At the same time, Reinecke et al. [62] built a two-level homogenisation model for nanofluids, which was similar to Prasher’s thinking. Han et al. [63] used this model to study the aggregate effect on the ETC of the composites in an electric field, which led to anisotropic particle distribution. They ignored the linear chain aggregate of particles and assumed that the aggregates are dispersed in the matrix (Fig. 5(b)). In the two-level homogenisation model, the first level is based on the B–H model to compute the ETC of the single aggregate:

3.2.2. Connection mechanism Some researchers argue that particles are in contact with each other or the particle phase is continuous, while others believe that particles are discrete. When the particles are in contact with each other, it will produce contact thermal resistance (CTR) and thermal conduction pathways, which have a profound impact on the ETC of the composites. Considering these problems, researchers introduced many models of the different connection mechanisms between particles, as well as between the particles and the matrix. From scanning electron microscopy (SEM) images, Zhou and coworkers [64] found that natural graphite flakes are highly oriented and formed conductive networks in natural flake graphite/polymer composites at a very high concentration. They established another model, simplifying the composite geometry into a section of randomly oriented graphite flakes that is placed in parallel with the section of the oriented graphite flakes (Fig. 6). Then, according to the representative elementary volume method, the keff is given by:

keff ¼ ko

qf m qð1 f m Þ þ kr qo qr

ð15Þ

where ko and kr are the TC of the layer with the oriented and random graphite flakes; q, qo and qr are the density of the composite and the oriented and randomly graphite flakes, respectively; f m is the mass fraction of the oriented graphite flake that represented the degree of orientation. Note that this model assumes that the heat flow is transferred through the graphite flake network.

366


Fig. 7. The schematic illustration of the connection mechanism of particles; the real particles contact on the left and the equivalent particles contact on the right.

Fig. 8. The schematic illustration of the H–J model and the two step H–J model.

Subsequently, Xu et al. [65] found that the derivation of the M-G model relied on the particle phase being continuous. Therefore, they reconstructed the M-G model using the linear combination of the mesoscale control volume and reconsidered the connection mechanism of particles by adding the RI between the continuous particles to form the new continuous elements (Fig. 7). The 1=kp;e ¼ 1=kp þ RI is substituted for 1=kp , where the kp;e is the TC of the element. The reconstructed M-G model of keff is expressed as:

Also, some researchers focused on the connection mechanism between the particles and matrix. For example, Xu et al. [67] divided the composites into three scales: the macroscopic scale, the mesoscale scale and the microscopic scale. Based on the statistical average of the TC at microscale volumes, the model of the ETC at the mesoscale is constructed, and then the ETCs at the mesoscale are integrated to obtain the ETC at the macroscale, i.e. the ETC of the composite. This model is expressed as:

keff km keff kp;e ð1 f Þ þ f ¼0 2keff þ km 2keff þ kp;e

ln keff ¼ ð1 f ÞC m ln km þ fC p ln kp

ð16Þ

It is noteworthy that this model is only suitable for the thermal equilibrium between the two phases. Furthermore, Agrawal et al. [66] considered the connection mechanism in which particles were independent of each other and modified the series model to compute the ETC of the composite.

ð17Þ

where C m and C p are the parameters that have a definite physical meaning. They contained the generalised parameters that characterise the effect of the thermal path across the two phases’ interface and the specific microstates, i.e. the connection mechanisms in the composite.

367


Fig. 9. The unit cell structure of (a) a GNP and (b) a CNT.


Theoretical basis

Range

Considerations

Prasher et al. [61] and Evans et al. [60] Han et al. [63] Zhou et al. [64]

Homogenization theory and EMA

Test content range of f was 0.5–4 vol% and 0–6 vol% by MC simulation Test content range of f was 11–23 vol% Application range was extra high f m

p; f ; RI ; aggregation

Xu et al. [65] Xu et al. [67] Chu et al. [68] Qian et al. [69]

Statistical approach and multiscale method Potential mean-field theory EMA Law of minimal thermal resistance and equal law of specific equivalent thermal conductivity Parallel model

Wang et al. [70] Kim et al. [71] Pan et al. [72]

Homogenization theory and EMA Fourier’s law

Test content range of f was 0–80 vol% Test content range of f m was 0–100 wt%

EMA

3.2.3. Pore structure Pore defects will be generated in the thermal conductive polymer composites during the production and application of the composites. Such defects can decrease the ETC of the composites. Considering the effect of the pore structure in composites, an improved two-step H–J model was proposed by Chu et al. [68] (Fig. 8). In the first step, they used the H–J model to obtain the ETC of the effective matrix that was formed by embedding pores 0 in the matrix (Fig. 8). The effective fraction of pores, f v oid , is given 0 by f v oid ¼ f v oid =ð1 f Þ, where f v oid is the volume fraction of the pore, which is called the porosity. The ETC of the effective matrix is given by: eff

0

km ¼ km

1 f v oid 0 1 þ 0:5f v oid

ð18aÞ

eff

Then the km takes the place of the km in the H–J model to obtain 0 the ETC of the porous composite, keff , and can be given as: 0 keff

Test content range of f was 0–65 vol% Test content range of f was 0–60 vol% Application range of f v oid was below 10% Test content range of f was 50–60 vol% and f v oid was 0–20 vol%

h i eff eff eff eff eff km 2km þ kp þ 2 kp km f ¼ eff eff eff eff 2km þ kp kp km f

ð18bÞ

where is the ETC of the particle adding the effect of RI using the same principle as in Fig. 9. This model may be suitable for the polymer composite when the km represents the TC of the polymer matrix and is also suitable for composites with multi-size particles.

f ; pore r; f ; pore f ; pore

Subsequently, Qian et al. [69] used a unit cell of a cube with a central sphere to represent the whole composite. In the first step, they treated the cube and sphere as a matrix and a particle to obtain the ETC of the composite without pores keff by the average integral method. Then, using the same geometric model and method, they treated cube and sphere as a composite and a pore to obtain the 0 ETC of the porous composite keff : 2=3

ðpv oid f v oid Þ 1 1 ðpv oid f v oid Þ þ 0 ¼ keff keff f v oid kv oid keff þ keff ðpv oid f v oid Þ1=3

1=3

ð19Þ

where pv oid is the shape factor of pores, which adds the shape factors in the x, y and z dimensions. The shape factor of the particles and the RI are considered in the first step. This model is similar to that of Chu et al. [68], but it considers the pore geometry in the second step. It is noteworthy that this model assumes that the spherical particles disperse homogeneously in the matrix and that heat flux transfers in a single direction. Wang et al. [70] reported a highly deformable sponge-like CNTbased polymer composite and considered the pore geometry to propose a simple parallel model: 0

eff kp

p; f ; RI ; aggregation p; f ; RI ; inclination angle; connection mechanism f ; RI ; connection mechanism r; f ; RI ; connection mechanism p; f ; RI ; pore shape factor; f ; RI ; pore

keff ¼ fkp þ f matrix km þ f v oid kv oid

ð20aÞ

where f matrix is the volume fraction of the matrix. The value of kv oid can be considered as 0, but f v oid obviously has an effect on f and f matrix . The f and f matrix can be described using the weight fraction

368



Theoretical basis

Range

Considerations

Chu et al. [78] Kim et al. [74] Huang et al. [80] Xue et al. [81]

Micromechanics Thermal resistance method EMA and percolation theory Average polarization theory

Test content range of f was 1–25 vol% Test content range of f was 1–10 vol% Test content range of f was 0–8 vol% Application range was whole range of carbon CNT f

p; p; p; p;

of the particle and matrix (f m and f m;matrix ) and the density of the particle and matrix (qp and qm ) as: f m ðor fm;matrix Þ

f ðor fm Þ ¼ ð1 f v oid Þ

qp

fm

qp

þ

f m;matrix

ð20bÞ

qm

Kim et al. [71] considered the nanoporous shell layer in a coreshell particle polymer composite to propose an ETC model of this composite. This model can be given by:

keff

1=3

b1 b keff

f max f ¼ 1 f max

0

2kv l þ ks 2ðkv l ks Þf s 2kv l þ ks þ ðkv l ks Þf s

f c; f c; f c; f c;

RI RI ; Particle orientation RI RI

3.3. Effect of volume fraction on ETC In the particle-filled polymer composites, the ETC of the composites will suddenly increase when the particles reach a certain concentration, which is known as the percolation phenomenon [74,75] and the concentration is called the percolation threshold, f c . The conventional EMA theory can be the same with the linear variation of the ETC with f , but not be valid at or near the f c . Therefore, some scholars have proposed a percolation theory [76] and some have built novel models for the ETC of the composites. Based on the percolation theory [76], the ETC of the composites tðpÞ

ð21Þ

where b is a complex parameter, which contains f v oid , the radius of the core and the TC of the core, the shell and the matrix; f max is the maximum volume fraction for the core-shell particles in this composite. A general model was proposed to calculate the ETC of a porous composite by Pan and co-workers [72]. According to the M-G model, the ETC of the porous composite can be written as:

keff ¼ kv l

f; f; f; f;

ð22Þ

commonly submits to the law of power given by keff ½f f c ðpÞ , where f c ðpÞ and tðpÞ are the f c and critical conductivity exponent involving the structural information for a given p. On the basis of these analyses, Wang et al. [77] put forward a semi-empirical island-network model, which means that the particles move from the island structure to the network chain structure in the matrix with the increasing number of particles. Recently, Chu and coworkers [78] proposed a simple model of the ETC for the GNP composites along the in-plane direction that included the effect of the percolation phenomenon. Then, they used a hypothesis that a GNP is covered with a thin interfacial thermal barrier layer to consider the RI , as shown in Fig. 9(a). Therefore, the ETC of the cell in-plane

c can be expressed as kx ¼ kp = 2RI kp =LGNP þ 1 . The ETC of the whole composite is given by: a

where ks and f s are the TC and f of the solid phase, and kv l is the TC of the gas and liquid phases, which can be calculated by the EMT model. Note that f s is equal to the maximum f . This model is suitable for heterogeneous composites with a high solid-phase mass fraction. Subsequently, Pan et al. added the percolation effect in this model [73]. Table 3 summarises of the theoretical models of microstructure effects, including the theoretical basis, application range, test content range and considerations of these models.

keff 2=3½f f c ðpÞ þ1 ¼ km HðpÞ þ 1= kcx =km 1

ð23Þ

where f c ðpÞ is approximately equal to 1=p and a is a fitting parameter from tðpÞ. The a was chosen in the range of 0.5–2. The flatness of the GNP was displayed by fitting parameter a. Thereafter, Xia et al. [79] used Chu’s model [78] to determine the ETC of graphene oxide (rGO)-reinforced aluminium nitride (AlN) composites. In this model, the HðpÞ is ignored because this value is very small. Kim et al. [74] experimentally obtained a similar power law relation [76]. They then built a two-dimensional connectivity model to

Fig. 10. The scales of the simulation method.

369


consider the percolating networks in the in-plane and throughthickness directions of the MWCNT polymer composites, contributing to the ETC, keff ;x and keff ;z , respectively, whence they derived:

ðhb Þ 1f 1 þ 2RI Lsin km kp keff ;x hb þ sin ðhb Þ cos ðhb Þ þ f CNT ¼ 2R sin ð h Þ 1f 1 keff ;z h sin ðh Þ cos ðh Þ þ þ I LCNT b km b b b kp f

ð24Þ

where hb is used for expounding the percolation like behaviour, which can be obtained by experimental measurements. Huang and collaborators [80] modified the relationship between the ETC of the composite and the f using the percolation theory to replace the volume fraction in Nan’s model. The model is expressed as: kp

keff p km ¼1þ 3 p þ RI km km r

kp km

½f f c ðpÞ

tðpÞ

ð25Þ

where r is the radius of the CNTs. Xue et al. [81] presented a novel model of the ETC for the CNTs composites based on the average polarisation theory, which uses an average value to denote the temperature gradient and heat flow over the system volume. This model includes the influence of the percolation threshold by the relation f c ¼ 0:7ð1=pÞ ¼ 0:7ð2r=LCNT Þ when p 1, which is obtained by the Monte Carlo simulation. Then, Xue et al. used a hypothesis that a CNT is covered with a thin interfacial thermal barrier layer to consider the RI (Fig. 9(b)). The model is expressed as:

9ð1 f Þ " þf

keff km 2keff þ km c

keff

c

keff kx keff kz c þ4 c þ 0:28r kx keff 2keff þ 0:5 kz keff LCNT

#

¼0

ð26Þ c kx

c kz

where and are the ETC of the cell in-plane and throughc c thickness. kx and kz are similar to the definition of Chu’s [78] model (Fig. 9(b)). This model is also suitable for the other transport properties of CNT composites. From another point of view, Zhu et al. [82] have researched the percolation phenomenon using the piecewise function method. Table 4 summarises the theoretical models of volume fraction effects, including the theoretical basis, application range, test content range and considerations of these models. 4. Simulation models of composites 4.1. Simulation models for interface resistance 4.1.1. Undressed interface The interfacial thermal resistance has been identified as a bottleneck factor influencing the ETC of the whole composite. With the rapid development of high-performance computers, some

academics have used simulation methods to study the ITR. The different scales of the simulation method are shown in Fig. 10. The interface sandwiched between the particle and matrix can be treated as the temperature gap [83,84], the interaction [85] and the third phase [86–88]. Hu et al. [83] used an MD simulation to reproduce the pumpprobe measurement method of measuring ITR. The different setup effects on ITR were also analysed. Bui et al. [84] calculated the phonon transmission probability by combining experiments with an off-lattice Monte Carlo (MC) simulation then put the results in an acoustic mismatch model to obtain the ITR. Zhou et al. [85] considered all states of the interface as interactions between the particle and the matrix. Mortazavi and co-workers [86] used the macro-threedimensional finite element method (FEM) to simulate the ITR of several composites filled with commonly shaped particles. They considered the interface as the third phase that was like a wrapping layer on the particle surface. The results showed that the interface effect of non-spherical particles is less than that of spherical particles on the ITR. Tsekmes et al. [87] also treated the interface as the third phase to consider the ITR. The interface has higher crystallinity and thus they set the TC of the interface as being somewhat higher than that of the polymer matrix. Core-shell nanoparticles, such as Ag/SiO2, ZnO/Zn and FeCr/Al2O3 nanoparticles, are a new type of particle that offers a new way to enhance the ETC of composites. In the core-shell nanoparticle composite, there are two kinds of ITR, the shell-matrix ITR and the coreshell ITR. Ngo et al. [88] treated the interface as a thin virtual phase to analyse both ITRs in the core-shell nanoparticle composite by FEM. The effect of different core-shell TC ratios on the ITRs was also analysed. 4.1.2. Modified interface Phonon scattering between the particle and matrix leads to substantial ITR, which has important implications for the ETC of the composite. Interface modifications can significantly reduce the ITR and common methods include the functionalisation of particles. The interface modification method is microscopic research, so the following papers all use the MD simulation method. The functionalisation of particles can be categorised as noncovalent and covalent. Non-covalent functionalisation is a physical modification process [89]. Tanaka and co-workers [90] used silane as a surface-coupling agent to reduce the ITR, then investigated the effects on the heat transfer and discussed the influence of the number and lengths of the surface-coupling molecules on the ITR. Subsequently, the effect of the matrix depth on the ITR was also analysed [91]. Lin et al. [92] designed alkyl-pyrene molecules and Wang et al. [93] considered 1-pyrenebutyl, 1-pyrenebutyric acid and 1-pyrenebutylamine to functionalise the graphene surface to reduce the ITR. Covalent functionalisation is a chemical modification that may reduce the original TC of particles but can also reduce the ITR. Ni

Table 5 List of details of micro-scale simulations in Section 4.2.1. Researchers

Material

Potential

Time steps

Model size

Tian et al. [99]

Nanocomposites

Lennard-Jones

0.001s

Wang et al. [100]

Functional graphene/epoxy nanocomposites Graphite flake/poly (p-phenylene sulfide) composites CNT/natural rubber composite

Polymer consistent force field

(s ¼ 2:115 1012 s) 1 fs

80a 5a 5a (a is the lattice constant of argon at 20 K)

COMPASS force field

1 fs

50 Å 50 Å for graphene

Adaptive intermolecular reactive empirical bond order potential Lennard-Jones

0.5 fs

50 Å 50 Å 52 Å for composite

Ju et al. [101] He et al. [102] Poliks et al. [103]

Aluminium oxide/polyethylene oxide composites

45 Å 50 Å

10/15 nm 10/15 nm 45/200 nm

370


et al. [94] designed the complex HLK5 molecule, which was a covalently functionalised CNT. The short-branched hydrocarbons were used to modify the graphene sheets to reduce the ITR [95] and the hydrocarbon chain linkers CH2 were used to modify the CNT to reduce the CTR [96]. In order to enhance interfacial connection, Wang et al. [97] used the end-grafted polymer chain method to functionalise the graphene, and they also investigated the effect of grafting density, chain length and initial morphology on the ITR. Shen et al. [98] proposed covalently functionalised graphene sheets using four different functional groups (AOH, AF, ANH2 and triethylenetetramine). They simulated the process involved in the pump-probe transient thermoreflectance method to calculate the ITR. 4.2. Simulation models for bulk composites 4.2.1. Microscale At the microscopic level, the common method of composite ETC simulation is the molecular dynamics methods, including equilibrium molecular dynamics (EMD) and non-equilibrium molecular dynamics (NEMD). The NEMD method is more commonly used for the ETC calculation because this method can account for arbitrary shapes and structures of the composites without any assumptions or simplifications and can describe the vibrational motion of phonons in detail. However, it also has many disadvantages, such as the large amount of computation and calculation time required. The prevalent software packages include the commercial software Material Studio from Accelrys Inc. and the open-source software LAMMPS (large-scale atomic/molecular massively parallel simulator). Based on the NEMD method, Tian et al. [99] determined the ETC of the nanocomposites and considered the comprehensive impacts including particle orientation and arrangement, interface mismatch, interface density and particle polydispersity. Wang et al.

Fig. 11. Schematic illustration of the equivalent CTR between the CNTs.

[100] used NEMD to calculate the TC of functional graphene with carboxyl and anime groups and the ETC of functional graphene/ epoxy nanocomposites. The results showed that although the TC of the functional graphene is smaller than raw graphene, the ETC of the composite is improved by adding functional graphene. Ju et al. [101] built the graphite flake/poly(p-phenylene sulfide) composite structure using Material Studio and then used reverse nonequilibrium molecular dynamics (RNEMD) to calculate the ETC of the composite. In this simulation, the graphite flake is reduced to the three graphene layers to save the computational time. He et al. [102] considered both aligned and randomly oriented CNTs of a natural rubber composite to calculate the ETC of this composite. Some researchers have used the MD method to study ceramic particle composites, e.g. aluminium oxide/polyethylene oxide composites [103]. During this simulation, using the RNEMD method, several types of composite structure were considered. Table 5 shows a summary of the microscale simulations models, including the material, application potential function, time steps and model size of these models.

4.2.2. Mesoscale From the mesoscopic point of view, the dissipative particle dynamics (DPD), lattice Boltzmann method (LBM) and off-lattice MC method are the most frequently used methods to calculate the ETC of composites. DPD originated from the lattice gas automata (LGA) method, which is an important mesoscopic scale simulation technique. DPD avoids the lattice problems of LGA and is combined with the MD method, which reduces the computational step-size and increases the computational speed [104]. Unlike the MD method, the expression of particle interactions is via phenomenological forces in the DPD [105]. DPD and MD are both based on Newtonian motion equations, but the particles in the DPD model do not represent the real particles but several aggregates of particles, i.e. coarse graining. Qiao et al. [106] used energy-conserving DPD to study the ETC of nanoparticle composites and then compared the results with those of the M-G model. The results of the DPD were in quantitative agreement with the M-G model. The effects of randomly dispersed, vertical alignment and horizontal alignment nanoparticles were considered. Recently, by coupling smoothed particle hydrodynamics (SPH) and the DPD method, Zhou and co-workers [107] set the forces and built the mesoscopic structure of the CNT/polymer composites in the DPD model and then inputted this configuration into the SPH simulation to calculate the ETC of CNT/polymer composites. The effects of the degree of dispersion, volume fraction and aspect ratio on the ETC of the random and aligned CNT composites were investigated. They also used DPD to study the dispersion morphologies of the CNT in a polymer matrix under conditions of equilibrated and Poiseuille flow [108]. The LBM is a commonly used mesoscopic approach that can implement complex geometry boundary conditions and various interactions between particles. For thermally conductive composites as thermal interface materials (TIM), Khiabani et al. [109] proposed a hybrid model that coupled the energy equation and the LBM to calculate the ETC of the composite and investigated the effects of f , particle size and the TC ratio. Chiavazzo and coworkers [110] reconstructed the structure of the disk-shaped nanoparticle composite and used the two-dimensional ninespeed (D2Q9) model of the LBM to obtain the ETC of the composite. Zhou et al. [85] subsequently suggested an LBM numerical model to predict the ETC of composites in two steps: first, they built the microstructure of the composite by the MC random sampling method, i.e. random distribution of particles; second, they used the LBM to analyse the heat flow through the microstructure. Note that this model uniformly treats multiple interface states as


interactions between particles and the matrix. This model is consistent with theoretical models and experimental results and many scholars have used LBM to study different composites [111–114]. The off-lattice method assumes that the space is continuous and can accurately reflect the actual situation, but the calculation of this method is more complicated. In contrast, the lattice method assumes that the space is discrete and thus greatly simplifies the computation, but some details of the model are omitted. When the details of the model can be ignored, the lattice method can be used to simplify the calculation process. The off-lattice MC method treats the heat flow as the Brownian motion of the discrete heat walkers. Duong et al. [115] used this simulation method to investigate both the ITR and CTR of the aligned CNT/polymer composites. The effect of the degree of CNT isolation and distribution on the ETC of aligned CNT/polymer composites was also considered. Bui et al. [116] subsequently researched the ETC of CNT/polymer composites using an off-lattice MC method. In their work, the effect of ITR, CTR and bundle configuration on the ETC were also taken into account. They found that the CTR must be considered at high f because the probability of CNT connection is increased. Note that the CTR can be ignored when f is lower than 10 wt%. In recent years, this method was used to analyse the ETC of the CNT/WS2/polymer composite [117,118]. Comparing the LBM and the MC method on phonon heat-transfer simulations, Han et al. [119] found that both of these methods produced good results but the computational time of LBM is shorter than that of the MC method. On the other hand, Fiedler et al. [120] presented an optimised lattice MC method for the thermal analysis of composites, which minimises computational time. They used the energy conservation law to calculate the effective thermal inertia instead of establishing the model of local thermal inertia.

4.2.3. Macroscale From a macroscopic perspective, the well-known methods are the FEM and the finite difference method (FDM). FEM is an effective numerical simulation method and commonly used software packages include ANSYS [121], ABAQUS [122] and COMSOL Multiphysics [87]. A random close-packed structure of micro-nano hybrid-particle composite was reproduced using the MC algorithm by Sanada et al. [123], and then they used the FEM to analyse the ETC of this composite and the potential of the nanoparticles to improve the ETC. Li et al. [121] divided phonons into three categories – phonons in the SWCNT, matrix and interface – then inputted the macro thermal properties of different phonons into the FEM to calculate the ETC of a randomly distributed composite. In this model, Li et al. treated contact thermal resistance between the SWCNTs as a thin matrix layer, as shown in Fig. 11. Tsekmes et al. [87] used the finite element software COMSOL Multiphysics to analyse the effect of particle shape, size, interconnectivity and agglomerations on the ETC of the SiC/epoxy composite. Jin et al. [124] investigated the ETC of polytetrafluoroethylene composites filled with graphite particles and carbon fibres by FEM simulation. The impact of f , p and distribution type on the ETC of composites was studied. The results of this model were in good agreement with theoretical models and experimental data. Recently, Zhang et al. [122] used the FEM to build a threedimensional model to simulate a special graphene foam structure, which considered both the ITR and CTR to calculate the ETC of this co-continuous composite. Tong et al. [125] considered the condition of high particle loading of particle-filled composites. Using FEM simulation, they obtained the ETC of composites and compared the results with several theoretical models to study the effect of the ITR. Zhang et al. [126] proposed that the heat transfer of a composite was the result of the interactions between the particles and the

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matrix, i.e. a conjugate heat-transfer problem between the particles and the matrix. Then, the FDM was used to solve the conjugate heat-transfer problem to obtain the ETC of the composite. This simulation model assumed that the particles were not aggregated. Another novel method is the element-free Galerkin (EFG) method. The EFG is a type of meshless simulation method that only requires a set of nodes to construct the functions. Thermal analysis of the CNT composites using the EFG method was proposed by Singh and co-workers [127] and they also used this method to analyse the effect of the interface on the ETC of the CNT composites [128]. 4.2.4. Multiscale The Multiscale model bridges various time scales and various length scales via a combination of various methods. Some scholars have combined the micro- and macroscales, i.e. they used a theoretical model at the macroscale with the MD at the microscale to build a multiscale model. Clancy et al. [129] used an MD simulation to estimate the ITR between the nanoparticles and amorphous and crystalline polymer matrixes. They then put the results of the ITR into Nan’s model to estimate the ETC of bulk composites. The results indicated that both types of matrix had the same ITR values. Tang et al. [130] researched the influence of hydrocarbon chain functionalisation on the ITR between the graphene and polymer matrix using NEMD. The ETC of the bulk composite was based on the theoretical model. The influence of graft density on the TC of the graphene and composite was also taken into account. Wang et al. [93] systematically investigated the effect of different noncovalent functionalisations on the ITR in graphene polymer composites using RNEMD. The ETC of the composites was obtained by Nan’s model. Mortazavi et al. [131] built a multiscale model to study the ETC of expanded graphite nanocomposites. In this model, the TC of expanded graphite was obtained by MD simulation and the ETC of the bulk composite by FEM. In order to calculate the ETC of the bulk composite, they built a representative volume element of the composite to receive the temperature gradient and then used the Fourier law to calculate the ETC of the bulk composite. In their following study [132], they improved the multiscale model in three steps. The first and second steps were to still calculate the TC and to build the representative volume element, but in the third step, they proposed using the finite element homogenisation method to handle the whole microscale representative volume elements to obtain the ETC of the composite. The ITR was also considered by MD. Other scholars have combined the micro- and mesoscales. Bui et al. [133] used a multiscale model to obtain the ETC of a nanocomposite, in which the ITR used the MD simulation at the microscale and the ETC of the bulk nanocomposite used the offlattice MC simulation at the mesoscale. However, the edge effects were ignored due to the coarse graining of the off-lattice MC method. Recently, many researchers have built novel multiscale models. Yu et al. [134] considered the Kapitza thermal resistance at the interface between the particle and matrix, and the formation of highly densified polymer sheathing (effective interphase) near the particle. They built a sequential multiscale model to characterise the effects of the particle size on the ETC of the composites. The effects of the particle size on the ETC of the composites were discussed via NEMD simulation, and then both the Kapitza thermal resistance interface and the effective interphase were demonstrated in a micromechanics model and defined as a function of the particle radius. Finally, the accuracy and the relative concentration influence of the particle, the Kapitza thermal resistance interface and the effective interphase were analysed using FEM. Subsequently, Shin et al. [135] improved this sequential multiscale model to propose a multiscale homogenisation model that

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considered the local temperature gradient fluctuation by the ITR and the more accurate polymer densification of the composites. 5. Conclusions and outlook Particle-filled polymer composites are a promising thermal conductive material and have aroused extensive interest. A great deal of attention has been paid to the model of particle-filled polymer composites, as these models can more accurately predict the ETC of composites and can quantitatively design the ETC of composites. A number of models that explain many of the impacting factors are reviewed in this article. Key challenges and opportunities from this review can be summarised as follows. For theoretical models: Firstly, theoretical models usually focus on one factor and thus lack universality. Further studies should focus on more generalised theoretical models. Secondly, the semi-empirical theoretical models and the rateable series parallel theoretical models are dependent on the experiment results. However, some experiments are hard to realise and the empirical parameters are difficult to obtain. Thirdly, the theoretical models are usually based on the EMA and micromechanics and innovative models have not yet formed systematic theories. For simulation models: Firstly, some researchers used Fourier’s law based on the temperature gradient to obtain the ITR and CTR through simulations, but Fourier’s law at the microscale may not be suitable. Secondly, using simulations to model realistic measurement processes can be used as a new reference for ITR and CTR simulations. Thus far, only the pump-probe realistic measurement method has been simulated to obtain the ITR. Future studies could focus on simulating other realistic measurement methods to obtain the ITR and CTR. Thirdly, although the microscale simulation models are more precise, the amount of calculation is relatively large. Fourth, multiscale simulations not only consider the characteristics of various scales of the particle-filled composites, but also save computing resources. Fifth, both theoretical and simulation models are based on many simplifications and assumptions, which tends to increase the prediction error and lack detailed information. Therefore, a precise and universal thermal conductivity model of composites to study the ETC of particle-filled composites is imminent. Acknowledgments This work was supported by the National Natural Science Foundation of China [Project No. 51506033] and the Guangxi Natural Science Foundation of China; the Innovation Project of GUET Graduate Education (Project No. 2016YJCX18) and the Guangxi’s Key Laboratory Foundation of Manufacturing Systems and Advanced Manufacturing Technology [Grant No. 15-140-30-005Z]. Conflict of interest The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.ijheatmasstransfer. 2017.09.067.

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