Thermal conductivity of particle-filled polymers - … conductivity of particle-filled polymers ......

Thermal conductivity of particle-filled polymers

I.L. Ngo1,2 and C. Byon1,* 1 School of Mechanical Engineering, Yeungnam University, Gyeongsan, 712-749, South Korea 2 School of Transportation engineering, Hanoi University of Science & Technology, Hanoi, 112-452, Vietnam * Corresponding Author: Telephone: +82-53-810-2452, Fax: +82-53-810-4627, E-mail: [email protected]

This study reviews the considerations on thermal conductivity (TC) of particle-filled polymers and develops a numerical approach. A comprehensive and extensive review on the TC of transparent and flexible polymer (TFP) composites is first presented. The discussion covers TFPs filled with many highly conductive materials using various experimental methods and measurement techniques. A variety of theoretical, empirical, and models based on numerical methods are discussed. Since each model is suitable for only specific composites, analytical models should be considered more extensively for general problems. Using a numerical approach, the TC can be improved effectively under appropriate conditions for polymers filled with single and hybrid micro-/nanoparticles. Various guidelines for achieving the maximum TC and enhancing it are provided. Particularly, novel correlations are proposed that can be widely used for accurately and effectively predicting the TC of such polymer composites. This study provides better understanding of research trends of polymer science, particularly the TFPs.

Keywords: Analytical model; Particle filler; Thermal conductivity; Transparent and flexible polymer

1. Background

1.1 Thermal conductivity of pure polymers

Thermal conductivity (TC or k hereafter) is an important property in material science and technology, electronics, building insulation, and related fields, particularly when high operating temperature is required. It characterizes the ability of a material to conduct heat flow. The TC has been studied extensively for both polymers and processing for potential applications. However, polymers typically have much lower intrinsic TC than metals or ceramic materials, making them good thermal insulators. As shown in Table 1, the TC of pure polymers is commonly very low (less than 0.5 W/(m•K)). In general, the TC of a material is primarily evaluated using Fourier’s law in steady-state thermal conduction. For polymers, however, the Debye kinetic equation, k = Cpvλ/3, is usually used for theoretical prediction. In this equation, Cp is the specific heat capacity per unit volume, v is the average velocity of a phonon. λ is the phonon mean free path which is the average distance that a phonon travels between two collisions. Lopes and Felisberti [3] indicated that a phonons’ average free path is the main parameter affecting the TC of composites. For amorphous polymers, λ is an extremely small constant (a few angstroms) due to phonon scattering from numerous defects, which is why the TC of pure polymers is so low.

Table 1 TCs of well-known polymers at room temperature [1, 2].

Name k [W/(m·K)] Name k [W/(m·K)]High density polyethylene (HDPE) 0.44 Poly(dimethylsiloxane) (PDMS) 0.25

Polyphenylsulfone (PPSU) 0.35 Polysulfone (PSU) 0.22 Poly(ethylene vinyl acetate) (EVA) 0.34 Polymethylmethacrylate (PMMA) 0.21

Poly(acrylonitrile-butadiene-styrene) (ABS) 0.33 Polycarbonate (PC) 0.20 Low density polyethylene (LDPE) 0.30 Polyvinyl chloride (PVC) 0.19

Polyphenylene sulfide (PPS) 0.30 Polyvinylidene difluoride (PVDF) 0.19 Poly(butylene terephthalate) (PBT) 0.29 Epoxy resin (ER) 0.19

Polytetrafluoroethylene (PTFE) 0.27 Poly(ethylene terephthalate) (PET) 0.15 Nylon-6.6 (PA66) 0.26 Polystyrene (PS) 0.14

Nylon-6 (PA6) 0.25 Polypropylene (PP) 0.12 Polyetheretherketone (PEEK) 0.25 Polyethylene (PE) 0.11

Polytetrafluoroethylene (PTFE) 0.25 Polyimide, Thermoplastic (PI) 0.11 According to quantum mechanics, the elastic waves (or sound waves) from lattice vibration of immobile positive ions and free electrons (electron gas) in a solid also behave as particles, which are called “phonons”. We can thus consider a solid as a dilute gas of phonons. For an isotropic solid polymer structure, the phonon mean free path is equivalent to that of a gas. Yang provides the TC of various polymers in a polymer handbook [4]. Tritt provides measurement techniques for determining the TC of a bulk material, such as the radial flow method, laser-flash diffusivity, and pulse-power method

Polymer science: research advances, practical applications and educational aspects (A. Méndez-Vilas; A. Solano, Eds.) _______________________________________________________________________________________________

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[5]. The membrane method, bridge method, and photothermal emission method for thin-film materials have also been considered. These approaches are very useful for understanding the physical properties and mechanical behaviors of polymers, particularly the TC. To enhance the TC of a material in applications requiring thermally conductive polymer, a common method is to add a filler with higher TC to the pure polymers while maintaining the other properties as much as possible.

1.2 Thermal conductivity of particle-filled polymers

Polymers reinforced with organic or inorganic micro-/nanoparticle fillers are becoming more and more common in applications that require highly conductive polymer. Examples include electronic packaging for flexible-polymer electronics, encapsulations, satellite devices, and areas where good heat dissipation, low thermal expansion, and light weight are needed [6]. The TC of polymers has traditionally been enhanced by the addition of thermally conductive fillers, including graphite, carbon black (CB), carbon fibers, and ceramic or metal particles. The fillers have much higher TC than that of pure polymers (Table 2). For example, the TC of planar graphite is as high as 3000 W/(m•K) [10]. It is evident that particle-filled polymers have many advantages due to their easy processability, low cost, and durability against corrosion. Tavman [11] provided the following reasons for using fillers: improving TC and diffusivity, electrical conductivity or magnetic permeability, stiffening the matrix material, reducing internal stresses in injection molded parts, reducing creep, reducing cost, and improving the appearance of the final product.

Table 2 TCs of thermally conductive fillers at room temperature [7–9].

Carbon-based fillers Metallic fillers Ceramic fillers Name k [W/(m·K)] Name k [W/(m·K)] Name k [W/(m·K)]

Carbon Nanotube (CNT) 2000–6000 Copper (Cu) 483 Boron Nitride (BN) 250–300 Diamond 2000 Silver (Ag) 450 Beryllium oxide (BeO) 260

Pitch-based Carbon Fiber (along the axis)

530–1100 Gold (Au) 345 Aluminum nitride (AlN)

200

PAN-based Carbon Fiber (along the axis)

8–70 Aluminum (Al) 204 Aluminum oxide (Al2O3)

30

Graphite (on plane) 100–400 Nickel (Ni) 158 Carbon black (CB) 6–174

Fillers can be classified as carbon-based, metallic, and ceramic fillers, as shown in Table 2. Carbon-based fillers are ideal thermally conductive fillers for polymers because they have a higher TC, excellent corrosion resistance, and a lower thermal expansion coefficient than metals. King et al. [12] studied the TC of a polymer composite by adding three different types of carbon to Vectra A950RX liquid crystal polymer: CB, synthetic graphite particles, and carbon fiber. Synthetic graphite (as a single filler) and a combination of CB and synthetic graphite (as hybrid fillers) caused the largest increase in TC. Polymer composites filled with vapor-grown carbon fiber (VGCF) are estimated to have a very high TC of 1260 W/(m•K) [13] and have tenfold improvement in TC compared to that of pure epoxy [14]. The TC of polymers can be enhanced by adding metallic fillers such as aluminum powder, copper powder, or other particles. Tekce et al. [15] indicated that the TC of copper-filled polyamide composites depends on the TC of the filler particles, as well as their shape, size, volume fraction (VF), and spatial arrangement in the polymer matrix. For ceramic fillers, Ishida and Rimdusit [16] achieved a TC of 32.5 W/(m•K) with a BN-filled polybenzoxazine at a maximum filler loading of 78.5% by volume (88% by weight). Zhou et al. [17] reported that the combined use of BN particles and short alumina fiber results in higher TC of the composites compared to when only BN particles are used. By adding spherical aluminum nitride (AlN) into polymers, Ohashi et al. [18] achieved an excellent TC of 8.2 W/(m•K). Yu et al. [19] also reported that the TC of composites filled with AlN reinforcement is higher with larger particle size. The TC is five times that of pure PS at about 20% VF of AlN for a composite containing 2-mm PS particles. Lee et al. [20] extensively studied a combination of the following fillers: SiC, ZnO, and BN. At a filler content of 60 vol%, the TC of the composites reached 2.85, 2.26, and 2.08 W/(m•K), respectively. Han et al. [21] considered various types of fillers such as BN, alumina, diamond, SiC, and silicon nitride, among which BN was the best material for enhancing the TC of epoxy composites. While many studies have used numerous methods to improve the TC of composites, this study is the first to review the development and research direction for enhancing the TC of transparent and flexible polymers (TFPs).

2. Thermal conductivity of transparent and flexible polymer composites

Advances in micro and nanotechnology have motivated a number of recent studies on TFP. This type of polymer has many advantages, such as significant reduction of optical absorption losses in objective observation, ease of fabrication, and biocompatibility. This review considers three major types of TFP: poly-dimethyl siloxane (PDMS), polyethylene terephthalate (PET), and polyimides (PI).


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2.1 TC of particle-filled PDMS

PDMS is the most widely used silicon-based organic polymer since it is optically clear, generally inert, non-toxic, and non-flammable. In addition, it is low cost, easy to fabricate, and biocompatible, it is also used in various microfluidic applications. The first column of Table 3 shows the TC of pure PDMS as a function of temperature. This table shows that the TC increases as the applied temperature decreases. Therefore, the temperature-dependent TC of PDMS should also be considered.

Table 3 TCs of PDMS, PI, and PET [4].

PDMS PI PET Temperature(K) k[W/(m·K)] Type k[W/(m·K)] Filler k[W/(m·K)]

230 0.25 Polyetherimide 0.07 30% Glass fiber 0.29 290 0.20 Thermoplastic 0.11 45% Glass fiber 0.31 340 0.20 Thermoplastic, 40%graphite 1.73 30% Graphite fiber 0.71 410 0.17 Thermoset 0.23–0.50 40%PAN carbon fiber 0.72

Thermoset, 50% glass fiber 0.41

Enhancing the TC of PDMS filled with multi-wall carbon nanotubes (MWCNTs) has received the most significant interest in recent years. This may be due to the very high TC of MWCNTs, as shown in Table 2. Composite materials were obtained by infiltrating PDMS into aligned MWCNT arrays grown by chemical vapor deposition on silicon substrates [22]. The anisotropic thermal diffusivity of aligned CNT-polymer composites was determined using a photothermoelectric technique. The effective room-temperature TC of the composite along the direction of CNT alignment is at least 6 times higher than that of the polymer matrix, which reflects effective medium theory (EMT) predictions. The infiltration of long aligned CNT arrays is currently the most efficient method for obtaining high-TC polymer composites. Hong et al. coated MWCNTs with a silica layer by three-step reactions [23]. The good wettability of the silica/MWCNTs in PDMS resulted in higher TC from the facile phonon movement at the interface, even with lower MWCNT content. Hong et al. also studied the TC enhancement of PDMS composites filled with uniformly dispersed MWCNTs [24]. Experimental results were compared with calculated results based on simple models. Composites using master-batch MWCNTs had 10% greater TC than those prepared with raw MWCNTs. In 2011, Hong et al. [25] continued developing this research by incorporating alumina particles into MWCNT-filled PDMS. They showed that the TC increased with the amount of MWCNTs. The thermal and electrical conductivities of the composites were increased with increasing alumina concentration. The alumina particles help to disperse the MWCNTs within the PDMS matrix due to the ball milling effect during compounding. Kong et al. produced PDMS composites filled with pristine and functionalized MWCNTs by two different methods: the solution mixing method and the mini-extruder method [26]. The composites produced by the mini-extruder have higher TC due to better nanotube dispersion. The functionalization of nanotubes increases the tensile strength, TC, and thermal stability of the composites due to the improved interfacial adhesion and nanotube dispersion. Quaglio et al. [27] reported better thermal behavior for devices made of the nanocomposite material and demonstrated the direct enhancing effect of the CNT content in the composite. Nanocomposites of PDMS filled with other fillers such as CB, carbon fiber, CNTs, and metallic powder were also reported. Chuang et al. [28] considered a PDMS prepolymer with metallic filler (silver-coated copper flakes), showing that the conducting PDMS may have higher TC than normal PDMS. Chen et al. studied PDMS filled with CB nanoparticles for micro fuel cells [29]. Their thermal gravimetric analysis (TGA) showed good thermal stability of the composite at up to 500°C. Chen et al. [30] continued this work by considering the TC and stability of CB-filled PDMS under the effects of temperature and filler VF. The results showed excellent improvement in the TC of CB-filled PDMS composites (by an order of magnitude compared with the pure PDMS). They also found that PDMS-CB composites with lesser CB loading (10 wt%) achieved TC on a par with the reported values for epoxy resin-CB composites (with 70 wt% loading). Hong et al. [31] focused on many of the physical properties of PDMS nanocomposites filled with both nano-sized CB and CNTs using a planetary mixer and two-roll mill methods. They demonstrated that the TC of the composites increased linearly with the filler concentration, while the electrical properties and the dynamic mechanical properties were also improved. Balachander et al. recently developed a high-TC PDMS nanocomposite filled with a low VF of gold nanowires (< 3 vol%) [32]. They reported a 30-fold increase in the TC of PDMS (up to 7.5 W/(m•K), which is six times higher than that of any nanocomposite with low nanofiller loadings and exceeds theoretical predictions. The nanowire diameter and aspect ratio are key for obtaining cold-welded networks that enhance the TC.

2.2 TC of particle-filled PI

Aromatic polyimides have been widely applied as matrix polymers for advanced applications in the aerospace and microelectronics industries due to their outstanding characteristics, such as good chemical and mechanical properties, excellent thermal stability, and radiation resistance. The mechanical and electrical properties of PI composites filled


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with MWCNTs, single-wall carbon nanotube (SWCNTs), and CNTs are significantly improved using various methods, such as solvent casting, mixture casting, evaporation, and thermal imidization [33, 34]. For example, Singh et al. [35] studied PI/MWCNTs nanocomposite films and showed remarkable improvement of thermal stability compared to pure PI film. PI/SWCNT composites have been studied for many years due to their unique combination of electronic, mechanical, chemical, and thermal properties [36]. However, the TC has not been investigated. The physical properties of PI film differ from that of bulk PI. Kato et al. [37] found that PI film has a higher TC of 0.74 W/(m•K) than literature values obtained for bulk solid materials. Kurabayashi et al. [38] measured the in-plane and out-of-plane TCs of spin-coated PI films with thicknesses between 0.5 and 2.5 µm using three experimental techniques (mesa vertical conductivity measurement, membrane lateral diffusivity measurement, and the IC compatible method). The anisotropy in the TC of the PI films was also investigated, and the in-plane conductivity was four to eight times larger, depending on the film thickness and temperature. Irwin et al. [39] reported that the TC of PI composites is most affected by the surface treatment of the nano-fillers. The TC enhancement of PI filled with ceramic fillers has been extensively studied in recent years. A new thermally conductive PI composite film has been developed based on the dispersion of BN with different particle sizes in the PI precursor polyacrylic acid [40]. The PI/BN composites exhibited different TCs with different proportions of BN particles and different sizes, and the TC was up to 1.2 W/(m•k) for a mixture containing 30 wt% micro- and nano-sized BN fillers in the PI matrix. Sato et al. [41] studied PI films containing hexagonal BN particles. A TC of 7 W/(m•K) was achieved at a solid loading of 60 vol% with the flexibility maintained. PI/AlN composites based on pyromellitic dianhydride and 4,4'-oxydianiline were prepared by in situ polymerization using gamma-glycidoxypropyltrimethoxysilane as a coupling agent for the composite preparation [42]. The TC of the composite was enhanced, while the dielectric constant increased slightly. The electrical properties were altered to a lesser degree with increasing AlN fraction. Recently, Zhou et al. [43] considered PI composites filled with modified AlN fillers. They indicated that at 40 wt% filler loading, the highest TC of the AlN@PI/epoxy composite reached 2.03 W/(m•K), approximately 10.6 times that of epoxy. Two years earlier, Zhou et al. [44] also used core-shell Ag@SiO2 nanoparticles embedded in the PI matrix, and the composites with 50 vol% fillers showed remarkable TC improvement. The maximum TC was 7.88 W/(m•K). Compared with self-passivated nanometer aluminum particle composites, the core-shell Ag@SiO2 nanocomposite is beneficial for increasing the TC and reducing the permittivity. Metallic fillers were also used for enhancing the TC of PI composites [45]. A composite was prepared using modified aluminum fibers and aluminum nanoparticles as fillers in PI matrix, which resulted in high TC and low relative permittivity. They found that the TC of composites was significantly increased up to 15.2 W/(m•K). Table 3 also shows that the TC of PI/filler composites is higher than that of the pure PI. Graphite has been the most promising filler for enhancing the TC of PI composites.

2.3 TC of particle-filled PET

Another type of TFP is PET, which is a semicrystalline thermoplastic polymer with high mechanical strength and solvent resistance. PET has become one of the most widely used polymers since it was first prepared in 1941. It is the third most produced polymer after PE and PP, and it has the highest recycling rate of all plastics in several countries, such as the U.S.A. and Australia. Torres et al. [46] extensively studied the thermal properties (glass transition, melting point, and crystallinity) and mechanical properties (Young's modulus, elongation at break, and impact strength) of post-consumer PET bottles. Many studies considered the performance properties of PET, such as gas barrier, stiffness, and strength properties [6]. The applications of PET include food and beverage packaging, textiles, and automotive components [47]. Many studies have focused on how to achieve important physical properties with PET, such as thermal stability, crystallization rate, optical transparency, and mechanical properties. Techniques used include differential scanning calorimetry (DSC), UV-visible spectroscopy, TGA, wide-angle X-ray diffractometry (WAXD), scanning and transmission electron microscopy (SEM and TEM), and universal testing machines (UTMs) [6]. The thermal stability of PET nanocomposites can be improved by adding montmorillonite (MMT) using a solution intercalation method [48], the in situ interlayer polymerization approach [49], and in situ intercalative polymerization [50]. However, very few studies consider the TC of PET composites. For example, Kato et al. [37] used modified AC calorimetry with a laser-heating Angstrom method to derive the TC for a broad range of materials in the form of free-standing thin sheets or films, wires including fiber bundles, and films on substrates. The TCs were found to be 8.9 W/(m•K) and 6.9 W/(m•K). Lopes and Felisberti [3] used modulated DSC to study the TC of PET and LDPE containing aluminum particles. The TC showed a very complex dependence on composition, which was attributed to morphological effects. The effect of orientation on TC was compared with that of the modified Maxwell model [51]. The final column of Table 3 shows the TC of PET with fillers such as glass fiber or graphite fiber. Consequently, the TC of PET composite increases with increasing TC of the filler. PET filled with MWCNTs was also investigated using TEM and SEM to examine the morphology of nanocomposites [52]. However, the TC property has not been considered. Although many studies have examined PET


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filled with different fillers, a majority of them have focused on other physical properties instead of TC. Therefore, the TC should be received more attention in future studies.

3. Thermal conductivity prediction

3.1 TC models

Many theoretical and empirical models have been reported for predicting the TC of composite materials. Some well-known TC models are listed in Table 4. keff is the effective thermal conductivity (ETC) of a composite material, km is the TC of a matrix material (continuous phase), kp is the TC of the filler (dispersed phase), and ϕ is the VF of the filler.

Table 4 Well-known models for predicting TC [6, 53].

Model Equation Classic models 1

eff m p eff m p eff p m1 / 1 / / series ; 1 parallel ; geometrick k k k k k k k k

Maxwell eff m m p m p m p m p2 2 2k k k k k k k k k k

Lewis-Nielsen p m m

eff m m2p m m

/ 1 11; ; 1 ; , depend on filler shapes, orientations

1 /

k kABk k B C A

BC k k A

Agari-Uno eff 2 m 1 p 1 2log log 1 log ; where areexperimental constantsk C k C k C ,C

Bruggeman 1/(1 )

p eff p m m eff1 , 2 for spheres; and 1 for cylindersx

k k k k k k x x

Cheng-Vachon

1/2m p m p m

eff m m p m p mp m m p m

0.51 1 1 3 4ln ; ;

20.5

k B k k B C k kBB C

k k Bk B k k B C k kC k k k B k k

Hashin-Shtrikman eff m p m p m1 1 1 ; 1 ; 3 for spheresk k d k k k d k d

Hamilton-Crosser

p m m p

eff m

p m m p

1 1 3 for spheres;

2for cylinders1

k n k n k k nk k

nk n k k k

EMT 2eff p m p m0.25 8 ; 3 1 3 1 1k k k k k

Russell 2/3 2/3

m peff m2/3 2/3

m p

(1 )

(1 )

k kk k

k k

Each TC model is shown to be valid for only one composite or just a few, depending on the filler VF, filler TC, the effects of particle shape, orientation, and packing arrangement. For example, Tavman [54] indicated that the Agari-Uno model effectively estimates the TC of HDPE composites with high content of aluminum powder filler. Boudenne et al. reported that the Agari model provides a good estimation of the TC of copper-particle-filled PP for all filler concentrations [55], but with larger error percentage [56]. He et al. [56] also indicated that both the EMT and Nielsen models can give good predictions of the TC at a low filler VF. The Lewis and Nielsen equation was shown to fit the experimental data, whereas none of theoretical models proved adequate for a gas-filled polymers according to Progelhof et al. [57]. Nejad discussed the validation and suitability of many models for predicting the TC of nanocomposites [58]. Other models include the Behrens theoretical model, Donea variational model, and Norton empirical model [57, 59]. Percolation models for contacted particles, dynamic models for unsteady heat conduction, and models for porous media have also been discussed recently [60]. Other TC models and numerical methods were also considered for predicting the TC of composites. For example, Nan et al. [61] used a simple formula for TC enhancement in CNT composites. Song and Youn [62] proposed an asymptotic expansion homogenization technique for predicting the ETC of polymeric composites filled with CNTs. The lattice Boltzmann method has also been a robust and effective way to study the TC of composites. For example, Wang et al. [63] used this method for accurately predicting the TC of natural fibrous materials. Zhou and Cheng [64] investigated the TC of a composite filled with randomly distributed particles based on Monte Carlo distribution. Singh et al. [65] used the element-free Galerkin method to consider the effect of the interfacial-thermal resistance on the TC of CNT composites. Recently, the finite element method (FEM) has been considered as one of the most promising for TC prediction. This method can be used for complex composite structures [66]. A wide range of two-phase composites have been investigated, including Cu/PP, AlN/PI [67], and nanocomposites based on epoxy with many effects [68]. Particularly, novel correlations based on FEM study were proposed for predicting the TC of single-filler [69] and hybrid-filler [70]


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polymer composites, which are discussed in sections 3.2 and 3.3. The TC and thermal behaviors of core-shell nanoparticle composites have been also studied by an FEM approach [71,72].

3.2 Novel correlations for single-filler polymer composites

3.2.1 Numerical method

A numerical model of a unit cell and its boundary conditions (BC) are shown in Fig. 1a. The unit cell can be used as a control volume since the number of spherical particles dispersed throughout the matrix material is very large. The particles are assumed to be isolated and homogenously distributed inside the matrix material, and the TCs are constant for both spherical particles (kp) and matrix materials (km). The TC tensor is assumed to be independent of temperature, and its magnitude is the same in all directions. The thermal flow is considered to be homogeneous with no heat source. Laplace equations can therefore be used to describe the heat transfer in the composite structure based on the non-dimensional variables x*=x/L, y*=y/L, z*=z/L, and Ti

*=(Ti-Tcw)/(Thw-Tcw):

2 * *2 2 * *2 2 * *2i i i/ / / 0 i=p,mT x T y T z (1)

The subscripts “p” and “m” denote the particle and the matrix, respectively. The dimension L of a unit cell is the characteristic length. The BCs are also shown in Fig. 1a, in which all side walls are considered as adiabatic walls, and isothermal conditions are applied for the top and bottom walls. The isothermal BC is investigated to be insignificant for the ETC compared to the heat flux BC, particularly at low VF (ϕ < 0.2). The thermal contact resistance (TCR) between the particle and the matrix is modeled as a virtual shell with uniform thickness ds and TC ks. The following BCs are imposed at the interface between two materials:

* * * * *

p m m p* * * c m s sm m p p c* * * * *

s s s s s

/ and where

/ / 2 / ( 1)

T T T T R k d d Ln T n T R

d k d k L k

(2)

In Eq. (2), κ is the TC ratio kp/ km between a spherical particle and the matrix. Another important parameter that must be considered is the particle VF, ϕ = Vp /Vcell, where V is a partial volume. ks is the harmonic mean TCs of the particle and the matrix. According to this definition, the TC of the interface is slightly higher than that of the matrix, which is in good agreement with reported assumption [68]. Based on the thermal flow field obtained in the computation, the ETC is determined by the following equation in dimensional and non-dimensional forms:

*

0.5 0.5* * * * *effeff eff 0.50.5 0.5

hw cw m

/z

kQk k T z dx dy

L T T k

(3)

where Q is the overall heat flux obtained by integrating the heat fluxes across the outlet surface of the unit cell.

a) b) c) Fig. 1 (a) Schematic diagram of numerical model, computational domain, and boundary conditions. (b) ETC as a function of VF at various TCs. (c) ETC as a function of TC ratio at various VFs.

3.2.2 Results and discussions

* Without the effect of TCR

Many studies indicate that the TC of polymer composites increases with the filler content. This was verified here, as shown in Fig. 1b. Indeed, there is a nonlinear increase in TC with increasing particle VF. The TC increases relatively slowly when the VF is less than approximately 40%, but it increases rapidly for larger VF. In principle, the TC is equal to unity as ϕ approaches zero, regardless of κ. That is why all curves in Fig. 1b start from a point with zero abscissa and unity ordinate. This is valid regardless of the TCR value and will be discussed further in the next section. Figure 1c shows that the TC increases with the TC ratio (κ), and it approaches a given value when κ approaches a critical value. This critical value depends on the VF, as shown in Fig. 1c. For illustration, the deviation between the


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obtained ETC and the ETC at κ = 108 is calculated. For VFs of 10%, 30%, and 50%, the critical values of κ are around 100, 400, and 4000 with less than 1% deviation. In principle, the two following conditions should be satisfied. Firstly, the TC is equal to unity when κ equals unity, regardless of the value of ϕ, as shown in Fig. 1c. Secondly, the TC approaches a given value when κ approaches infinity, as evident in both Figs. 1b and 1c. In other words, the ETC becomes a function of the VF only when κ increases beyond a threshold value. In general, the TC of a composite increases with either the VF or the TC ratio. This characterizes the TC enhancement in composite materials with thermally conductive fillers. A generalized correlation is proposed based on extensive FEM numerical study. The non-linear regression method was used to find all unknown coefficients in a general mathematical function that satisfies all conditions from a physical viewpoint. As a result, Eq. (4) is obtained for predicting the TC of particle-filled composite materials in terms of the particle VF and the TC ratio between the spherical particles and the matrix. Another generalized correlation with the effects of TCR is presented in the next section. Notably, this correlation is valid for a wide range of VF (0 ≤ ϕ ≤ 0.52) and TC ratio (1 ≤ κ ≤ 108).

/0.2276 15 /0.0149

*eff

21 /0.0104 /0.1364

0.4918 1 2.161 10 111 ;

1 3.2826 1.76 10 0.1792

A e eAk

B B e e

(4)

* With the effect of TCR

In the previous section, the variation of ETC as a function of VF and TC ratio was analyzed without consideration of the TCR effect. When no TCR is imposed, neither a temperature difference nor a heat flux difference is generated across the interface between the particle and the matrix. Therefore, the TC of composites generally increases with the particle VF and TC ratio. However, this trend may be affected by the TCR under particular conditions. The composite TC decreases as the TCR increases, which accelerates sharply with larger VF, as shown in Fig. 2a. In addition, the ETC increases with the VF when the TCR is less than about 0.5, but this trend changes for a larger value of TCR. For large TCR, the TC of the polymer composite decreases, even beyond unity. Therefore, the effect of VF on the ETC depends significantly on the value of TCR.

a) b) c)

Fig. 2 ETC as a function of the TCR for (a) various VFs at κ = 103 and (b) various TC ratios at ϕ = 0.5. (c) ETC from the poroposed correlation compared with that of Hasselman and Johnson’s model [73], λ = Rc×kp/a, where a is the particle radius. Figure 2b shows the effect of the TCR on the ETC of the composite at various TC ratios. This figure shows that when κ is high, the ETC decreases more significantly, and all curves converge as the TCR increases. These trends are consistent with previous results [65]. It should be noted that the TC in Fig. 2b approaches a plateau when κ approaches a critical value. In other words, the effect of κ on the ETC of the composite becomes negligible for large values of κ. Overall, the TC of composite materials decreases more significantly when the VF and TC ratio are higher. This result provides useful insight for TC enhancement of single-filler polymer composites.

1 2* 2eff 1 2

1 2

1 11

1 1

B Bk A A

C C

(5)

* * *

* *

* *

/0.6617 /0.1298 /0.07681 2

/0.0557 /0.22531

/0.0618 /0.27511

1.3091 0.6249 0.0805 0.9412 0.5434 1.36

where 0.0929 1.6138 1.8026 and

0.6661 0.8352 0.8314

c c c

c c

c c

R R R

R R

R R

A e e A e

B e e

C e e

*

*

* *

/0.2291

/0.28442

/0.2556 /0.04762

37

0.4847 4.8489

1.0043 2.7754 2.5558

c

c

c c

R

R

R R

e

B e

C e e

Based on an extensive numerical study over a wide range of non-dimensional parameters, a generalized correlation for predicting the ETC was developed using a non-linear regression method to fit all numerical data. The parametric ranges of the correlation are as follows: TC ratio: 1 ≤ κ ≤ 108, VF: 0 ≤ ϕ ≤ 0.2, and TCR: 10-10 ≤ Rc

* ≤ 1. This covers the


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entire range typically found in practical applications. Consequently, a novel correlation for these parameters is obtained as shown in Eq. (5). The mean absolute deviation (MAD) of the correlation with respect to the numerical data is estimated to be less than 1% over the entire parametric ranges. Equation (5) shows that the non-dimensional TC of the composite is always unity when the VF equals zero, regardless of κ and Rc

*. Two important features can also be found from this correlation: (1) there exists a threshold value of κ over which the ETC is not affected by variation of κ, and (2) the ETC always decreases as the TCR increases due to the negative sign. The ETC from Eq. (5) was validated by comparison with Hasselman and Johnson’s model [73], as shown in Fig. 2c. The results indicate good agreement, and the maximum relative deviation is less than about 0.35% when Rc

* or λ equals zero (no effect of TCR).

3.3 Novel correlation for hybrid-filler polymer composites

3.3.1 Numerical method

Figure 3a shows the unit cell and BCs used for the numerical study. It is assumed that the nanoparticles (dispersed phase) are isolated from each other and homogenously distributed in pairs inside the matrix (continuous phase), and the TCs are constant for all spherical particles (kp1, kp2) and matrix materials (km). The numerical method is similar to that presented in section 3.2.1. The Laplace equations are also expressed in non-dimensional form as shown in Eq. (1), and the ETC is determined by Eq. (3). However, there are two nanoparticles with the same or different size (VF) and TC. Therefore, instead of “p”, particle 1 and particle 2 are denoted by the subscripts “p1” and “p2”, respectively. Four non-dimensional parameters that significantly affect the ETC of the hybrid-filler composite are the TC ratios κ1 and κ2

between particles and the matrix, and the particle VFs ϕ1 and ϕ2. These parameters are defined as follows: κ1 = kp1/ km, κ2 = kp2/ km, ϕ1 = Vp1 /Vcell, and ϕ1 = Vp1 /Vcell. The parameters are part of a correlation proposed for predicting the TC of a hybrid-filler polymer composite. The TCR between particles and the matrix material is neglected here. The major goal is to reduce the extremely large working time required to obtain the correlation with many effective parameters. Furthermore, the effect of TCR is not significant when the filler VF is less than a threshold value ϕ1 or ϕ2 < 0.04. The four effective parameters can also provide qualitatively understanding of the thermal behaviors of such a polymer composite. At present, developing a correlation with four independent variables is always a considerable challenge that requires much efforts in post-processing to fit all numerical data.

a) b) c)

Fig. 3 (a) Schematic diagram of numerical model. ETC as a function of (b) VF of particle 1 for various VFs of particle 2, κ1 = 200, κ2 = 4κ1 and (c) TC ratio of particle 1 for various TC ratios of particle 2, ϕ1 = 0.01, ϕ2 = 4ϕ1.

3.3.2 Results and discussions

Figure 3b shows the variation of ETC with respect to the VF of particle 1 (ϕ1) for various VF of particle 2 (ϕ2). It was found that the ETC of nanoparticle-filled composite materials depends on ϕ1 only if ϕ2 is very small (ϕ2 = 10-4, solid line in Fig. 3b). Therefore, the ETC increases with ϕ1, which is similar to the behaviors of a single filler discussed in section 3.2. In addition, the ETC always equals unity when the VF of both fillers approaches zero. The ETC curve is then shifted up, depending on ϕ2. This explains why the addition of heterogeneous fillers in the matrix materials can gain higher TC than single filler in the same testing conditions. This result is also in good agreement with the results obtained by King et al. [12]. The gradient of the ETC curve also increases as VF increases, suggesting that there is a nonlinear behavior of ETC with respect to VF, particularly when VF is large. Figure 3c shows the ETC as a function of TC ratio of one particle (κ1) with the ratio of the other particle (κ2) kept constant. The presence of single particle (particle 1) is only effective for TC enhancement when κ2 is equal to unity (solid line in Fig. 3c). The ETC increases with the TC ratio (either κ1 or κ2), and it approaches a plateau as the TC ratio approaches a threshold value of approximately 103. This is in good agreement with previous results [69].

*eff,max 1 2 1 20.952 0.058exp 2.475 1k (6)


561

To gain useful insights about the correlation between ETC and participating parameters, the variation of ETC is considered with respect to the sum of two-filler VFs and their various ratios for particular relationships between the TC ratios: κ1 < κ2 (Fig. 4a), κ1 = κ2 (Fig. 4b), and κ1 > κ2 (Fig. 4c). The sums of TC ratios of the two fillers are all the same in Fig. 4. The figure shows that the ETC of composite materials containing heterogeneous fillers varies as a function of the VF sum and VF ratio, and it depends significantly on the TC ratio between the two fillers. The ETC increases as the VF sum increases for all TC ratios. When κ1 < κ2, the ETC increases with the decrease of the VF ratio, as shown in Fig. 4a. However, this trend is reversed when κ1 > κ2 (Fig. 4c). Particularly, the ETC is nearly insensitive to the VF ratio when κ1 = κ2, as shown in Fig. 4b, where small deviation between curves represents computational error. This means that the role of the two fillers is identical when they have the same TC, which corresponds to the case of single-filler polymer composites.

a) b) c)

Fig. 4 ETC as a function of the sum of two filler VFs and their various ratios. (a) κ1 = 2, κ2 = 500κ1, (b) κ1 = κ2 = 501, (c) κ1 = 500κ2, κ2 = 2.

a) b) c)

Fig. 5 ETC as a function of the quotient between two TC ratios for various sums of ratios. (a) ϕ1 = 0.01, ϕ2 = 4ϕ1, (b) ϕ1 = ϕ2 = 0.025, (c) ϕ1 = 4ϕ2, ϕ2 = 0.01.

Each part of Fig. 5 represents a set of VFs ((a) ϕ1 < ϕ2, (b) ϕ1 = ϕ2, and (c) ϕ1 > ϕ2). The sum of VFs (ϕ1 + ϕ2) is constant for three cases. This figure covers an entire parametric range that is typically found in thermal applications (2 ≤ κ1 + κ2 ≤ 104, 10-3 ≤ κ1 / κ2

≤ 103). The maximum ETC exists for each sum of TC ratios, and it was found as a function of the TC ratio between two fillers. This is due to the competing effect of hybrid fillers. When κ1 is much less than κ2, or κ1 / κ2 is very small, the thermal flows in the top-down direction are forced away from particle 1, while some or none of the flows move over particle 2. This results in much less heat transfer between the two particles and the surrounding medium, hence reducing the ETC. Particle 1 acts as an obstacle to thermal flow through the composite structure. The results are similar when κ1 is much larger than κ2, or κ1 / κ2 is very large. The ETC is also reduced since particle 2 acts as an obstacle to thermal flow in this case. The correlation between the maximum ETC and TC ratio approximately obeys an exponential law, as in Eq. (6). Figs. 5a and 5c also indicate that the ETC curves are asymmetric with respect to the TC ratio between two fillers when ϕ1 ≠ ϕ2. When ϕ1 < ϕ2 (Fig. 5a), the maximum ETC lies on the left side of the plot, and the ETC increases with κ1 / κ2. This trend is reversed when ϕ1 > ϕ2, as shown in Fig. 5c, which shows a symmetric ETC contribution of particles 1 and 2. When ϕ1 = ϕ2, these asymmetric curves become symmetric, as shown in Fig. 5b. The line of maximum ETC coincides with the vertical axis where κ1 / κ2 = 1. Based on a comprehensive and extensive study, a novel correlation for predicting the ETC of composite materials with hybrid fillers is proposed in Eq. (7). This equation is valid for 1 ≤ κ1 ≤ 104, 1 ≤ κ2 ≤ 104, and 0 ≤ ϕ1 ≤ 0.04, 0 ≤ ϕ2 ≤ 0.04, which covers the entire range typically found in practical applications. The MAD of the correlation with respect to the numerical data is estimated to be less than 2% over the parametric range. The following constraints were used to derive the novel correlation, which has a physically reasonable mathematical form. (i) The two fillers have the same role in the ETC of the hybrid-filler polymer composite: κ1 is comparable to κ2, and ϕ1 is comparable to ϕ2. (ii) The ETC of the composites always equals unity when the VF of both fillers is equal to zero, regardless of the TC ratio values. (iii)


562

There exists a threshold value of κ beyond which the ETC is nearly unaffected by variation of κ. (iv) The TC always equals unity when the TC ratio of both fillers equals unity, regardless of the VF values. The ETC from Eq. (7) is investigated to be in line with experimental results [74].

1 2*eff 1 2

1 2

0.204; 13.15731 11 exp 1 ;

and 2.84211 1

D EE Ek D

FF F

(7)

4. Conclusions

The TC of particle-filled polymers was considered by reviewing the literature and developing a numerical approach. First, TC enhancement of TFP composites was thoroughly reviewed. The topics included TFP filled with many highly conductive fillers using various experimental methods and measurement techniques. PDMS filled with carbon-based fillers including CB, CNTs, and particularly MWCNTs have attracted attention in numerous studies. The threshold value of the TC of PDMS composite was approximately 7.5 W/(m•K) (a 30-fold increase) when using gold nanowire filler. The TC enhancement of PI filled with ceramic fillers has been studied extensively in recent years. A very high TC of up to 15.2 W/(m•K) was obtained for PI composites using modified aluminum fibers and aluminum nanoparticles. Many studies on PET polymer have focused on finding methods to control physical properties such as thermal stability, crystallization rate, optical transparency, and mechanical properties. However, the TC enhancement of PET polymer filled with highly conductive fillers deserves more attention in future studies. Next, a variety of theoretical, empirical, and models based on numerical methods were discussed. Since each model is suitable for only specific composites, analytical models should be considered more extensively for general problems. The TC of polymers filled with single and hybrid micro-/nanoparticles was then examined numerically using FEM. The numerical results indicate that the ETC increases with either the particle VF or the TC ratio in general. This trend changes when a non-dimensional TCR is present and greater than approximately 0.5. These results provide very good guidelines for enhancing the TC of such polymer composites. Novel correlations with and without the effect of TCR were also proposed in Eq. (4) and Eq. (5). These correlations are very useful for accurately and effectively predicting the TC of a wide range of single-filler polymer composites. Finally, for polymer composites containing hybrid filler, the numerical results indicate that the ETC varies as a function of the VF sum and VF ratio and depends significantly on the TC ratio between the two fillers. The ETC increases with the sum of VF for all TC ratios. When κ1 < κ2, the ETC increases with the decrease of the VF ratio. However, this trend changed when κ1 > κ2. The ETC is not affected by the VF ratio when κ1 = κ2. Excellent improvement of TC can be obtained using hybrid-fillers instead of single filler with very high TC. Moreover, a maximum ETC exists for each sum of TC ratios, and it obeys an exponential law of the TC ratio between the two fillers in Eq. (6). Another useful correlation was proposed as a function of four non-dimensional parameters in Eq. (7).

Acknowledgements This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) with funding from the Ministry of Science, ICT and Future Planning (2014R1A2A2A01007081).

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