Numerical analysis for the effects of particle distribution and particle size on effective thermal conductivity of hybrid-filler polymer composites

International Journal of Thermal Sciences 142 (2019) 42–53

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Numerical analysis for the effects of particle distribution and particle size on effective thermal conductivity of hybrid-filler polymer composites

T

Ich Long Ngoa,∗, Chan Byonb,∗∗, Byeong Jun Leec,∗∗∗ a

School of Transportation Engineering, Hanoi University of Science and Technology, No. 01, Dai Co Viet, Hai Ba Trung, Hanoi, Vietnam CTO, R&D division, Recensmedical, Inc. Ulsan 44919, South Korea c School of Mechanical Engineering, Yeungnam University, Gyeongsan, 38541, South Korea b

A R T I C LE I N FO

A B S T R A C T

Keywords: Hybrid filler Optimal condition Polymer composite Thermal conductivity

This study describes the thermal conductivity (TC) enhancement of polymer composites with randomly distributed hybrid fillers using both numerical and theoretical approaches. The effective thermal conductivity (ETC) of such composites is examined as a function of the TC ratios, volume fraction (VF), and particularly particle distribution and particle size. Effects of particle distribution were first explained comprehensively with combination of both numerical and theoretical analyses. Analyzed results show that particle distribution and particle size become important and affect significantly to the maximum ETC at high VF, high TC, and large size of hybrid fillers. However, the optimal TC ratio between two fillers is shown to be unaffected by both particle distribution and particle size. Particularly, the TC is generally enhanced when more particles are stacked and concentrated along the thermal flow direction. These results are very good in practical optimization process to achieve the maximum ETC of composites, thus total cost of composite synthesis can be reduced significantly.

1. Introduction

TCs of silicone rubber filled with ZnO increased with the volume content of ZnO. Polymer composites based on vapor-grown carbon fiber are estimated to have a very high TC of 1260 Wm−1K−1 [8]. Tenfold improvement in TC was demonstrated in epoxy composites with vaporgrown carbon nanofiber, as reported in Ref. [9]. Additionally, Tekce et al. [10] indicated that the TC of copper-filled polyamide composites depends on the TC of the fillers, their shape and size, volume fraction (VF), and spatial arrangement in the polymer matrix. Yu et al. [11] also indicated that the TC of composites filled with AlN reinforcement was higher for larger particle size used. The TC obtained at about 20% VF of AlN was five times higher than that of pure polystyrene. Other kinds of filler which has recently received an interest in the TC enhancement for pure polymers, are core-shell nanoparticles. Nanoparticles (cores) are first synthesized, then they are covered by one-layer or multi-layers materials (shells) with enhanced properties (higher TC), finally these core-shell nanoparticles are added into the polymer matrix. As a result, the TC of polymer composites can be improved due to the higher TC of shells compared to using core-nanoparticle with lower TC only. Indeed, Zhou et al. [12] indicated that the TC was remarkably improved by adding core-shell Ag/SiO2 nanoparticles into polyimide matrix. Kim et al. [13] reported the significant

Advanced polymer composite materials have received significant interest due to their advantages in potential applications that require higher thermal conductivity (TC). Some examples are flexible polymers in electronic packaging and encapsulations, satellite devices, and areas where good heat dissipation, low thermal expansion, and light weight are required [1–4]. In order to get these requirements, one of the effective methods is using fillers. In this method, the highly conductive fillers (dispersed phases) such as graphite, carbon black, carbon fibers, and ceramic or metallic particles are added into matrix materials (continuous phase). Therefore, the TC of formed composite can be enhanced effectively. The use of planar graphite with very high TC, up to 3000 Wm−1K−1 is a particular example [5]. The polymer composites filled with thermally conductive particles have advantages due to their easy processability, low cost, and durability against corrosion. More details in the use of filler were reported by Tavman [6]. Single kind of, or homogenous mono-fillers were first used for enhancing the TC of composites. Many studies demonstrated that the TC of polymer composites depends significantly on properties and features of the fillers used, such as filler content. Mu et al. [7] reported that the

∗

Corresponding author. Corresponding author. ∗∗∗ Corresponding author. E-mail addresses: [email protected] (I.L. Ngo), [email protected] (C. Byon), [email protected] (B.J. Lee). ∗∗

https://doi.org/10.1016/j.ijthermalsci.2019.03.037 Received 21 August 2017; Received in revised form 13 October 2017; Accepted 29 March 2019 1290-0729/ © 2019 Published by Elsevier Masson SAS.

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Nomenclature

Greek symbols

a Length of cubical filler [m−1] A Area [m2] k Thermal conductivity [Wm−1K−1] L Length of unit cell [m] n Interface normal [m−1] N Number of particle [−] q” Heat flux [Wm2] Q Overall heat flux [W] r Particle radius [m] Rc Thermal contact resistance [m2KW−1] T Temperature [K] x Coordinate in x direction [m] y Coordinate in y direction [m] z Coordinate in z direction [m] B, C, D, E, F, G, H, I, J, K, M Coefficients in Equations from (11) to (15) [∼]

κ π ϕ

Thermal conductivity ratio [−] PI number [−] Volume fraction [−]

Subscripts, Superscripts 1, 2 d eff in m out p pro s *

Index of particle 1 and 2 [-] Division of two quantities [−] Effective [−] Inlet [−] Matrix [−] Outlet [−] Particle [−] Projected [−] Sum of two quantities [−] Non-dimensional form

provided. In addition, the thermal behaviors of ETC due to the synergic effects of hybrid fillers are also considered and discussed.

enhancement of the TC using FeCr metal core–aluminum oxide shell particles with a highly mesoporous shell layer compared to the use of uni-modal particles. Recently, Ngo et al. have performed an extensive study and a numerical analysis on the TC of core-shell nanoparticle polymer composites, and provided optimal conditions for enhancing and achieving the maximum TC [14,15]. Heterogeneous fillers or hybrid fillers have been regarded as another effective method for enhancing TC of composites, as reported in literature. Two kinds of filler with the difference in the TC, size, or shape are filled together into the polymer matrix. It was demonstrated that there is a positive synergic effect on the TC of composite when different fillers are combined [16]. This combination can effectively cause thermally conductive pathway/chain/network, thus much heats are transferred between fillers and surrounding, hence the TC can be improved significantly. Lee et al. [17] found that the TC of composites filled with spherical particles and fibrous fillers are higher than the matrix material at low and intermediate filler content. Xu et al. [18] also indicated that using a hybrid filler at a volume ratio of 1:6 gives the TC higher than using each single filler. The highest TC of polyvinylidene fluoride (PVDF) composites was up to 11.5 Wm−1K−1 at total filler VF of 0.6. Chen et al. [19] indicated that the combination of fillers at a certain condition can result in higher TC than using single filler for the same total filler loading. The optimal ratio of hybrid filler was shown to be valid for epoxy composites filled with single-walled carbon nanotubes and graphite nanoplatelets. Furthermore, the TC of polymer composites can be improved by adding nanofillers into the polymer-matrix containing the micro fillers, as reported in Ref. [20]. Gao et al. [21] also indicated that the epoxy composites filled with aluminum nitride/graphene nano-hybrids get the highest TC enhancement compared to the pure epoxy. Although previous studies provided some results in enhancing the TC of composites filled with hybrid fillers, very limited number of studies have examined the influences of many effects on the TC of hybrid-filler polymer composites, particularly anisotropic particle distribution, particle size of hybrid fillers. Furthermore, no research explained in detail the effect of particle distribution and considered how to maximize the TC of hybrid-filler polymer composite under this effect. The objective of this study is to describe and examine extensively the effects of particle distribution and particle size as well as effective parameters on the effective thermal conductivity (ETC) of polymer composites with randomly distributed hybrid fillers. An extensive study of literature review on the existing-related models is also performed in the present study. The characteristics of ETC in such composites are studied comprehensively under many effects. Therefore, the good guidelines obtained for optimizing or maximizing the TC are first

2. Numerical methodology Schematic of numerical model and boundary conditions (BC) are shown in Fig. 1. A unit cell can be used as a control volume since the number of particles dispersed throughout the matrix is very large. In Fig. 1, hybrid filler particles, particle-1 (green and big) and particle-2 (red and small) are assumed to be isolated each other and randomly distributed inside the matrix. Geometry of unit cell was first built by user-defined code in MATLAB. Particles-1 were first distributed randomly until all of them do not contact each other, and then particles-2 were distributed. At this point, particles-2 required do not contact with either itself or particles-1. Geometry was finally tested to make sure the inputs parameters identified, then it was imported into COMSOL using Livelink for MATLAB. In this regard, the particle distribution is changed whenever the different hybrid fillers are considered. This is to include the effects of anisotropic particle distribution resembling to that in experimental process. The TCs are constant for all spherical heterogeneous-particles (kp1, kp2) and matrix materials (km).

Fig. 1. Schematic of numerical model and boundary conditions. 43

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I.L. Ngo, et al.

where Rc∗ is non-dimensional parameter characterized for an effect of thermal contact resistance. Sine a single filler is considered in this case, κ1 = κ2 = κ and ϕ1 = ϕ2 = ϕ are set in our simulation for comparison purpose. When thermal contact resistance is neglected, Rc∗ = 0 as discussed above, Eq. (7) reduces to a simple one, Eq. (8).

The thermal flow is considered to obey the Fourier's law with no heat generation. Thus, Laplace equations can be used to describe the conductive heat transfer in a composite structure, which are given by

∂ ⎛ ∂Ti ⎞ ∂ ⎛ ∂Ti ⎞ ∂ ⎛ ∂Ti ⎞ + + =0 ∂x ⎝ ∂x ⎠ ∂y ⎝ ∂y ⎠ ∂z ⎝ ∂z ⎠ ⎜

⎟

i = 1, 2, m

(1)

∗ keff =1+

The subscripts “1”, “2” and “m” denote the particle-1, particle-2 and matrix, respectively. The following non-dimensional variables are used to convert Eq. (1) into a non-dimensional form.

y x z k T − Tout ; y∗ = ; z ∗ = ; k i∗ = i ; Ti∗ = i km qin" L/ k m L L L

x∗ =

⎜

⎟

⎜

⎟

⎜

⎟

i = 1, 2, m

(2)

3. A review on prediction models The TC of pure polymers with very low TC can be effectively enhanced using hybrid fillers, as mentioned in Introduction section. Sanada et al. [20] investigated numerically and experimentally the TC of polymer composites with nano and micro fillers, and concluding that the TC can be improved by the addition of nanofillers into the polymermatrix containing the micro fillers. Very high TC was reported by adding both modified aluminum fiber and aluminum nanoparticle into polyimide (PI) matrix (from around 0.11 Wm−1K−1 for pure thermoplastic PI [26] up to 15 Wm−1K−1), in addition to low relative permittivity of such composite [27]. However, very limited number of theoretical models and generalized/empirical or semi-empirical correlations has been available in literature reported for predicting the TC of such polymer composites effectively and accurately. A potential idea is to develop the two-phase models available in literature to multiphase ones. At this point, the ETC of composites can be first established for one filler, then the single-filled composite can be considered as a new matrix at fixed filler content, thus the ETC of a desired composite can be established for another filler. In the present study, the following Eqs. (912) are considered for this purpose. Notably, all equations are expressed in non-dimensional forms with the terms and scales defined by Eqs. (2), (4) and (6) in Section 2.

(3)

The effects of thermal contact resistances (TCR) between particles and the matrix material are not considered due to the scope of present study. Although this effect was not include in the numerical analysis, the present results show very good agreement with those from available models in literature (Figs. 2 and 3) and even experimental results obtained by Sanata et al. [20] (Fig. 4). Four significant dimensionless parameters in this study are TC ratios between the particles and the matrix κ1, κ2, and the particle VFs ϕ1, ϕ2. These parameters are defined as:

κ1 =

k1 k NV N V ; κ2 = 2 ; ϕ1 = 1 1 ; and ϕ2 = 2 2 km km Vcell Vcell

(4)

where N1 and N2 denote for number of particle-1 and particle-2 per unit cell, respectively. Based on the temperature field computation from simulation, the ETC is given by:

k eff = −

Q L (Tin − Tout )

(8)

The comparison between the ETCs from present work and those obtained by Nan's model, Eq. (8) was shown in Fig. 2 with VF ranging from 0 to 0.3 for two cases of TC ratio, κ = 10 and κ = 50. Based on Fig. 2, the ETCs obtained by two approaches are in very good agreement with each other, particularly when VF is less than 0.2. A minor deviation between them is due to other effects such as the effects of particle distribution and particle size that are being considered in our present study.

where the characteristic length L, is the dimension of a unit cell, as shown in Fig. 1. It is highlighted that the heat flux was specified at the inlet (top surface of unit cell) to define the characteristic temperature since the heat flux BC should be used. This kind of BC is similar to that applied in most experiments of TC measurement. Furthermore, heat flux in the main thermal flow direction (z direction) was kept to be constant from the inlet to outlet of unit cell using Pointwise constraint in COMSOL. This is to ensure the steady state of this problem, which is reasonable for the Fourier's law mentioned above. Therefore, the final results of ETC are expected to be more accurate than those using isothermal BC reported in literature. Therefore, the non-dimensional form of Eq. (1) is given by:

∂ ∂T ∗ ∂ ∂T ∗ ∂ ⎛ ∂Ti∗ ⎞ + ∗ ⎛ ∗i ⎞ + ∗ ⎛ ∗i ⎞ = 0 ∂y ⎝ ∂y ⎠ ∂z ⎝ ∂z ⎠ ∂x ∗ ⎝ ∂x ∗ ⎠

3ϕ (κ − 1) 2 + ϕ + κ (1 − ϕ)

i) Empirical model (parallel and series models)

(5)

In Eq. (5), Q is the total heat transfer rate, which is defined by the top surface area of unit cell multiplied by heat flux through it. As a result, Eq. (5) is reduced to Eq. (6) with which the average temperature (T¯in∗ ) determined at the inlet. ∗ keff =

k eff 1 = ∗ km T¯in

(6)

The mesh independence test can be found elsewhere [14,22–24]. It was found that the ETC depends insignificantly on the mesh size since this is well-known as a steady-state problem of pure conductive heat transfer. However, the number of mesh elements needs to be larger for larger VFs and number of particles per unit cell. This is to ensure an accuracy in predicting the final ETC of a composite structure. The computational code mentioned above was first validated by comparing with the results from a model developed by Nan et al. [25] since this model has been used widely in theoretical analysis of TC of polymer composites. Nan's model applied for spherical fillers can be expressed in non-dimensional form as follows: ∗ keff =

κ (1 + 2Rc∗) + 2 + 2ϕ [κ (1 − Rc∗) − 1] κ (1 + 2Rc∗) + 2 − ϕ [κ (1 − Rc∗) − 1]

Fig. 2. Comparison between the ETCs from present work and those by Nan's model [25], Eq. (8) for two TC ratios, κ = 10 and κ = 50.

(7) 44

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Fig. 3. ETC of hybrid filler polymer composites obtained from various models and present work, (a) ϕ1 = ϕ2, κ1 = 0.5, κ2 = 675, (b) ϕ1 = 2ϕ2, κ1 = 1125, κ2 = 675, N1 = 15 and N2 = 25. Error bars show the deviation of ETC due to particle distribution. n

∗ keff =

1 + (E − 1) ∑i = 1 Ei ϕi 1−

n ∑i = 1 Ei ϕi

κi − 1

where

⎧ Fi = κi + (E − 1) ⎨ E = 3 for spherical particles ⎩ (12)

In the above equations, ϕi and κi are the VFs and TC ratios of fillers, respectively (n = 2 for hybrid nanofiller), which were defined in Eq. (4). First, Eq. (9) is derived from an empirical model with the exponent α ranging from −1 to +1. When α = 1 and α = -1, Eq. (9) is wellknown as parallel and series models, respectively [28]. These models can give the upper and lower bounds of the ETC of composites, as expected. Second, Eq. (10) is based on a geometric mean model, which has been widely utilized in literature in a certain condition. Third, Eq. (11) was modified from Lewis-Nielsen model applied for predicting the TC of single-filler composites [22]. For spherical-shaped fillers, Bi equals 1.5, and the maximum packing fraction of fillers ϕm,i is specified at 0.637, as suggested in previous studies [22,28,29]. Finally, Eq. (12) was modified from original Hashin-Shtrikman model [30]. Although it is simple, it was validated to predict well the TC of a randomly dispersed, particle-in-matrix, two-phase composites. Furthermore, the modified Hashin-Shtrikman model is also validated to predict well the TC of hybrid-filler composites, as evident in the next paragraphs.

Fig. 4. Validation with experimental benchmarks obtained by Sanada et al. [20], ϕ1 = 8ϕ2, κ1 = κ2 = 173.1, error bars show a deviation of ETC due to particle distribution. n ∗ α (keff ) =1+

v) Agrawal's model

∑ ϕi (κiα − 1)

(9)

i=1

ii) Geometrical mean model

∗ keff

n ∗ keff =

∏ κiϕi

(10)

i=1

n

1−

n ∑i = 1 Ci Di ϕi

⎧Ci =

κi − 1 ; κi + Ai

where

Di = 1 +

1 − ϕm, i 2 ϕm ,i

ϕi

⎨ B and ϕ depend on filler shape and orientation m, i ⎩ i

2

+

1/3

⎛ 2π ⎞ ⎝ 3ϕ1 ⎠

⎜

12ϕ2 1/3

( ) π

⎟

4ϕ

;

1/3

+ ⎛ 1⎞ ⎝ 9π ⎠

π (κ1 − 1)

2

+

1/3

⎛ 2π ⎞ ⎝ 3ϕ2 ⎠

⎜

⎟

4ϕ

1/3

+ ⎛ 2⎞ ⎝ 9π ⎠

π (κ2 − 1)

In 2015, Agrawal et al. [31] proposed a theoretical model based on the laws of both minimal thermal resistance and the specific equivalent TC. The series model was used for heat transfer in the heat conduction element. In this regard, the ETC was calculated from two parts of element, one part for particle-1 (G) and another for particle-2 (H). As a result, Agrawal's model is re-written in non-dimensional form, as shown in Eq. (13). This model shows that the role of two particles is equivalent, and the ETC always equals to unity when the VFs of both particles are zero, regardless of the TC ratios. However, the shortcomings of this model are given as: (1) the ETC does not always equal unity when the TC of both particles equals unity, thus this condition is not reasonable in physical viewpoint with the TCR assumed to be neglected; (2) the thermal interaction between two particles (part G and part H) was not

∗ keff

1 + ∑i = 1 Bi Ci ϕi

12ϕ1 1/3 π

( )

(13)

iii) Modified Lewis-Nielsen's model

=

2 = G+H

⎧G = 1 − ⎪ ⎪ where ⎨ ⎪H = 1 − ⎪ ⎩

(11)

iv) Modified Hashin-Shtrikman's model

45

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Fig. 5. ETCs as a function of the TC ratio between two fillers for various sums of VFs, (a) ϕd = 10−3, (b) ϕd = 0.1, (c) ϕd = 1.0. κs = 100, N1= N2=15 were used for all cases.

Fig. 6. ETCs as a function of the TC ratio between two fillers for various ratios of VFs, (a) ϕs = 10−2, (b) ϕs = 0.1. κs = 100, N1= N2=15 were used for all cases.

46

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Fig. 7. Effects of particle distribution on the maximum ETC, (a) ϕd = 0.5 and (b) ϕd = 1.0. ϕs = 0.15, κs = 103, and N1= N2=15 were used for all cases.

bound (series model) and upper bound (parallel model), as expected. In addition, the ETC from present work is shown to be in good agreement with that obtained by modified Lewis-Nielsen model, Eq. (11). Particularly, it matches well with the results obtained by modified HashinShtrikman model, Eq. (12) for all considered range of VF and differently studying cases (Fig. 3a and b). Therefore, the numerical results from the present work were well validated in comparison with other well-known models. Additionally, the present work was also validated by comparing with both experimental and numerical results from Sanada et al. [20]. According to Sanada's work, alumina micro fillers with perfect spherical shape were used. Those fillers were DAW05 and DAW10 with a mean diameter of 4.5 μm and 9.0 μm, respectively, thus particle size ratio of 2 was defined. In addition, two fillers with the same TC, 36 Wm−1K−1 were used while TC of matrix material (epoxy resin) was 0.208 Wm−1K−1. Consequently, the results obtained are shown in Fig. 4. It was demonstrated from this figure that numerical results from the present work are consistent with the experimental ones obtained by Sanada et al. [20], particularly when total VF is less than 0.2, a typical range encountered in practical applications [36–38]. Furthermore, they were shown to be more close to experimental data compared to analytical results obtained from the same work. Therefore, it can be used for further analyses.

obviously considered in this model due to the assumptions originated from a series model. vi) Ngo and Byon's correlation-1

{

J (κ − 1)

}

J (κ − 1)

∗ keff = 1 + I exp ⎡ K + 1κ − 1 ϕ1 + K + 2κ − 1 ϕ2 ⎤ − 1 1 2 ⎣ ⎦ where I = 0.204; J = 13.1573; K = 2.8421

(14)

vii) Ngo and Byon's correlation-2

{

M (κ − 1)

M (κ − 1)

}

∗ keff = 1 + I exp ⎡ K + κ1 − 1 ϕ1 + K + κ2 − 1 ϕ2 ⎤ − 1 1 2 ⎣ ⎦ where I = 0.204; M = 13.3347 exp( −13.2701Rc∗); K = 2.8421

(15)

More recently, Ngo et al. [23] have studied numerically synergic effects of hybrid nanofillers using finite element method. The thermal interaction between two particles was taken into account. Consequently, a correlation with four important parameters (κ1, κ2, ϕ1, ϕ2) was proposed, as shown in Eq. (14). It is clear to see that the shortcomings from Agrawal model are resolved by this correlation. However, the TCR was not included in all equations mentioned above, which was reported as a significant influence to the ETC of a particle-filled polymer composite [22,32–35]. If the TCR between the particles and the matrix reaches a threshold value, then the ETC can be less than that of the matrix, hence it can override the TC enhancement of fillers even with very high TCs and VFs used [22]. Therefore, Ngo et al. [24] continued developing another model with the effects of TCR based on an extensively numerical study. That model is represented by Eq. (15). It was found that the coefficient J in Eq. (14) becomes M as a function of TCR (Rc∗). Notably, coefficient M (Eq. (15)) is 13.3347 when Rc∗ approaches zero, and it slightly differs from J of 13.1573 (Eq. (14)) due to the errors in regression method. The interfaces between both particle-1, particle-2 and the matrix was assumed to have the same TCR. Fig. 3 shows the comparison between the ETC from present simulation and the results obtained by aforementioned models with total VF of hybrid filler ranging from 0 to 0.3. In Fig. 3a, κ1 and κ2 were set to 0.5 and 675 that correspond to TC ratios of hollow glass microspheres (HGM) (≈0.2 Wm−1K−1) and aluminum nitride (AlN) (≈270 Wm−1K−1) to that of epoxy resin as a matrix material (≈0.4 Wm−1K−1). In Fig. 3b, the silver (≈450 Wm−1K−1) was used as particles-1 instead of HGM. The purpose is to examine the effects of high TC of filler (κ1 = 1125 and κ2 = 675), in addition to the difference in VFs (ϕ1 = ϕ2 and ϕ1 = 2ϕ2) on the ETC. The effects of particle distribution were also included here, as represented by error bars. Based on Fig. 3, the ETC from all models and present work lies between lower

4. Results and discussions The synergic effect is well-known to form when adding more than two kinds of filler into the matrix material, thus it can lead to the significant enhancement of TC at the optimal conditions. This was demonstrated in some previous studies. However, it should be noted that the maximum ETC exists only at an optimal condition with respect to the TC ratio between two fillers rather than the VF ratio [23]. Although the ratio of TCs depends on the filler and it is not easy to adjust freely in practice, it is valuable at least in principle and in advanced/smart polymer composites with controllable TC of fillers (i.e. temperaturedependence-TC fillers). In addition, this optimal ratio can be used as a good reference point for choosing appropriate fillers. On the other hand, if giving ϕd (=ϕ1/ϕ2) and ϕs (=ϕ1+ϕ2), it was evident that the ETC increases or decreases with increasing the ϕd at a given ϕs depending on the TC ratio of hybrid filler. This was demonstrated by Zhu et al. [39], particularly by Ngo et al. [23]. This is because in principle, the ETC always increases with increasing VF of a filler, and in general, it continues increasing when adding another filler. Notably, the spherical particles were assumed to be isolated (no contact each other), thus there is no thermal bridge or pathway that can results in further TC 47

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the ETC is reduced. In this case, particles-1 act as an obstacle of thermal flow through the composite structure. It is similar for the case that κ1 is much larger than κ2, or κd is very large. The ETC is also reduced since the particles-2 act as an obstacle of thermal flow or total resistance of thermal flow also increases. Total thermal resistance depends on relative TC between two fillers (κd). In addition, the variation of ETC with respect to κd is expected to be continuous. It implicates that there is an intermediate value of κd at which the ETC achieves maximum value. At this value, the heat transfer between fillers and surrounding medium becomes most effective in order that the thermal flow can move through easily. It means the composite has highest ability to conduct heat through its body. The arguments mentioned above is also demonstrated in the present study for polymer composites reinforced with randomly distributed hybrid filler, as shown in Figs. 5 and 6. In these figures, κs = κ1+κ2 is defined as the sum of TCs of hybrid filler. The optimal value of κd (κd,opt) obtained at which the maximum ETC (k*eff,max) is achieved. Notably, although the effect of particle distribution has not been considered yet, the lines in Figs. 5 and 6 are obtained by averaging the ETC data at each operating condition. It is evident from Fig. 5 that the maximum ETC (i.e. extreme points) specifies at a κd,opt for a given ϕs, but κd,opt does not depend on ϕs while the ETC of composites still increases with increasing the ϕs. This is valid for all cases of ϕd. When ϕd is less than unity, the extreme points are on the left side of vertical line κd = 1, and they move to the right when increasing ϕd, thus the κd,opt is less than unity, as shown in Fig. 5a and b. In addition, Fig. 5c shows that the curves of ETC become obviously symmetrical through vertical line κd = 1 when ϕd equals unity, and extreme points coincide with this vertical line. It implicates that hybrid filler becomes single filler with which the maximum ETC can be achieved. Therefore, it is suggested that κd,opt of hybrid filler increases with increasing ϕd; if ϕd < 1 then κd,opt < 1 and vice versa, if ϕd = 1 then κd,opt = 1. This is a good guideline for maximizing the TC of hybrid filler composites in a certain condition. Fig. 6 shows the ETCs as a function of the TC ratio between two fillers (κd) for various VF ratios (ϕd). It was found from this figure that at low κd (e. g. κd = 10−2), the ETC decreases as the ϕd increases, however, it increases with increasing the ϕd at high κd (e. g. κd = 102). This conclusion is valid for both low (Fig. 6a) and high (Fig. 6b) VF of hybrid filler. It implicates that the ETC always increases or decreases with VF ratio (ϕd) depending on the TC ratio (κd). Therefore, there is no existence of extreme point in terms of ϕd in a certain condition. It was confirmed the argument mentioned previously that the maximum ETC exists only at an optimal condition with respect to the TC ratio between two fillers rather than the VF ratio. Notably, there is a slight deviation for each ETC curve shown in Figs. 5 and 6 because of the effects of particle distribution, which will be taken into account in the next paragraphs. Fig. 7 shows the variation of ETC under the effects of particle distribution for two particular cases of VF ratio, ϕd < 1 (Fig. 7a) and ϕd = 1 (Fig. 7b). In order to account for this effect, 50 random cases are considered for each operating conditions of effective parameters (TCs and VFs). In this regard, the particle arrangement or particle distribution is randomly changed at the same operating condition. Therefore, the variation of ETC limited by lower bound and upper bound, the mean ETC (mean line) and maximum ETC obtained are shown in either Fig. 7a or Fig. 7b. It was found from these figures that the variation of ETC is larger with the higher ETC obtained, it maximizes at the maximum ETC, and it depends on TC ratio between two fillers. The relative deviation between maximum and minimum values at the optimal condition was approximate 4.5% and 4% for Fig. 7a and b, respectively. It is suggested that more testing cases are required to find out the maximum TC of composite in practice, even at the same operating condition. In addition, the variation of ETC is shown to increase generally with increasing both VF and TC ratio of hybrid filler. In order to explain explicitly the effects of particle distribution on the ETC of polymer composites, it is potential to carry out a theoretical

Table 1 Theoretical analysis of cubic-filler model, N1= N2=2.

enhancement. This assumption is reasonable for the low VF considered in the present study. In physical viewpoint, the existence of maximum ETC with respect to the TC ratio between two fillers due to the synergic effect can be explained as follows: when κ1 is much less than κ2, or κd (=κ1/κ2) is very small, the thermal flows are forced away from the particles-1 with very lower TC, thus a few or none of them moves over the particles-2, particularly those placed behind particles-1, this process continues, resulting in much less total heat transferred between two particles and surrounding medium, or total resistance of thermal flow increase hence 48

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Fig. 8. ETC from theoretical equations (Figs. a and b) and simulation results for both cubic (Fig. c) and sphere (Fig. d) shapes of fillers, ϕs = 0.015, ϕd = 0.5, κs = 103, N1= N2=2.

one, and there is a mutual effect between these two parts. Higher values in transverse part corresponds to lower one in longitudinal part and vice versa. However, the sum of two parts shown in Fig. 8b indicates that Case-5 with smallest Apro (a2∗2 ) and Case-6 with largest Apro (2a1∗2 + 2a2∗2 ) result in highest and lowest ETCs, respectively. It means the TC of composites can be maximized with the particle arrangement resembling to Case-5. The ETCs obtained from all theoretical equations were also compared with those from simulation for both cubic (Fig. 8c) and sphere (Fig. 8d) shapes of fillers. All parts of Fig. 8 confirm the above conclusion, highest and lowest ETC obtained for Case-5 and Case-6, respectively. Notably, all particles in Case-5 are in a line along thermal flow direction while those in Case-6 are in a line perpendicular to thermal flow direction. It can be explained as: in Case-5, particles make a line along thermal flow direction, resulting in the longer chain of thermal network that heat transferred maximally between particles with surrounding medium since the contact line is longer, furthermore, total thermal resistance reduces due to small resistance area (projected area), hence the ETC gets maximum value. It is similar for Case-6, yet in contrary direction. It implicates that the ETC is higher when more particles are stacked along the thermal flow direction. It is very useful for enhancing the TC of synthesized polymer composites, suggesting that the additive fillers need to be stacked along the thermal flow

analysis regardless of simplifying geometrical model. This analysis is shown in Table 1 and Fig. 8. Cubical fillers were considered with two particles for each one (thus N1]N2=2). Although the morphology of cubical fillers differs from that of actual fillers with spherical or irregular shape, the trend and behavior in the ETC of composites containing all those fillers are expected to be similar. This is demonstrated in the results shown in Figs. 8 and 9. Furthermore, using cubical fillers is to obtain accurately theoretical equations according to a generalized method of thermal resistance network. In this regard, one-dimensional steady thermal flow was assumed, thus heat transfer occurs in only one direction (top to bottom). As an example, the size of filler-1 was assumed to be less than that of filler-2. Therefore, nine capable cases of particle arrangement were shown in Table 1, which corresponds to nine theoretical equations. These equations indicate that there are two main parts contributing to the ETC of composite: Longitudinal part (terms in front of “1”) and transverse part (remaining terms) represented by Case1 in this table. Transverse part refers to projected area (PA or Apro) of hybrid filler on a plane normal to thermal flow direction, otherwise longitudinal part refers to effective length of fillers along the thermal flow direction and it also depends on the TC of hybrid filler. The ETCs contributed by both longitudinal and transverse parts were shown in Fig. 8a. It is obvious to find that the transverse part affects overwhelmingly to the final ETC of composite compared to longitudinal 49

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Fig. 9. Particle distribution on the ETC enhancement, (a) maximum, (b) medium, (c) minimum ETC and (d) ETC versus total Apro. ϕs = 0.15, κs = 103, and N1= N2=10.

effective length of fillers as well as their PA. In this regard, the bottom surface (xy plane) was divided with 102 × 102 points (called as prescribed points) uniformally distributed in x and y directions, repsectively. From each prescribed point, searching point is moved along z direction with small interval (502 points per unit length). If searching point reaches fillers, then a unity point is set and traced, otherwise a zero is set. The sum of unity points can be converted into effective length at each prescribed point, while the sum of prescribed points that unity points are available can be converted into PA. The effective length of hybrid filler is normalized by unity length L, and PA is normalized by L2. Therefore, the results are shown in Fig. 9 for three particular cases: maximum ETC (Fig. 9a), randomly medium ETC (Fig. 9b), and minimum ETC (Fig. 9c). In addition, Fig. 9d prepresents the ETC as a function of total PA of hybrid filler on a plane normal to thermal flow direction (top to bottom). It was found from Fig. 9 that more PAs are distributed at highly normalized length (up to 0.7) shown in Fig. 9a. It means more particles are stacked or concentrated along thermal flow direction, and total PA obtained is lowest, resulting in maximum ETC, as shown in Fig. 9d. On the other hand, high PAs are distributed at lowly normalized length (around 0.2) shown in Fig. 9c for case of minimum ETC. It means particles are sparsed, and PA formed is higest (Fig. 9d). The results shown in Fig. 9b are in medium position between maximum and minimum cases. Notably, the medium cases with the ETC ranging from 1.55 to 1.57 shown in Fig. 9d do not likely obey this law. This may be

direction to get highest TC. Notably, the ETCs from theory (Fig. 8b) were shown to be less than those obtained from simulation (Fig. 8c). This is due to the assumption of one-dimensional steady problem from the theory, as mentioned previously. In addition, it also depends on the boundary conditions used in simulation. It was found that cubic shape of fillers (Fig. 8c) were resulted in higher ETC than sphere shape (Fig. 8d) with the same VFs and other conditions. This is in good agreement with the result reported in previous studies [22,40]. The increase in the ETC using cubic shape can be explained to be due to the interaction between particles to form thermal conductive chains. Cubical particles have a larger contact along its sides, face-to-face contact, thus the particle-particle interaction becomes stronger and more thermal conductive chains are formed accordingly, resulting in larger heat transferred between particles and surrounding medium when thermal flow move through their bodies, hence the formed ETC increases. On the other hand, spherical particles exhibit point-to-point contact that forms weaker particle-particle interaction. However, the results represented in Fig. 8 also indicate that the effects of filler shape was insignificant when the VFs is small, which is typically encountered range of practical application. Explaining effects of particle distribution on the TC of composite was extended for larger number of particles, ten for each filler. Hybrid filler was randomly distributed in unit cell, and 50 testing cases were considered for this purpose. The ETCs were predicted from COMSOL while geometrical model imported into MATLAB for computing the 50

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Fig. 10. Effects of particle size (as a function of number of particle per unit cell) on the maximum ETC, (a)–(b) ϕd = 0.5, (c)–(d) ϕd = 1.0, and (e)–(f) ϕd = 4.0. ϕs = 0.15 and κs = 103 were used for all cases.

by controlling number of particles per unit cell (N stands for N1 and N2) while keeping the filler VFs constant (ϕ stands for ϕ1 and ϕ2). Particle radii of two fillers are defined by Eq. (16) as:

due to the following reasons: (1) the dominance of longitudinal part according to theoretical analysis mentioned previously, (2) effects of particle packing result in formation of more thermal conductive chains although the projected area is still large, (3) not excluded the errors in computation process. However, these cases are still in same trend from maximum to minimum cases, as shown in Fig. 9d. In conclusion, the TC of composite generally increases with decreasing the PA of hybrid filler. The effects of particle size are also considered in the present study

rp∗ =

rp L

=

3

3ϕ 4πN

(16)

According to this definition, the particle size is inversely proportional to the number of particles per unit cell. As an example, N1 was 51

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Acknowledgments

assumed to equal to N2. The results obtained are shown in Fig. 10 for three particular cases of ϕd: ϕd < 1 (Fig. 10a and b), ϕd = 1 (Fig. 10c and d), and ϕd > 1 (Fig. 10e and f). Based on this figure, it was found that the variation of ETCs due to particle distribution are significant at large particle size (small N), but it gradually shrinks at smaller size (high N). Maximum relative deviation at optimal condition was estimated to be approximate 14.8%, 4.8%, 2.6%, and 1.4% for N of 2, 10, 30 and 60, respectively. According to the geometrical viewpoint, for large particle size, the deviation of PA between maximum (Case-5) and minimum (Case-6) cases shown in Table 1 and Fig. 9 becomes large. This results in high variation in the ETC of composite with large filler size. However, when particle size becomes smaller, the spacing between particles can be narrow, then the particle distribution is limited by maximum packing fraction (volume ratio of real filler to apparent filler), thus the ETC is not affected significantly by changing particle position. In summary, the effect of particle size becomes important at its large size. Fig. 10 also indicates that at optimal condition, the upper bound of maximum ETC generally decreases and its lower bound generally increases with decreasing particle size, which are in good agreement with the argument mentioned previously. However, the mean value is shown to be affected insignificantly by changing particle size. This is in good agreement with the numerical and experimental results reported in literature [7,41]. In addition, it was also found from Fig. 10 that the optimal TC ratio between two fillers κd,opt is shown to be unaffected by both particle distribution and particle size. This is good for practical optimization process to achieve the maximum ETC of composites, thus the cost of composite synthesis can be reduced significantly. In summary, the effects of both particle distribution and particle size become important and affect significantly to the maximum ETC at high VF, high TC, and large size of hybrid fillers, thus it should be considered when synthesizing the hybrid filler polymer composites in particular, and particle-filled composites in general.

This research is supported by the Hanoi University of Science and Technology (HUST) under project number T2018-TT-006. This research was also supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2017R1A2A2A05000770). References [1] I.H. Tavman, In Nanoengineered Nanofibrous Materials, NATO Science Series II, Mathematics, Physics and Chemistry, Dordrecht, Netherlands, 2004. [2] C. Liu, T. Mather, ANTEC 2004, Society of Plastic Engineers, ANTEC 2004, Society of Plastic Engineers, 2004. [3] I.L. Ngo, C. Byon, A review on enhancing thermal conductivity of transparent and flexible polymer composites, Sci. Adv. Mater. 8 (2016) 257–266. [4] M.R. Wang, J.H. He, J.Y. Yu, N. Pan, Lattice Boltzmann modeling of the effective thermal conductivity for fibrous materials, Int. J. Therm. Sci. 46 (2007) 848–855. [5] S. Stankovich, D.A. Dikin, G.H.B. Dommett, K.M. Kohlhaas, E.J. Zimney, E.A. 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5. Conclusions An analytical investigation on the TC of hybrid filler polymer composites has been performed. The Livelink for MATLAB with COMSOL was first used to generate geometrical models with randomly distributed hybrid fillers, and then the ETC is predicted accurately and effectively. The numerical results were thoroughly validated by comparing with well-known models, another numerical results and experimental results for both single and hybrid-fillers. The characteristics of ETC in such composites are studied comprehensively with many effects, particularly the particle distribution and particle size. Consequently, the existence of maximum ETC has been explained and discussed thoroughly. In this regard, maximum ETC exists only at an optimal condition with respect to the TC ratio between two fillers rather than the VF ratio. This optimal TC ratio was shown not to depend on the sum of VFs while the ETC still increases with increasing this sum. In order to explain explicitly the effects of particle distribution on the ETC, theoretical analysis was used in combination with simulation results. It was found that the TC of composite generally increases with decreasing the projected area of hybrid filler. Particularly, the TC is generally enhanced when more particles are stacked and concentrated along the thermal flow direction. It is very useful for the TC enhancement of synthesized polymer composites. The effects of particle distribution and particle size become important and affect significantly to the maximum ETC at high VF, high TC, and large size of hybrid fillers. The optimal TC ratio between two fillers was shown to be unaffected by both particle distribution and particle size. These results are very good in practical optimization process to achieve the maximum ETC of composites that total cost of composite synthesis can be reduced significantly. In addition, many good guidelines for optimizing and maximizing the TC were also provided. 52

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