A modified Hashin-Shtrikman model for predicting the thermal conductivity of polymer composites reinforced with randomly distributed hybrid fillers

International Journal of Heat and Mass Transfer 114 (2017) 727–734

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

A modified Hashin-Shtrikman model for predicting the thermal conductivity of polymer composites reinforced with randomly distributed hybrid fillers Ich-Long Ngo a,⇑, S.V. Prabhakar Vattikuti a, Chan Byon b,⇑ a b

School of Mechanical Engineering, YU, Gyeongsan 712-749, South Korea School of Mechanical and Nuclear Engineering, UNIST, Ulsan 689-798, South Korea

a r t i c l e

i n f o

Article history: Received 19 June 2017 Accepted 26 June 2017

Keywords: Computational material Hybrid filler Polymer composite Thermal conductivity

a b s t r a c t This paper describes an extensive study on thermal conductivity (TC) of polymer composites with randomly distributed hybrid fillers. Finite element method in combination with user-defined code is used to predict accurately the TC of these composites under many effects and effective parameters such as volume fractions (VFs) and TC ratios of fillers to that of the matrix. A literature review on the TC prediction models of hybrid-filler polymer composites is studied and discussed. The effects of particle distribution and particle size of hybrid filler are also taken into account and analyzed. It was found that these effects become important and affect significantly to the effective TC, particularly at high VF, high TC, and large particle size. Remarkably, a modified Hashin-Shtrikman model is first proposed based on an extensively numerical results. It can be widely utilized for predicting the TC of polymer composites with randomly distributed hybrid fillers accurately and effectively, regardless of non-spherical filler shape. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Advanced polymer composites 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 become important [1–4]. One of the effective methods is of adding a filler (dispersed phase) into the matrix material (continuous phase). However, it is well-known that other properties (e.g. mechanical properties, optical properties) of formed polymer composite can be reduced significantly due to the change in structure and material behaviors when fillers are added. There are three common types of fillers: carbon-based fillers, metallic fillers, and ceramic fillers. The planar graphite with very high TC, up to 3000 W m
1 K
1 is a particular example of highly conductive fillers in carbon-based filler group [5]. The polymer composites filled with thermally conductive particles have many advantages due to their easy processability, low cost, durability against corrosion, etc. [6]. Various theoretical and empirical models have been suggested for many years to predict the TC of polymer composites. Maxwell model [7], Lewis and Nielsen model [8,9], Agari and Uno model ⇑ Corresponding authors. E-mail addresses: [email protected] (I.-L. Ngo), [email protected] (C. Byon). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.06.116 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.

[10], or Bruggeman model [11], are well-known models which widely used in literature. Tavman indicated that the Agari and Uno model effectively estimates the TC of aluminum-powderfilled high-density polyethylene (HDPE) composites at high filler contents [12]. He et al. [13] reported both the effective medium theory (EMT) and Nielsen models can give good predictions of the TC at a low filler volume fraction (VF). More details of the TC prediction models were discussed in [14,15]. However, the above-mentioned models are case-sensitive, and do not provide a reliable and universal prediction on the TC since they do not incorporate variously effective parameters such as the filler VFs, the TC ratios of fillers to that the matrix. In addition, these models were considered only for predicting the TC of composite materials with single fillers (or mono-fillers). Other fillers 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 advanced 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 the use of core-nanoparticle only. Zhou et al. [16] indicated that the TC was remarkably improved by adding core-shell Ag/SiO2 nanoparticles into polyimide matrix. Kim et al. [17] also reported the significant enhancement of the TC using FeCr metal core–alu-

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Nomenclature k thermal conductivity [kg-m/s3-K] L length of unit cell [m] N number of particle [–] q00 heat flux [kg/s3] Q overall heat flux [kg m
2/s3] Rc thermal contact resistance [s3-K/kg] T temperature [K] x coordinate in x direction [m] y coordinate in y direction [m] z coordinate in z direction [m] A, B, C, D, E, F, G, H, I, J, K, M coefficients in Eqs. from (9) to (14) [] Greek symbols a exponent in Eq. (7) [–] j thermal conductivity ratio [–]

minum oxide shell particles with a highly mesoporous shell layer compared to the use of uni-modal particles. Recently, Ngo et al. [18,19] 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. Heterogeneous fillers or hybrid fillers have been regarded as another effective method for enhancing TC of polymer composites, as reported in literature. Two fillers with the difference in the TC, size, or shape are mixed together into the polymer matrix. It was demonstrated that there is a positive synergic effect on the TC of composite when combining different fillers [20]. This combination can effectively cause thermally conductive pathway/chain/network, thus a significant heat can be transferred between fillers and surrounding, hence the TC can be improved significantly. Lee et al. [21] 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. [22] also indicated the use of hybrid filler at a volume ratio of 1:6 gives the TC higher than using each single filler. Furthermore, the TC of polymer composites can be improved by the addition of nanofillers into the polymer matrix containing the micro fillers, as reported by Sanada et al.

Fig. 1. Computational domain and boundary conditions.

k

p

/

correction factor in Eq. (14) [–] PI number [–] volume fraction [–]

Subscripts, superscripts 1, 2, i index of particle 1 and 2 [–] d division of two quantities [–] eff effective [–] in inlet [–] m matrix [–] max, min maximum and minimum [–] out outlet [–] p particle [–] q number of particles [–] s sum of two quantities [–] ⁄ non-dimensional form

[23]. However, very limited number of studies has mentioned about the influences of many effects on the TC of polymer composites such as anisotropic distribution, particle size of hybrid fillers. Particularly, no study considers a valuable model for predicting accurately and effectively the TC of polymer composites reinforced with randomly distributed hybrid fillers under the aforementioned effects. This paper focuses on describing and examining extensively many effects and effective parameters on the effective thermal conductivity (ETC) of polymer composites with randomly distributed hybrid fillers. An extensive study of literature review on the TC prediction models is first performed in this present study. In addition, a modified Hashin-Shtrikman (HS) model based on extensively numerical results is well validated by comparing with experimental results in literature. This novel model can be widely utilized for predicting the TC of polymer composites with randomly distributed hybrid fillers accurately and effectively. In addition, thermal behaviors of a polymer composite under the synergic effects of hybrid fillers are also considered and discussed. 2. Numerical methodology Fig. 1 shows the unit cell and boundary conditions (BC) used for numerical model. In this model, hybrid-filler particles consisting of particle-1 (green) and particle-2 (red) are assumed to be isolated each other. This assumption is particularly reasonable for low VF of fillers, less than 0.15 used in this study. In addition, hybrid filler are randomly distributed inside the cubic unit cell with characteristic length L shown in Fig. 1. To attain this polymer composite structure, a geometrical model was built by user-defined code in MATLAB. First, a particle with a given diameter was distributed randomly within unit cell using a random function, then second particle was created in such a way that it does not contact with the first one, implicating that the distance between two particles was constrained to be larger than a sum of their diameters. This step was iterated until the constrain mentioned above is satisfied. It is similar for other particles, both particle-1 and particle-2. The geometrical model is done when the number of hybrid-filler particles per unit cell created was equal to that given. Finally, this geometrical model was transferred to COMSOL Multiphysics using Livelink for MATLAB. As a result, the distribution of hybrid filler is automatically changed whenever the VFs and number of particles per unit cell are set. This is to consider the effects of particle size and anisotropic particle distribution resembling to that in experimental process. The TCs are assumed to be constant for all

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spherical heterogeneous-particles (kp1, kp2) and matrix materials (km). Thermal flow is considered to obey Fourier’s law with no heat source. Thus, Laplace equations can be used to describe the conductive heat transfer in such composite structure.




 @ @T i @ @T i @ @T i þ þ ¼0 @x @x @y @y @z @z

i ¼ p1 ; p2 ; m

ð1Þ

The subscripts ‘‘p1”, ‘‘p2” and ‘‘m” denote the particle-1, particle-2 and matrix, respectively. The following nondimensional variables are used to normalize Eq. (1).

x x ¼ ; L

y y ¼ ; L

z z ¼ ; L



ki ¼

ki ; km

T i ¼

T i
T out q00in L=km

ð2Þ

where the characteristic length L, is the dimension of a unit cell, as shown in Fig. 1. It is worth to know that the heat flux specified at the inlet (top wall) was used to define the characteristic temperature. This is because the heat flux BC is obviously practical in measuring the TC of a composite, and it is similar to that applied in most related experiments. Furthermore, heat flux in the main thermal flow direction (z direction) was kept constant from the inlet to outlet of unit cell using Pointwise constraint. This is to ensure the steady-state problem, which is reasonable for the Fourier’s law mentioned above. Therefore, the final results of ETC are expected to be more adaptive than using isothermal BC reported in literature. Consequently, Eq. (1) can be rewritten in the non-dimensional form as:




 @ @T i @ @T i @ @T i þ þ ¼0 @x @x @y @y @z @z

i ¼ p1 ; p2 ; m

kp1 ; km

j2 ¼

kp2 ; km

/1 ¼

N 1 V p1 ; V cell

and /2 ¼

N 2 V p2 V cell

ð4Þ

where N1 and N2 are the number of particle-1 and particle-2, respectively. The ETC through a testing material can be determined by the following equation:

keff ¼


Q LðT in
T out Þ

ð5Þ

In Eq. (5), Q is the total heat transfer rate, which is as a function of the top surface area of unit cell multiplied by heat flux through it. As a result, Eq. (5) becomes Eq. (6), in which the average temperature at the inlet (Tin bar) is determined from computation. 

keff ¼

keff 1 ¼ km T in





keff

a

¼1þ

n X /i ðjai
1Þ

ð7Þ

i¼1



keff ¼

n Y

j/i i

ð8Þ

i¼1



P 1 þ ni¼1 Ai Bi /i Pn 1
i¼1 Bi C i /i 8 < Bi ¼ ji
1 ; C i ¼ 1 þ 1
/2 max;i /i ji þAi /max;i where : A and / i max;i depend on filler shape and orientation

keff ¼

ð3Þ

Four significant dimensionless parameters in this study are TC ratios between the particles and the matrix j1, j2 and the particle VFs /1, /2. These parameters are defined in Eq. (4). The effects of thermal contact resistance (TCR) is assumed to be negligible.

j1 ¼

containing the micro fillers. Adding both modified aluminum fiber and aluminum nanoparticle into polyimide (PI) matrix gained very high TC (from around 0.11 W m
1 K
1 for pure thermoplastic PI [26] up to 15 W m
1 K
1) and low relative permittivity [27]. However, very limited number of theory models and generalized 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 models. Indeed, the ETC of composites can be first established for one filler, and then the single-filled composite can be considered as a new matrix at fixed filler content, thus the final ETC can be established for the remaining filler, and so on. In the present study, the following Eqs. (7-10) are considered for this purpose. Notably, all equations are expressed in non-dimensional form using Eq. (4).

ð6Þ

The mesh independence test can be found elsewhere [15,18,24,25]. 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 increased with the increase of both VFs and number of particles per unit cell. This is to ensure enough accuracy in predicting the final ETC of a composite structure.

ð9Þ 

P 1 þ ðD
1Þ ni¼1 Ei /i Pn 1
i¼1 Ei /i ( ji
1 Ei ¼ ji þðD
1Þ where D ¼ 3 for spherical particles

keff ¼

In the above equations, /i and ji are the VF and TC ratio of ith filler (n = 2 for hybrid filler), respectively. First, Eq. (7) was derived from an empirical model with the exponent a ranging from
1 to +1. When a = 1 and a =
1, Eq. (7) is well-known as parallel and series models, respectively [28]. These models can predict the upper and lower bounds of the ETC of composites, as reported in both literature and in this study. Second, Eq. (8) is based on a geometric mean model, which has been widely used in literature under a certain condition. Third, Eq. (9) is developed from LewisNielsen model [15]. For spherical-shaped fillers, Ai equals 1.5, and the maximum packing VF of filler /m,i is specified at 0.637, as suggested by previous studies [15,28,29]. Finally, Eq. (10) is developed from an original form of HS model [30]. This model was also called as lower bound equation. Although it is simple, it was validated to predict well the TC of a randomly dispersed, particle-in-matrix, two-phase composite. Particularly, the modified form of this model shown in Eq. (10) will be also validated to predict well the TC of three-phase composites, as discussed in the next paragraph.



keff ¼ 3. Literature review and model validation It has been well-known that the TC of pure polymers can be effectively enhanced using hybrid fillers. Sanada et al. [23] 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 polymer-matrix

ð10Þ

2 FþG

8  1=3 > 1 > þ  1=3 2 ; F ¼ 1
12/ > p > 4/ 1=3 > 2p < þð 9p1 Þ pðj1
1Þ 3/1 where  1=3 > > 2 > þ  1=3  2 1=3 G ¼ 1
12/ > p > 4/2 : 2p þ pðj2
1Þ 3/2

9p

ð11Þ 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

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heat conduction element. In this regard, the ETC was calculated from two parts of element, one part for particle-1 (F) and another for particle-2 (G). As a result, Agrawal’s model was re-written in non-dimensional form, as shown in Eq. (11). This model shows that the role of two particles is equivalent and the ETC always equals to unity when the VFs of both fillers are zero, regardless of the TC ratios. However, the shortcomings of this model are given as: (i) the ETC does not always equal unity when the TC of both particles equals unity, thus this condition is not reasonable in physical viewpoint; (ii) the thermal interaction between two particles (part F and part G) is not obviously considered in this model due to the assumptions originated from a series model.

  Iðj1
1Þ Iðj2
1Þ  keff ¼ 1 þ H exp /1 þ /2
1 J þ j1
1 J þ j2
1 where H ¼ 0:204;

I ¼ 13:1573;

ð12Þ

J ¼ 2:8421

More recently, Ngo and Byon [24] have studied numerically synergic effects of hybrid filler using finite element method. The thermal interaction between two particles was taken into account. Consequently, a correlation with four important parameters was proposed, as shown in Eq. (12). 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 one of the factors influenced significantly to the ETC of a particle-filled polymer composite [15,32–35]. 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 TC and VF used [15]. Therefore, further model needs to be introduced to overcome this shortcoming. Therefore, Ngo et al. [25] continued developing another model with the effects of TCR based on an extensively numerical study. That model is represented by Eq. (13). It was found that the coefficient I in Eq. (12) becomes K as a function of TCR (R⁄c). Coefficient K in Eq. (13) is 13.3347 when R⁄c approaches zero, and it slightly differs from I of 13.1573 from Eq. (12) due to the errors in regression method. The TCRs between particle-1 and the matrix was assumed to be the same to that between particle-2 and the matrix.

n h i o  j1
1Þ j2
1Þ keff ¼ 1 þ H exp Kð / þ Kð /
1 Jþj1
1 1 Jþj2
1 2

where H ¼ 0:204; K ¼ 13:3347 expð
13:2701Rc Þ; J ¼ 2:8421 ð13Þ

Fig. 2. Validation of various models used for predicting the ETC of hybrid-filler polymer composites, /1 = 2/2, j1 = 0.5, j2 = 675.

Fig. 2 shows the comparison between the ETC from the aforementioned models with total VF of hybrid filler ranging from 0 to 0.3. As an example, j1 and j2 were set to 0.5 and 675 that correspond to TC ratios of hollow glass microspheres (HGM) (0.2 W m
1 K
1) and aluminum nitride (AlN) (270 W m
1 K
1) to that of epoxy resin as a matrix material (0.4 W m
1 K
1). Notably, the TCR is negligible here, which is used for comparison purpose. Based on Fig. 1, it was found that the ETCs from all models were covered by lower bound (series model) and upper bound (parallel model), as discussed above. These models underestimate or overestimate the ETC compared to simulation result. On the other hand, the ETC from present work is shown to be in good agreement with that obtained by modified Lewis-Nielsen model (Eq. (9)), by Ngo’s model (Eq. (13)), and by Eq. (8) with a =
0.23. Particularly, it matches very well with the results obtained by the modified HS model (Eq. (10)) for all considered range of VF. The results obtained here become a motivation to develop a novel model for predicting the TC of hybrid-filler polymer composites. This will be shown in Section 4.2. In summary, it is evident that the present work was well validated, and it can be used for further analyses.

4. Results and discussion 4.1. Effects of particle distribution and particle size Fig. 3 shows the effects of particle distribution of hybrid filler on the ETC for three particular cases of VF, low (Fig. 3a), intermediate (Fig. 3b), and high VFs (Fig. 3c). The ETCs were automatically determined for each monitoring point at which the hybrid filler was randomly re-distributed. A hundred of monitoring points were assumed to be large enough so that the effects of particle distribution of hybrid filler can be captured, thus the ETC varies in a valid range named ‘‘band width”. Therefore, the band width of ETC is defined as the subtraction between maximum ETC and minimum one, as shown in Fig. 3d. It was found from Fig. 3 that the particle distribution of hybrid filler has less effect on the ETC of composite at low VFs and TC ratios (square and circle points from Fig. 3a–c). However, it becomes important and affects significantly to the ETC at high VF and TC of hybrid filler (triangular and diamond points from Fig. 3a–c). Therefore, the band width of ETC increases generally with increasing both VF and TC ratio, as represented in Fig. 3d. Based on an extensive study, the maximum band width of ETC obtained is up to 0.2 that corresponds to 8% of mean ETC. Worse and best cases of the ETC in terms of particle distribution can be explained as follows: in the best case when particles make a line along thermal flow direction, the longer chain of thermal network can be formed that results in much heat transferred between particles and 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 worst case, but in contrary direction. More details of this explanation can be found in further study [36]. In summary, the effects of particle distribution should be taken into account when synthesizing the polymer composites reinforced with high VF and TC ratio of hybrid fillers. The effects of particle size were also considered in the present study. The number of hybrid-filler particles per unit cell (N) was changed while keeping the VFs constant. Therefore, the particle size is inversely proportional to the number of particles. As an example, the number of particle 1 (N1) was assumed to equal to that of particle 2 (N2). The results obtained are illustrated in Fig. 4. In this figure, js is the sum of TC ratios of two fillers, js = j1 + j2. In addition, the regions limited by upper bounds and lower bounds in Fig. 4 are the variation of ETC due to the effects of par-

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731

Fig. 3. ETCs under effects of anisotropic hybrid filler distribution, (a) /1 = /2 = 0.01, (b) /1 = /2 = 0.1, (c) /1 = /2 = 0.15, and (d) band width of ETC for various TC ratios and VFs.

Fig. 4. ETCs as a function of number of particles used, /1 = 0.075, /2 = 0.025, j1 = 2j2.

ticle distribution. It was found that the effects of particle distribution is large at small numbers of particles per unit cell (N < 10), but it becomes small at higher ones (N > 30), in general. For example, if the VFs of hybrid filler are /1 = 0.075, /2 = 0.025, then the band width of ETC are large when diameters of particles are 0.24 and 0.17 times greater than the dimension of unit cell L. This result

implicates that the ETC is also affected significantly when the particle sizes are large enough, thus this effect should be considered in practice. If the mean ETCs are computed for all studying cases in Fig. 4, it generally increases as the particle size decreases, particularly at the particle size with N < 10. The decrease of particle size leads to the increase in surface area of hybrid filler at fixed VFs, resulting in the increase of thermal contact area, then the heat transferred between fillers and surrounding increases hence the ETC enhances accordingly. This explains why adding smaller size of particle into the matrix material can gain higher TC of polymer composite, as reported in literature [23]. However, the ETC is shown to increase insignificantly with further decrease of particles size (corresponds to N > 30). It means there is a saturation state of ETC in terms of particle size. This is in line with the result obtained by Mu et al. [37]. Notably, since the hybrid particles were assumed to be isolated for both same and different type of particles, there is no additional effects, for example particle aggregation which can enhance the TC of polymer composites reported in literature [38]. It was also found from Fig. 4 that the band width of ETC increases with increasing the TC ratio of hybrid filler (js) for all considered range of number of particle per unit cell. This is due to the effects of particle distribution, and it is in line with the conclusion mentioned previously. Based on this study, it was concluded that the effects of particle distribution and particle size of hybrid fillers need to be considered when synthesizing the polymer composites reinforced with high VF, high TC, and large particle size of hybrid fillers.

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4.2. Modified Hashin-Shtrikman model As discussed in Section 3, the present results match well with those obtained by the modified HS model (Eq. (10)) for all considered range of VF. Therefore, this modified model is expected to predict well the TC of hybrid-filler polymer composites. However, since the TC of composites also affect by other effects, for example the particle distribution and particle size mentioned in Section 4.1, Eq. (10) needs to be developed to include aforementioned effects. Consequently, a modified HS model Eq. (14) is first proposed. 

keff ¼ 1þ2kM 1
kM

ð14Þ

where M ¼ jj11
1 / þ jj22
1 / þ2 1 þ2 2

where k is a coefficient or correction factor owing to many effects, for example the effects of particle distribution, particles size mentioned in Section 4.1, phonon scattering defects, filler shape, and even thermal resistances. The valid range of k can be found based on an extensive study. Many testing cases were conducted under various conditions. The coefficient M in Eq. (14) was known from those conditions and k⁄eff was then predicted from simulation for each case. Minimum and maximum values of k were derived from those of k⁄eff using Eq. (14). Results obtained was therefore shown in Table 1 for some particular cases. Comparing case-1 and case-2 to find that kmax decreases and kmin increases with increasing the TC ratios, while keeping VFs and number of particles constant (particle sizes fixed). It should be noted that the maximum and minimum values of k⁄eff still increase with increasing the TC ratios. The conclusion obtained above is similar to a case of comparing case2 and case-3. However, the number of particles was changed while keeping other effective parameters constant. The effects of the hybrid-filler VFs were also represented in Table 1, from case-4 to case-6. The same conclusion was found here. In summary, kmax decreases and kmin increases with increasing the TC ratios, VFs, and number of particles per unit cell. Particularly, the valid range of k is approximate from 0.525 to 4.321, as highlighted in Table 1. This covers the range of k valid for other studying cases with the hybrid-filler VFs less than 0.3 and the TC ratio less than 105. Based on these results and considered the safety factor due to other effects on the ETC of these polymer composites, the valid range of k was finally proposed as k = 0.5–5. Notably, a modified HS model shown in Eq. (14) is still met the following constraints in physical viewpoint: (i) The ETC always equal to unity when both /1 and /2 approach zero or both j1 and j2 approach unity (assumed that the effects of TCR is negligible); (ii) the ETC approaches a plateau value when the TCs of hybrid fillers beyond the critical values. In addition, it is remarkable to find that the ETC is independent on the VF ratio (/d) between two particles when hybrid filler has the same TC, j1 = j2, otherwise, it depends on the sum of VFs. It means there is no need to consider the VF ratio between two fillers in practice

when they have the same TC. On the other hand, the variation of ETC is now only due to other effects, for example the anisotropic distribution of hybrid filler, and particle size effects included in correction factor k. Eq. (14) was again validated by comparing with experimental results obtained by both Zhu et al. [39] and Yung et al. [40], as shown in Fig. 5. From Zhu’s study, the low density polyethylene (LDPE) with the TC of 0.317 W m
1 K
1 was used as the matrix and HGM with the TC of 0.21 W m
1 K
1 and AlN with TC of 270 W m
1 K
1 were used as the fillers, respectively. It is similar for Yung’s study, however, the epoxy resin with 0.4 W m
1 K
1 in the TC was used as the matrix instead of LDPE. It was evident from Fig. 5 that the ETCs from Eq. (14) match very well with experimental results obtained by both Zhu’s study and Yung’s study. The coefficient of determination (R2) is greater than 0.992. Notably, the coefficient k in Eq. (14) was used as a parameter to get a best fit to experimental results. As a result, k = 1.98 and k = 3.84 were obtained, as shown in Fig. 5a and b, respectively. These values are in the valid range of k mentioned previously. Fig. 5 also shows that the ETC from Eq. (14) with k = 1 (i. e. Eq. (10)) is close to the experimental results only at low and zero filler VFs. However, it underestimates and becomes much lower than that of experiment at high VFs. This is because of the HS model is well-known as a lower bound equation, as mentioned in Section 3. In summary, a modified HS model, Eq. (14) is first proposed here and it can be utilized to predict the TC of polymer composite reinforced with randomly distributed hybrid fillers accurately and effectively under many effects. Furthermore, this model can be developed for predicting the TC of not only hybrid filler polymer composites, but also multi-phase ones. In addition, it is possible to extend it to polymer composites with nonspherical-shaped fillers. First, many studies reported that numerical model with spherical particles can predict the ETC of polymer composite well even with nonspherical-shape of filler, and the VF can be used as a sole effective parameter. This statement was discussed in many previous studies [11,23,31,41]. Furthermore, the effects of filler shape were shown to be insignificant to the TC of composites when the VF is less than 0.1, a typically encountered range of practical applications. This has been also discussed by Tekce et al. [42], Kochetov [43], Ngo and Byon [15], and Tsekmes et al. [44]. Particularly, regardless of the filler shape effects at higher VFs, the results in Fig. 5 was also concentrated for the remarkably good agreement between the ETC from Eq. (14) and experimental results obtained by Zhu’s and Yung’s studies. Notably, the shape of fillers used in these experimental studies was not really spherical, HGM was assumed to be spherical, but AlN has polygon shape by Zhu et al. [39] and rough sphere shape by Yung et al. [40]. This modified model, Eq. (14), is very flexible for predicting the TC of a polymer composite since many effects and even safety factor were included in the proposed coefficient k, as discussed previously.

Table 1 Determination of correction factor k in Eq. (14).

N1 N2 /1 /2

j1 j2

M k*eff,max k*eff,min kmax kmin

case-1

case-2

case-3

case-4

case-5

case-6

1 1 0.075 0.025 2 1 0.01875 1.08322 1.03843 1.43953 0.67448

1 1 0.075 0.025 1000 500 0.09963 1.43197 1.23252 1.26338 0.72200

10 10 0.075 0.025 1000 500 0.09963 1.3817 1.30913 1.13296 0.93767

15 25 0.01 0.01 2 1 0.0025 1.03276 1.00395 4.32072 0.52547

15 25 0.1 0.1 2 1 0.025 1.10529 1.07192 1.35637 0.93646

15 25 0.15 0.15 100 50 0.28693 2.47928 2.23683 1.15096 1.01739

Bold values show the maximum and minimum values of correction factor k obtained from simulation.

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Fig. 5. ETCs from present work Eq. (14) compared with that obtained by (a) Zhu et al. [39] j1 = 0.66, j2 = 851.74, /d = 1 and (b) Yung et al. [40], j1 = 0.525, j2 = 675, /d = 1.

Fig. 6. (a) The use of modified HS model, Eq. (14), for predicting ETC and (b) coefficient k as a function of number of particles.

Modified HS model, Eq. (14) was also applied for re-predicting the ETC under the effects of particle size (number of hybrid-filler particles) and even particle distribution. It is reflective and convenient to re-use the results shown in Fig. 4 for this purpose. The ETC from Eq. (14) was calculated with same testing conditions given in Fig. 4 while varying coefficient k. The results were illustrated in Fig. 6a. Based on these results, the maximum variation of ETC in each studying cases is covered by the range of ETC obtained from Eq. (14). Coefficient k is shown to still be in its valid range. Furthermore, Eq. (14) can be used to predict accurately the ETC with an appropriate k. As an example, average ETCs obtained for a case of js = 15 was considered here, as shown in Fig. 6b. Determining k is similar to that in Table 1. The best fit was also obtained for scattering points. It is obvious to found from this figure that coefficient k depends on and be a function of particle size (number of particles). Notably, k is still in its valid range mentioned previously. It can be applied for other testing conditions with an appropriate k. This implicates a widely utilized possibility of modified HS model, Eq. (14) for predicting the TC of various polymer composites.

erature review on the TC prediction models of such polymer composites was studied and discussed extensively. Consequently, particle distribution of hybrid filler has less effect on the ETC of composite at low VFs and TC ratios. However, it becomes important and affects significantly to the ETC at high VF and TC of hybrid filler. Therefore, the effects of particle distribution should be taken into account when synthesizing the polymer composites reinforced with high VF and TC ratio. The effects of particle size were also taken into account. The results show that the ETC is affected significantly when the particle sizes are large enough, thus this effect should be considered in practice. Particularly, a modified HS model by Eq. (14) was first proposed. Correction factor k in this equation was found based on numerical results, i. e. k = 0.5–5. This model was well validated with both numerical results in Fig. 4 and experimental results obtained in literature. Furthermore, it is possible to extend Eq. (14) for predicting the TC of composites with even nonspherical-shape fillers. Therefore, this novel model can be widely used for predicting the TC of polymer composites with randomly distributed hybrid fillers accurately and effectively.

5. Conclusions Conflict of interest A numerical investigation on the TC of polymer composites with randomly distributed hybrid fillers has been performed. A lit-

The authors declared that there is no conflict of interest.

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