Analytical modeling and characterization of heat transfer in thermally conductive polymer composites filled with spherical particulates

Composites: Part B 45 (2013) 43–49

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Analytical modeling and characterization of heat transfer in thermally conductive polymer composites filled with spherical particulates Siu N. Leung b, Muhammad O. Khan a, Ellen Chan a, Hani Naguib a,⇑, Francis Dawson b, Vincent Adinkrah c, Laszlo Lakatos-Hayward c a b c

Department of Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto, Ontario, Canada M5S 3G8 Department of Electrical and Computer Engineering, University of Toronto, 10 King’s College Road, Toronto, Ontario, Canada M5S 3G4 AEG Power Solutions Inc., 2680 Fourteenth Avenue, Markham, Ontario, Canada L3R 5B2

a r t i c l e

i n f o

Article history: Received 28 July 2011 Accepted 1 October 2012 Available online 12 October 2012 Keywords: C. Analytical modeling B. Physical properties A. Polymer–matrix composites (PMCs) B. Thermal properties Thermal conductivity

a b s t r a c t The continuous development of electronic devices requires optimum solutions for heat dissipation. This prompts the needs to the development of novel polymer composites that possess high thermal conductivity and high electrical resistivity. This paper discusses the challenges in this research area and some potential strategies to solve the problems. In this work, a thermal conductivity analyzer was designed and implemented to measure the effective thermal conductivity of composites with heterogeneous structures. An analytical model was developed to simulate the effect of contents of spherical fillers on the effective thermal conductivity of the composites. Using the developed model, together with experimentallymeasured effective thermal conductivity of various composites, a semi-empirical approach was developed to estimate the thermal interfacial resistance at the polymer–filler interface. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction The continuous miniaturization of microelectronic devices to boost their power over the past decades has generated urgent needs for novel multifunctional materials with good thermal conductivity and enhanced dielectric properties [1–3]. As a result, advanced thermally conductive materials have drawn extensive interests from researchers and manufacturers of electronic components. These materials can be subdivided into five main categories: (i) monolithic carbonaceous materials [4–6], (ii) metal matrix composites [6–8], (iii) carbon–carbon composites [6,9,10], (iv) ceramic matrix composites [6] and (v) polymer matrix composites (PMCs) [6,11–14]. Previous studies revealed that the effective thermal conductivities (keff) possible with PMCs are not as high as those of the other aforementioned advanced materials [6]. However, their excellent chemical and dielectric properties in addition to their good processability and low cost [3] are beneficial to various electronic packaging applications. Therefore, a technological breakthrough that allows the fabrication of PMCs with dramatic improvement in keff, without the compromise of the electrical insulating property and the processability of polymers, will be vital to the technological advancement in the field of heat management for electrical and electronic devices.

⇑ Corresponding author. Tel.: +1 416 978 7054; fax: +1 416 978 7753. E-mail address: [email protected] (H. Naguib). 1359-8368/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compositesb.2012.10.001

For PMCs, conventional polymers have excellent electrical insulating (i.e., electric resistivity 1012–1016 X cm) and dielectric properties; however, they are also thermal insulators [15]. Although recent work suggested that individual chains of polyethylene can have extremely high thermal conductivity (i.e., 104 W/m K) [16], a cost-effective fabrication method for ideal single crystalline fibers before it becomes practical to replace conventional metallic heat-transfer materials has yet been developed. Therefore, the addition of appropriate fillers in the polymer matrix seems to be the most plausible solution to meet the requirements in keff. In this context, extensive studies have been conducted to promote PMCs’ keff. Various kinds of fillers, such as metal [17–20], ceramics [21–26], and carbonaceous fillers [27–30] have been applied to prepare PMCs. The thermal conductivity of these composites would be in the range of 0.5–2.0 W/m K when less than 33.3 vol.% of fillers were added into the polymer matrix. Although composites with much higher thermal conductivity (i.e., 5.0– 32.5 W/m K) were achieved in some of the previous research [20–22], it required either the addition of 60–90 vol.% of thermally conductive fillers or the uses of electrically conductive metal fillers. These would compromise the processability or the electrical insulating property of PMCs, which are two key advantages provided by PMCs in thermal management applications. In the research and development of PMCs that suit the needs demanded by various thermal management applications, it is important to accurately measure the effective thermal conductivity (keff) of the composites. However, this is not a trivial task.

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Although a number of methods and ASTM standards have been developed to address this, commercially available thermal conductivity analyzers either have very narrow measurement ranges or are restricted to homogeneous materials. Considering the thermal conductivity (k) of neat plastics is in the range of 0.1–0.2 W/m K and assuming researchers are targeted to develop PMCs to replace metals, which have their effective thermal conductivity (keff) as high as 200–300 W/m K, the task can be achieved by three of the methods – (i) laser flash method [31]; (ii) photothermal method [32] and (iii) comparative technique [33]. Among the three methods, the laser flash method is not applicable in the research of PMCs because it can only be used for homogeneous materials [31]. Therefore, the photothermal method and the comparative technique are the two appropriate methods to measure k. In addition to the experimental studies, various numerical and theoretical models have been developed by various researchers to predict the thermal conductivity of composites. Some wellknown analytical expressions include Maxwell’s equation [34], Maxwell–Eudken equation [35], Bruggeman’s equation [36], and modified Bruggeman’s equation [37]. For numerical models, four of the most commonly used methods are Nelsen Model [38], effective unit cell model [39], Agari’s semi-empirical model [21], and percolation model [40]. However, the results obtained by the aforementioned models did not agree well enough with the experimental measurements [40,41], especially when the filler contents in the composites were high. This paper aims to provide guidelines for researchers to develop multifunctional PMCs with the required properties. Firstly, a thermal conductivity analyzer was designed and implemented to measure keff of composites. Secondly, an analytical model was derived to estimate keff of polymer composites filled with spherical fillers. The theoretically-determined results were compared with the experimentally-measured keff of polyphenylene sulfide (PPS)hexagonal boron nitride (hBN) composites and other data from literatures. Finally, a semi-empirical approach, which employs the derived analytical model and empirical keff measurements, was developed to investigate the effect of thermal contact resistance at the polymer–filler interface on keff. 2. Design and implementation of thermal conductivity analyzer 2.1. Design and implementation Among the two plausible ASTM standards (i.e., photothermal method and comparative method), the comparative method, in accordance to ASTM E1225-04 [33] was adopted in this study. According to the ASTM standard, a thermal conductivity analyzer was designed and it is illustrated in Fig. 1.

This method measures the sample’s keff by comparing the temperature gradient across the sample to that across the reference bars with known k. Temperatures at 1 mm below the top surface, in the middle, as well as at 1 mm above the bottom surface of the two reference bars and the sample are measured by ultrafine thermocouples with diameters of 0.076 mm. These thermocouples are pushed into small radial holes drilled part way through the sample or the reference bar. Heat sink silicone compound is applied at the interfaces between the sample and the reference bars to reduce the thermal contact resistance. In order to measure k or keff from 0.2 W/m K to 200 W/m K, the analyzer was designed to allow easy configuration with reference bars made of different materials (e.g., Pyrex 7740 and various metal alloys) with a wide range of k. Two band heaters have been installed to the external container of the analyzer in order to allow the setting of a thermal gradient that is similar to the thermal gradient along the referencesample-reference assembly. Furthermore, another layer of insulation cladding are used to surround the external container for avoid the excessive heat loss from the analyzer to the surrounding. Most importantly, the above design measures will promote the unidirectional heat flow along the reference-sample-reference assembly, which is a crucial design criterion of the analyzer. 2.2. Design validation A series of verification experiments were conducted to validate the performance of the thermal conductivity analyzer. Firstly, the effect of the applied torque to the set screw used to fix the reference bars-sample assembly on k measurements was investigated. A flexible Teflon sheet was placed between the set screw and the top heating block to accommodate the thermal expansion of the reference bars and the sample during the measurements. A 10 mm tall stainless steel 304 standard disc sample was loaded between two 50 mm tall stainless steel 304 cylindrical reference bars in the analyzer. Consequently, the apparatus was used to measure k for the standard sample, which is 16.8 W/m K, at 150 °C [33]. The values of k were measured by tightening the set screw at the top plate at three torque levels (i.e., hand-tight, 5 N m, and 7.5 N m). Measurements show that the average k was measured to be 17.67 W/m K ± 0.63 W/m K. The measurements were found to be unaffected by the applied torque’s magnitude. In other words, hand-tightening the set screw was found to be sufficient to fix the reference bars-sample assembly. In order to verify the system’s ability to measure k of a sample using reference bars made of materials, with known values of k, that are different from the sample, a 10 mm tall Pyrex 7740 standard disc sample was loaded between two 50 mm tall stainless steel 304 cylindrical reference bars in the apparatus. Similar to

Fig. 1. CAD drawings of the thermal conductivity analyzer: (left) interior components; (middle) insulation layer; and (right) exterior components.

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the previous case, the analyzer was set to measure k at 150 °C. The average k was found to be 1.33 W/m K ± 0.13 W/m K. Compared to literature’s value of k of Pyrex 7740 at 150 °C (i.e., 1.27 W/m K) [42], the error was about 0.06 W/m K or 4.7%. Nevertheless, the existing percentage error (i.e., about 5%) is still considered to be satisfactory in this research.

Table 2 Physical properties of various grades of hBN. PTX25 Density (q) Thermal conductivity (k) Shape Size

PTX60 3

2280 kg/m 300 + W/m K Spherical agglomerate 25 lm

2280 kg/m3 300 + W/m K Spherical agglomerate 60 lm

3. Materials and methodology 3.1. Materials Commercially available polyphenylene sulfide (PPS) (Ticona, Fortron 0203B6) was used as the matrix material in this work. The thermally conductive fillers used were spherical agglomerates of hexagonal boron nitride (hBN) (Momentive Performance Materials, PolarTherm, PTX25 and PTX60). PPS was chosen as the matrix material because of its high service temperature (i.e., 200 °C), which is needed in various heat management applications. For the filler, hBN was chosen for two reasons: (i) it is electrically insulating and (ii) it resembles the layered structure of graphite, making it extremely soft, and thereby easier to be compounded at high loading. All materials were used as received without any further modification. The physical properties of the polymers and fillers are summarized in Tables 1 and 2. 3.2. Sample preparation PPS and hBN were first dry-blended and then meltcompounded in a micro-compounder (DSM Xplore 15) at 300 °C and 50 rpm for 6 min. The extruded composites were cooled in a water bath at room temperature. The extrudates were then pelletized and grinded into fine powders using a pelletizer and a mill freezer (SPEX CertiPrep Group, 6850 Freezer/Mill), respectively. For keff measurements, the composites were compression-molded, at 310 °C, into 10 mm thick disc-shaped samples of 20 mm diameter. 3.3. Characterization The keff and polymer–filler morphology of each sample were studied in this work. The values of keff were measured by the thermal conductivity analyzer designed and constructed in this work. The average and standard deviation of keff were determined from measuring keff of three samples for each material composition. The cross-sections of the composites, fractured in liquid nitrogen, were analyzed with the scanning electronic microscopy (SEM, JEOL, model JSM-6060) to study the morphology of the composites.

It is assumed that the spherical fillers are evenly-distributed in the polymer matrix and are located in a face-centered cubic arrangement. The face-centered cubic arrangement was chosen since it is the most efficient way to closely pack spherical particles in lattice. Consider such arrangement, the model would be applicable to composites filled with upto 74 vol.% of spherical particles. Suppose the side length of the cube and the particle diameter is a and 2r, respectively, an expression to determine the filler volume fraction (uf) was derived:

uf ¼

16pr 3 3a3

ð1Þ

4.2. Analytical models to predict keff of polymer composites Depending on the filler contents in the polymer composites, the physical and the corresponding mathematical models would be different. Fig. 3a–c shows the heat transfer models of a heat conduction element in a polymer composite at three filler contents: lower than 26.18 vol.%, equal to 26.18 vol.%, and greater than 26.18 vol.%. It must be noted that when the loading of spherical fillers equals to 26.18 vol.%, a will be equal to 4r. For filler content lower than 26.18 vol.%, there are layers of polymer matrix without any filler particle (i.e., layer 2 and layer 5 in Fig. 3a). In contrast, for filler content higher than 26.18 vol.%, filler particles at the top or bottom surface and filler particles at the centre of the side faces coexist in some layers of the polymer matrix (i.e., layer 2 and layer 5 in Fig. 3c). In this model, each element is subdivided into sections with their thermal resistances equal to Ri, where i is the layer number as indicated in Fig. 3a–c. Hence, the thermal resistance network consists of thermal resistors connected in series along the heat flow direction. With the incorporation of the thermal contact resistance (Rint) in the model, the total thermal resistance (Rtotal) is given by the following equation:

Rtotal ¼

n X Ri þ Rint

ð2Þ

i¼1

where n is 4 when the filler content is equal to 26.18 vol.% and 6 otherwise.

4. Theory and modeling 4.1. Heat transfer elements The magnitude of a polymer composite’s keff was modeled by considering a series of thermal resistors along the heat flow direction. Fig. 2 illustrates an element used to model the keff of a polymer composite filled with spherical fillers. Table 1 Physical properties of PPS. Physical properties

Values

Density (q) Melting temperature (Tm) Service Tmax Thermal conductivity (k) Dielectric strength

1350 kg/m3 280 °C 200 °C 0.22 W/m K 22–28 kV/mm

Fig. 2. A heat transfer element in a polymer composite filled with spherical fillers in a face-centered cubic arrangement.

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Fig. 3. Side views of physical models for polymer composites filled with (a) <26.18 vol.%; (b) 26.18 vol.% and (c) >26.18 vol.% spherical fillers.

From the Fourier’s law of heat transfer, the amount of heat flow (q) across the element can be expressed as the following equation:

q ¼ keff a2

DT ¼ keff aDT a

ð3Þ

and the element’s thermal resistance is

Rtotal ¼

1

ð4Þ

keff a

Considering a thin layer of the element with thickness dy as along the upward direction shown in Fig. 3a–c and according to the Fourier’s law, the heat flow across the matrix and the filler can be determined by Eqs. (5) and (6), respectively.

q m ¼ k m Am

q f ¼ k f Af

dT dy

ð5Þ

dT dy

ð6Þ

where km and kf are the thermal conductivities of the polymer matrix and the filler, respectively; and Am and Af are the cross-sectional areas of the polymer matrix and the filler, respectively. The total heat flow (qtot) across each layer of thickness dy is the sum of qm and qf:.

qtot ¼ qm þ qf ¼ km Am

dT dT  dT þ k f Af ¼k A dy dy dy

ð7Þ

where A is the cross-sectional area of the element, and 

k ¼ km

Af Am þ kf A A

ð8Þ

Using geometry and calculus, the expressions to calculate the thermal resistances for each of the three cases were derived. Case 1: Filler content < 26.18 vol.%. The thermal resistances of different polymer composite layers can be determined by the following equation:

1   R1 ¼ R3 ¼ R4 ¼ R6 ¼ 2p k f  k m a  4r R2 ¼ R5 ¼ 2km a2

Z 0

r

1 dy u2  y2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k a2  m  þ r2 2p k f  k m



2p

1  kf  km

Z 0

r

1 dy u2  y2

ð12Þ

where

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k a2  m  þ r2 u¼ 2p kf  km

ð13Þ

Case 3: Filler content > 26.18 vol.%. The thermal resistances of different polymer composite layers can be determined by the following equation:

R1 ¼ R6 ¼

1   2p k f  k m

R2 ¼ R5 ¼

1   2p k f  k m

R3 ¼ R4 ¼

1   2p k f  k m

Z

ar 2

0

Z

0

ar 2

Z

1 dy u2  y2

1 dy 2y2 þ ay þ m

a 2

r

u2 

1 a 2

2 dy y

ð14Þ

ð15Þ

ð16Þ

where

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k a2  m  þ r2 u¼ 2p kf  km

m¼

k a2 a2  m  þ 2r 2  4 2p k f  k m

ð17Þ

ð18Þ

Taking the integration and substituting the results of Eqs. (9)– (18) into Eq. (2), the expressions to determine Rtotal for Cases 1–3 can be expressed as Eqs. (19)–(21), respectively. This can be substituted into Eq. (22) to determine keff of the element.

Rtotal ¼ Rint þ 

u þ r  a  4r 1   ln  þ ur km a2 kf  km u

ð19Þ

Rtotal ¼ Rint þ 

u þ r  1   ln   ur kf  km u

ð20Þ

p

ð9Þ

p

ð10Þ

  u þ 2a  r 1   ln  p kf  km u u  2a þ r   4r  a þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 a2 þ 8m  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln  p kf  km a2 þ 8v 4r  a þ a2  8m

Rtotal ¼ Rint þ 

where

u¼

R1 ¼ R2 ¼ R3 ¼ R4 ¼

ð11Þ

Case 2: Filler content = 26.18 vol.%. The thermal resistances of different polymer composite layers can be determined by the following equation:

keff ¼

a 1 ¼ Rtotal a2 Rtotal a

ð21Þ

ð22Þ

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4.3. Semi-empirical approach to determine thermal contact resistance A challenge for developing thermally conductive composites is the suppression of the thermal contact resistance between the polymer matrix and the thermally conductive fillers. Polymers usually have hydrophobic surfaces. Therefore, fillers with polar surfaces, such as boron nitride, will lead to poor interfacial bonding to the polymer [43], leading to high thermal contact resistance at the interfacial region. It is believed that the thermal contact resistance at the interface (Rint) plays a dominant role in the limitation on promoting the composite’s keff, and its value is extremely difficult to be measured experimentally. Therefore, it would be useful to develop a semi-empirical approach to estimate Rinf. This can be achieved by substituting the experimentally measured keff, together with Eqs. (19)–(21), into Eq. (22). The estimation of Rint would improve the accuracy of theoretical prediction of composite’s keff as well as to promote the development of thermally conductive polymer composite. 5. Results and discussion 5.1. Measurement of keff of PPS-hBN composites Fig. 4 shows the measured keff for PPS-hBN composites, loaded with 33.3 vol.% of either PTX25 or PTX60. keff of PPS-PTX25 composite and PPS-PTX60 composite were measured to be 1.83 W/m K and 1.89 W/m K, respectively. With the addition of 33.3 vol.% of hBN spherical agglomerates, keff of the PPS-BN composites exhibited more than 800% improvement over neat PPS (i.e., 0.22 W/m K). However, the values of keff were still significantly lower than that of the hBN fillers (i.e., >300 W/m K), suggesting the full potential of them to promote keff had not been achieved. It is believed that the interfacial phonon scattering, which can be mathematically related to Rint at the PPS-hBN interfaces, limited the potential increase in keff in the fabricated composites. Fig. 4 also shows that keff of the composites were virtually independent of the size of the BN particulates despite the increased interfacial area-to-volume ratio (Aint/V) ratio, which can be determined by Eq. (23), with smaller filler size.

Aint 16pr 2 ¼ V a3

47

The values of Aint/V for the PPS-PTX25 composite and the PPSPTX60 composite were calculated to be 8.00  104 m2/m3 and 3.33  104 m2/m3, respectively. The Aint/V ratio of the composite with 25 lm BN fillers is about 2.4 times larger than that of the composite with 60 lm BN fillers. In other words, the total interfacial area, between the matrix and the filler is smaller when the filler size is larger. The measurements seem to be counter-intuitive because it is generally expected that larger Aint/V should result in a larger Rint and thereby lower keff. An explanation to the measured results in this work would be provided in the later section of this paper. 5.2. Composite’s morphology Fig. 5 illustrates a SEM micrograph of the compression-molded PPS-hBN composite sample with 33.3 vol.% of PTX60. The SEM micrograph reveals that some of the spherical agglomerates had been debonded from the matrix, demonstrating the weak adhesive force between the filler and the matrix. Moreover, microfibrils, which would demonstrate interfacial adhesion, were not observed from the micrographs. This indicated that the interfacial adhesion between the hBN agglomerates and the PPS matrix was weak. Therefore, it is believed that the hBN particles were surrounded by air voids in the composites, and there appeared to be nothing attaching them to the polymer matrix. Without interfacial adhesion, the thermal conductive hBN fillers failed to induce significant improvement in keff of the composites. The SEM micrograph also shows that a portion of spherical agglomerates of hBN particulates could not sustain either the high shear stress during the melt compounding or the high applied pressure during the compression molding of the sample. As a result, they were broken into platelets. Since the spherical agglomerates of hBN have more isotropic properties than typical BN platelets, the failure to sustain the agglomerates would be detrimental to heat transfer ability of the composite. In addition, by comparing the density of the compression molded sample with the theoretical density of the PPS-hBN composite, it was found that the sample exhibited a void fraction of 15%, which would also reduce the keff of the composite. 5.3. Theoretical calculation of keff of PPS-hBN composites

ð23Þ

Fig. 4. The thermal conductivity of PPS-hBN composites filled with hBN spherical agglomerates of different sizes.

Eqs. (19)–(22) were employed to calculate the keff for PPS-hBN composites with 33.3 vol.% of hBN. First, assuming there was no Rint at the PPS-hBN interface, keff of both PPS-PTX25 and PPS-PTX60 composites were calculated to be 72.4 W/m K as shown in Fig. 6. It is obvious that, if Rint is negligible, the composite’s keff is

Fig. 5. SEM micrographs of compressed-molded PPS-hBN (PTX60) composite sample at a magnification of 500.

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independent of the size of the spherical fillers. Furthermore, the theoretically predicted keff increases dramatically when the filler content is greater than 26.18 vol.%. This suggests that it is important to have sufficient amount of filler such that every single layer of the polymer matrix has fillers embedded in it. A similar percolation behaviour was also observed experimentally with polymer composites filled with hBN agglomerates [44]. Most importantly, it was observed that the theoretically-predicted keff was significantly higher than the experimentally measured keff of these composites, which were 1.83 W/m K and 1.89 W/m K for PPS-PTX25 and PPS-PTX60 composites, respectively. These differences were attributed to the Rint at the PPS-hBN interfaces, which is believed to be caused by the incomplete bonding and phonon acoustic mismatch [45,46]. 5.4. Thermal contact resistance Substituting the experimentally measured keff of the PPS-BN composites into Eq. (22), the values of Rint for PPS loaded with 33.3 vol.% of PTX25 and PTX60 were estimated semi-empirically to be 1.09  104 K/W and 4.38  103 K/W, respectively. In other words, Rint for the PPS-PTX25 composite was about 2.5 times larger than that for the PPS-PTX60 composite. This increase was virtually consistent with the 2.4 times increase in the Aint/V ratio when the filler diameter reduced from 60 lm to 25 lm. Therefore, the semiempirical calculations suggested that Rint seems to be directly proportional to the Aint/V ratio of the composite.

Rint a

Aint Aint ) Rint ¼ C V V

ð24Þ

where C is the proportionality constant between Rint and Aint/V ratio. Rewriting Eq. (24) to obtain Eq. (25), it can be found that the product of Rint and a is independent of the diameter of the spherical fillers. Using Eqs. (19)–(22), it was computed that keff of polymer composites filled with spherical fillers was independent of the filler size, regardless of the presence of Rint at the polymer–filler composite.

Rint a ¼ C

1 16pr 2  a ¼ Cð144pu2f Þ3 a3

ð25Þ

In this research, the discrepancy between the theoreticallypredicted and experimentally-measured keff, which was caused by the weak interfacial adhesion between hBN agglomerates and PPS matrix in the composites, revealed that the high Rint at the

Fig. 7. Effect of reducing Rint on keff of the PPS-hBN composite loaded with 33.3 vol.% of spherical hBN agglomerates.

PPS-hBN interfaces significantly hindered the promotion of the composite’s keff by the thermally-conductive hBN fillers. Moreover, the existence of voids and the breakage of the hBN agglomerates in the compression-molded samples also contributed to the lower than predicted keff of the PPS-BN composite. Therefore, in order to design and fabricate novel thermally conductive polymer composites, it is important to develop strategies to minimize Rint, the void fraction, and the breakage of hBN agglomerates. Assuming the problems of voids and hBN agglomerates breakage have been solved, using Eqs. (19)–(22), the extent to which the reduction of Rint would promote keff was simulated, and the result is illustrated in Fig. 7. In the figure, Rint,treated represents the reduced Rint at the polymer–filler interface as a result of some potential surface treatment applied to the fillers. The simulation result indicated that the improvement in keff becomes obvious only when Rint has been reduced to less than 20% of its original level. In other words, an effective compatabilizer or an effective surface-treatment technique would be essential to make a breakthrough in the research and development of thermally-conductive polymer composites. Therefore, further investigation on the development of strategies to improve interfacial adhesion is currently conducted. 6. Conclusion

Fig. 6. Theoretical prediction of keff of PPS-hBN composites with different filler contents.

The challenges to research and develop novel thermal conductive polymer composites were discussed in this paper. In order to measure the thermal conductivity (keff) of composites with a wide range of values, a thermal conductivity analyzer was designed and constructed in this work. Moreover, a new theoretical model of heat transfer, incorporating the thermal contact resistance at the polymer–filler interface, was established to predict keff of a polymer composite filled with spherical fillers. Analyses on the polymer–filler morphology and comparison between the theoretically-predicted and experimentally-measured keff revealed that the weak interfacial adhesion between the filler and the polymer (i.e., high thermal contact resistance (Rint)) is a key factor that hinders the promotion of keff by the thermally-conductive fillers. Using the analytical model developed in this work, together with experimental measurements of keff, a semi-empirical approach was established to study the effects of polymer–filler interfacial properties on keff. Simulation result showed that an effective strategy to minimize Rint is critical to make a break-

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