Effect of CNT coating on the overall thermal conductivity of unidirectional polymer hybrid nanocomposites

International Journal of Heat and Mass Transfer 124 (2018) 190–200

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

Effect of CNT coating on the overall thermal conductivity of unidirectional polymer hybrid nanocomposites M.K. Hassanzadeh-Aghdam a, M.J. Mahmoodi a, J. Jamali b,⇑ a b

Department of Civil, Water and Environmental Engineering, Shahid Beheshti University, P.O. Box 167651719, Tehran, Iran Department of Mechanical Engineering, American University of the Middle East, Kuwait

a r t i c l e

i n f o

Article history: Received 8 February 2018 Received in revised form 18 March 2018 Accepted 19 March 2018

Keywords: Polymer hybrid nanocomposite Carbon nanotube Thermal conductivity Interfacial thermal resistance Micromechanics

a b s t r a c t The role of carbon nanotube (CNT) coating on the carbon fiber (CF) surfaces in the effective thermal conductivities of the unidirectional polymer hybrid nanocomposites is investigated by a newly presented multi-stage micromechanical method. The constructional feature of the hybrid nanocomposite is that randomly oriented CNTs grown on the CF surfaces. For simulating, a new version of the semi-empirical Halpin-Tsai (H-T) model is appropriately coupled with an analytical unit cell micromechanical model developed in the present research. The model captures the influences of the CNTs random dispersion, waviness, length, diameter, volume fraction and the CNT/polymer interfacial thermal resistance and also the CF cross-section shape parameters. The predicted results for the thermal conductivities of fibrous composites and polymer nanocomposites containing CNTs are verified with the available experimental data and a very good agreement is found. The results show that the longitudinal thermal conductivity of CF-reinforced hybrid nanocomposites is not affected by the CNTs coating. However, the nanocomposites transverse thermal conductivities are significantly enhanced over those of the conventional fibrous composites without the CNTs coating. An improvement in the nanocomposites transverse thermal conducting behavior can be observed with (i) increasing the CNTs volume fraction and length (ii) using straight CNTs and (iii) forming a perfect bonding interface. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction Heat dissipation from microelectronic packaging and especially advanced aircraft systems is increasingly important because of the rapid rise in thermal load expected to reach 10,000 kW [1–3]. The severe thermal load can significantly influence the reliability of aircraft and limit its service life. Also, the electronic packing materials must disperse the heat produced to lower temperature for providing a better performance. Therefore, because the improvement in eliminating the heat of such devices is related to their efficiency, lengthening of half-life time operating and preventing the premature failures, a well-organized thermal management seems to be essential. Generally, thermal management has become a vital task of academic and industrial interest for device design and application. Carbon fiber (CF)-reinforced polymer composites are widely used in many high-advanced industries such as aerospace due to their high strength, high stiffness, low mass density, good fatigue resistance and corrosion resistance [4–8]. Also, the axial thermal ⇑ Corresponding author. E-mail address: [email protected] (J. Jamali). https://doi.org/10.1016/j.ijheatmasstransfer.2018.03.065 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.

conductivities of these fibrous polymer composites is very high dominated by the thermally conductive CFs [1,3,9] which can effectively promote heat diffusion. However, the thermal conductivities of CF-reinforced polymer composites are greatly poor in the transverse direction (the direction perpendicular to the fiber) dominated by the thermally insulating polymer matrix [1,3]. It hinders heat transfer in the transverse direction and therefore, limits their application in various fields such as the satellite structure panels [3]. To resolve this issue, a secondary reinforcement with a high thermal conductivity such as carbon nanotube (CNT) has been added into the fiber-reinforced polymer composites [10–13]. A polymer matrix reinforced with conventional microscale fibers and CNTs forms a hybrid nanocomposite system. Commonly, two methods exist to consider the CNTs into the conventional composite materials for manufacturing high performance hybrid composites. In the first type, the CNTs are grown on the surface of the fibers. Then, the CNT-coated fibers as the reinforcements are embedded into the polymer matrix [14–17]. In the second type, the polymeric material is enriched with CNTs and then used as the matrix of fibers [18–20]. Amongst, CNTs is one of the most exciting nanostructural materials which have been used to design the high-performance

M.K. Hassanzadeh-Aghdam et al. / International Journal of Heat and Mass Transfer 124 (2018) 190–200

composite systems frequently due to their outstanding mechanical and thermal properties with high aspect ratio and low density [21–25]. The thermal conductivity of CNTs can be in the range of 2500–6000 W/mK [26,27]. A theoretical study existing in literature showed the thermal conductivity of (10, 10) single-walled CNT is about 6600 W/mK [28]. So, it is expected a significant increase in the thermal conductivity of nanocomposites and hybrid nanocomposites with adding even a small content of CNTs into polymer matrixes and traditional fibrous composites. For instance, Biercuk et al. [29] used the single-walled CNTs to augment the thermal transport properties of epoxy resin. The epoxy nanocomposite containing 1 wt% unpurified single-walled CNTs showed a 125% increment in thermal conductivity as compared with that of neat epoxy resin [29]. Also, Bryning et al. [30] measured the thermal conductivities of purified single-walled CNT-epoxy nanocomposites prepared by means of suspensions of CNTs in N-NDimethylformamide and surfactant stabilized aqueous CNT suspensions. It was reported an improvement in thermal conductivities of both types of the nanocomposites [30]. However, surfactant processed coupons revealed greater overall enhancement. This discrepancy was attributed to CNT/matrix interfacial thermal resistance [30]. Also, Guthy et al. [31] observed that the thermal conductivity of the poly (methylmethacrylate) (PMMA) nanocomposite containing 9 vol% single-walled CNTs improved by 250% compared with that of the pure PMMA resin [31]. Bouchard et al. [32] measured the effective thermal conductivity of polyethersulfone (PES) nanocomposites reinforced with different volume fractions of multi-walled CNTs. Also, the thermal conducting response of styrene-based shape memory polymer (SMP) nanocomposites containing CNT was evaluated by Yu et al. [33]. They experimentally examined the effects of temperature and CNT volume fraction on the SMP nanocomposite thermal conductivities [33]. Zhang et al. [34] found that at CNT content of 10 wt %, the thermal conductivity of poly(vinylidene fluoride) (PVDF) nanocomposite increases up to 0.47 W/mK, which is higher than twice that of the pure PVDF sample. In the field of hybrid nanocomposites, Kim et al. [19] experimentally evaluated the thermal conducting behavior of the CF-reinforced phenolic hybrid nanocomposites containing CNTs. It was shown that the addition of 7 wt% CNTs into a phenolic nanocomposite matrix reinforced with 50 vol% CFs results in an enhancement of the axial thermal conductivity from 250 W/mK to 393 W/mK [19]. Also, Wang and Qiu [35] fabricated some samples of polyester/vinyl ester hybrid nanocomposites reinforced with glass fibers and CNTs, and measured the thermal conductivity of them. They reported that adding 3 wt% CNTs into a glass fiber-reinforced polyester/vinyl ester hybrid nanocomposite yields 1.5-folds improvement of thermal conductivity as compared with that of traditional glass fiberreinforced polyester/vinyl ester composite [35]. Moreover, Liang et al. [36] experimentally evaluated the thermal conductivity of CF-reinforced epoxy hybrid nanocomposites containing CNTs. On the other hand, some micromechanical methods have been employed to predict the thermal conductivities of polymer hybrid nanocomposites. For example, Kundalwal and Ray [37] determined the effective thermal conductivities of a unidirectional polymeric hybrid composite reinforced by radially aligned CNTs-coated carbon fibers called ‘‘fuzzy fiber” as shown in Fig. 1 [37,38]. The constructional feature of fuzzy fiber-reinforced hybrid composite is that the uniformly aligned CNTs are radially grown on the circumferential surfaces of the fibers [37,38]. To simulate the hybrid nanocomposite, they used the effective medium (EM) approach in conjunction with the composite cylinder assemblage (CCA) approach [37]. The influences of volume fractions of the CF and CNT, temperature, the CNT/matrix interfacial thermal resistance on the hybrid nanocomposite thermal conducting response were investigated [37]. In another study, by employing the method of

191

Fig. 1. Schematic diagram of a fuzzy fiber [37,38].

cells (MOC) and EM approach, Kundalwal et al. [38] studied effect of the CNT non-straight shape on the thermal conducting behavior of unidirectional fuzzy fiber-reinforced polymer hybrid nanocomposites. Nevertheless, current processing techniques lead to a randomly dispersed state for the CNTs on the fiber surfaces [39,40]. Fig. 2 illustrates a scanning electron microscope (SEM) image of CFs with CNTs grown on their surfaces [39]. Therefore, fabrication of a fiber coated with radially aligned CNTs as indicated in Fig. 1, is very difficult. A literature survey clearly proves that no substantial work exists that accurately predicts the thermal conductivities of the CNT-coated CF-reinforced polymer hybrid nanocomposites. Note that the thermal conductivity plays a key role in choosing the optimum materials to meet specific design requirements in many aerospace applications [41]. The main objective of the current work is present a multi-stage micromechanical approach to analyze and predict the effective thermal conductivities of randomly dispersed CNT-coated CFreinforced polymer hybrid nanocomposites. The organization of the paper is following: in Section 2, a newly presented version of the semi-empirical Halpin-Tsai (H-T) model is presented. Also, in Section 3, the extended simplified unit cell (ESUC) model is developed. Then, the simulation procedure of the hybrid nanocomposites is presented in Section 4. In Section 5, parametric studies are conducted to numerically evaluate the effects of the CFs volume fraction and cross-section shape, the CNT/matrix interfacial thermal resistance, volume fraction, length and non-straight shape of the CNTs grown on the CF surface on the thermal response of the CNT-coated CF-reinforced polymer hybrid nanocomposites. Also,

Fig. 2. Scanning electron microscope image of carbon fibers with CNTs grown on their surfaces [39].

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in Section 5, the comparative studies are conducted with the experimental data available in the literature. The concluding remarks are given in Section 6. The new developed model with reported results could be actually useful to guide the design of hybrid nanocomposites.

Also, in the CNT-reinforced polymer matrix nanocomposites, the CNT waviness is of particular interest [45–50]. The CNT waviness is inherent to the manufacture process of polymer nanocomposites. Consequently, an acceptable model must be consider the non-straight shape of CNTs. Hence, in the present study, a waviness efficiency factor d is added to b in Eq. (2), as follows

2. Modifying the semi-empirical H-T model In this section, the modified version of semi-empirical H-T model [42] is proposed to extract the thermal conductivities of CNT-polymer matrix nanocomposites. The polymer nanocomposites thermal conductivity dependence on the CNT volume fraction, length and diameter can be evaluated by the initial form of semiempirical H-T model, as follows [42]

1 þ abV CNT ; 1  buV CNT

K ¼ Km

ð1Þ

b¼

a¼

2L ; D

b¼

1 þa

;


 1  um V CNT u¼1þ 2

um

b¼ ð2Þ

where K m and K CNT are the thermal conductivity of polymer matrix and CNT, respectively. Also, V CNT , L and D specify the CNT volume fraction, length and diameter, respectively. The value of um is equal to 0.82 [42]. In the CNT-reinforced polymer matrix nanocomposites, a large interfacial region between the CNT and polymer matrix is formed due to the large surface area to volume ratio of CNTs [26,37,43,44]. Therefore, the CNT/polymer interface plays a key role in predicting the polymer nanocomposite thermal conductivity. In the field of studying the nanocomposite thermal conducting behavior, the interface effect has been incorporated to be the interfacial thermal resistance between the polymer matrix and the CNTs [26,37,43,44]. The CNT/matrix interfacial thermal resistance identifies a heat flow barrier associated with a weak interface, and differences in phonon spectra based on the atomic arrangements and densities of the two phases [37,44]. Fig. 3 displays the schematic of a CNT coated with a thin interfacial thermal barrier layer. Herein, the CNT and surrounding interfacial thermal barrier layer is considered as an equivalent nanofiber. The effective thermal conductivity of the nanofiber along the CNT direction can be calculated by the mixture rule for a simple series model of the barrier layer/CNT/barrier layer, as follows

K eff CNT ¼

K CNT 1 þ 2rLK

K CNT Km

;

rK ¼ RK m

eff

dK CNT Km

1 þa

;

d¼1

dlK eff CNT Km

1

dlK CNT Km

þa

eff

The CNTs are supposed to be randomly dispersed in three dimensions leading to l ¼ 1=6 [51]. 3. The ESUC method In the present work, the micromechanical formulations of the ESUC model [52–55] are developed for observing heat conducting behavior of composite materials. Fig. 4 shows the representative volume element (RVE) of the ESUC model. The numbers of subcells present in the RVE along the x and y directions are c and r, respectively. Thus, the RVE includes c  r sub-cells. Each sub-cell is labeled by ij, with i and j denoting the counter of the sub-cell along the x and y directions, respectively. The lengths of each sub-cell in the x and y directions are ai and bj, respectively. Also, Lc and Lr are the lengths of the RVE in the x and y directions, respectively. A unit length in the z direction is considered to apply generalized plane strain condition on the RVE corresponding to the long fiber reinforced composites [38,52,54]. For extracting the governing equations, it is assumed that the temperature within each sub-cell of the RVE varies linearly [38]. Also, the generation and loss of heat energy within each sub-cell are ignored [38]. Based on the energy balance between the global heat flux components Q l (l ¼ x; y; z) over the RVE faces opened in

y

where r K and R are the Kaptiza radius and resistance, respectively [26,37,43,44]. The magnitude of R has been reported to be between 0.77  108 m2 K/W and 20  108 m2 K/W [37]. Therefore, K CNT in

Fig. 3. Schematic diagram of a CNT coated with a very thin interfacial thermal barrier layer.

Qy

1

br

Qx Qz

bj

Lr

CNT Interface

ð4Þ

ð5Þ

ð3Þ

Eq. (2) is replaced with K eff CNT mainly to consider the effect of interfacial thermal resistance. In general, interface region includes a significant contribution to the overall properties of polymer matrix or metal matrix composite materials [37,43,45–47]. In the case of metal matrix composite systems, a comprehensive study on the interface influence has been conducted in Refs. [46,47].

A W

where A and W are the amplitude and half-wavelength of a wavy CNT, respectively. As mentioned above, the presented model by Eq. (1) is only able to predict the axial thermal conductivity of aligned CNT-reinforced nanocomposites. So, an orientation factor l is added to Eq. (4) herein for accounting the random orientation of CNTs into the nanocomposites, as follows

in which K CNT Km K CNT Km

dK eff CNT Km

x

ijk

b1 z

a1

ai

ac

Lc

Fig. 4. Representative volume element of the extended simplified unit cell model.

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the x, y and z directions and the local heat flux components qijl flows within each sub-cell ij, one can arrive r X bj q1j x ¼ Q x Lr ;

Carbon fiber

j¼1 c X

ai qi1 y i¼1 r X c X

CNT

¼ Q y Lc ;

ð6Þ

Fig. 6. Schematic general view of a hybrid fiber (CNT-coated carbon fiber).

bj ai qijz ¼ Q z Lr Lc :

b)

a)

j¼1 i¼1

Equivalence of the local heat flux components along the interfaces of the RVE sub-cells yields ij q1j x ¼ qx ði > 1Þ;

ij qi1 y ¼ qy ðj > 1Þ:

Polymer CNT Carbon fiber

ð7Þ

Also, following relations are obtained from the temperature compatibility based on the RVE definition [38]

8 Pc P a T i1 ¼ c a T ij ¼ Lc T
;x > > < Pi¼1 i ;x Pi¼1 i ;x r r 1j ij
j¼1 bj T ;y ¼ j¼1 bj T ;y ¼ Lr T ;y > > : ij T ;z ¼ T
;z

ðj > 1Þ; ði > 1Þ

ð8Þ

ði > 1; j > 1Þ

where T ij;l represents the local temperature gradient within the subcell ij and T
;l indicates the global temperature gradient on the RVE. According to the Fourier’s law, following relation is expressed [38]

Tij; ¼ Kij

1

: qij

ð9Þ

ij

where K is the thermal conductivity tensor of the sub-cell ij. By substituting Eq. (9) into Eq. (8) in conjunction with Eqs. (6) and (7), a system of cr þ c þ r linear equations with the same number of unknowns can be arranged in a matrix form, as follows

Amm qm1 ¼ Fm1

where m ¼ cr þ c þ r

ð10Þ

where q and F are the sub-cells heat flux vector and external heat excitation vector, respectively. Also, A is the coefficients tensor formed by the geometrical parameters and material properties of the sub-cells. 4. Multi-stage micromechanical simulation of the hybrid nanocomposites Fig. 5 shows the schematic diagram of a lamina made of the CNT-coated CF-reinforced polymer hybrid nanocomposite being investigated here. The CNTs have been grown on the CF surfaces while they are randomly dispersed on the CFs circumferential surfaces. Such a resulting hybrid fiber; i.e. CNT-coated CF, is displayed in Fig. 6. When this hybrid fiber is embedded into the polymer material, the gap between the CNTs is filled with the polymer. Therefore, the randomly dispersed CNTs reinforce the polymer matrix surrounding the carbon fiber along the direction transverse to the length of the carbon fiber, as displayed in Fig. 7. It can be viewed as a cylindrical effective fiber (CEF) in which a carbon fiber

Fig. 7. (a) 3D schematic general view and (b) cross-section of cylindrical effective fiber.

is embedded in the CNT-reinforced polymer matrix nanocomposite and the radius of the CEF equals the sum of the carbon fiber radius of and the straight CNT length. A 3D schematic general view and cross-section of such a CEF is demonstrated in Fig. 7a and b, respectively. The hybrid nanocomposite RVE can be treated as being composed of two phases wherein the reinforcement is the CEF and the matrix is the monolithic polymer material. In fact, the CEFs reinforce the remaining polymer matrix. Consequently, the entire polymer material is divided into two domains, (1) to fill the gap between the CNTs grown on the surface of carbon fibers totally named as the CEFs, forming CNT-polymer nanocomposite and (2) the remaining polymer as the matrix phase for CEF reinforcements into the hybrid nanocomposite, as shown in Fig. 8. A regular arrangement of CFs with square packing is considered in Fig. 8. Note that it can be also seen in Fig. 5 with a random arrangement of CFs. Predicting the effective thermal conductivities of polymer nanocomposite containing randomly dispersed CNTs is the first step of the hybrid nanocomposite simulation. Note that the configuration of CNTs is a hollow, cylindrical structure consisting of hexagonal carbon rings which cannot satisfy the requirement of micromechanical models. Generally, the actual structure of CNT has been replaced with a cylindrical solid in the micromechanical simulation, as extensively employed in previous studies [37– 39,56]. Thus, in the first attempt using the modified version of semi-empirical H-T model presented in the current work, the effective thermal conductivities of the polymer nanocomposite considering the CNT/polymer interfacial thermal resistance, the CNT non-straight shape and randomly dispersed CNTs into the matrix are determined. In the second part, considering the polymer nanocomposite material as the matrix phase and the CF as the

Polymer

CNT

Cylindrical effective fiber Carbon fiber Carbon fiber

CNT Polymer

Fig. 5. Schematic general view of a CNT-coated carbon fiber-reinforced polymer hybrid nanocomposite.

Fig. 8. Cross-section of the representative volume element of a hybrid nanocomposite with regular arrangement of carbon fibers.

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reinforcement, effective thermal conductivities of the CEF are to be evaluated using the ESUC model according to the RVE shown in Fig. 9. In this part, in order to simulate the CF circular crosssection, the RVE of the ESUC model consists of 50  50. Finally, by means of the estimated thermal conductivities of the CEF and the monolithic polymer matrix, the effective thermal conductivities of the hybrid nanocomposite can be predicted by the ESUC model. Fig. 10 represents the multi-stage modeling procedures of the randomly oriented CNT-coated CF-reinforced polymer hybrid nanocomposites. 5. Results and discussion Verifying the validity of the semi-empirical H-T model proposed in this research is significant. To this end, the predictions of the new version of H-T model are compared with the available experimental data for multi-walled CNT-polycarbonate nanocomposites [26]. Two types of nanocomposite systems, including a polycarbonate resin containing randomly dispersed long CNTs and a polycarbonate resin containing randomly dispersed short CNTs are

Polymer nanocomposite

Carbon fiber Fig. 9. Representative volume element of the extended simplified unit cell model for simulating cylindrical effective fiber.

Polymer

CNT

Carbon fiber

H-T model

Polymer nanocomposite Extended simplified unit cell model

Cylindrical effective fibers

Extended simplified unit cell model

CNT-coated carbon fiber-reinforced polymer hybrid nanocomposite Fig. 10. The multi-stage nanocomposite.

modeling

procedures

of

the

polymer

hybrid

selected. The average length of the long and short CNTs is equal to 200 mm and 20 mm, respectively as reported in Ref. [26]. The average diameter of the long CNT is equal to that of the short CNT; i.e. D ¼ 60 nm [26]. The values of K m , K CNT , R, d and l are equal to 0.21 W/mK, 150 W/mK, 8.3  108 m2 K/W, 0.5 and 1/6, respectively [26,47,50,57]. The comparisons of the thermal conductivities of the long CNT-polycarbonate nanocomposite and the short CNT-polycarbonate nanocomposite predicted by the presented H-T model with those of the experimental measurements [26] have been indicated in Fig. 11a and b, respectively. The semi-empirical H-T model considering only random orientation factor tremendously over predicts the experimental data of the nanocomposite thermal conductivities [26]. When the CNT waviness and random orientation factors are simultaneously incorporated in the simulation of the long CNT-polycarbonate nanocomposite, the predictions are closer to the experimental data [26], as shown in Fig. 11a. However, in the case of short CNTpolycarbonate nanocomposite, a great difference exists between the experimental data [26] and calculations of the H-T model considering both CNT waviness and random orientation factors, as can be seen from Fig. 11b. It may be due to the critical role of the CNT/ matrix interfacial thermal resistance in the thermal conducting behavior of short CNT-reinforced polymer nanocomposites. An excellent agreement between the predictions and experimental measurements [26] for both long CNT-polycarbonate nanocomposite and short CNT-polycarbonate nanocomposite systems can be observed when the random orientation, non-straight shape of CNTs and the CNT/matrix interfacial thermal resistance factors are taken into account in the semi-empirical H-T model. The results of Fig. 11 reveal that the predictions of the H-T model without interfacial thermal resistance are higher than those of the H-T model including interfacial thermal resistance. Additionally, the thermal conductivity of a polymer nanocomposite reinforced with straight CNTs is higher than that of a polymer nanocomposite reinforced with wavy CNTs. As an outcome, forming a fully bonded interface between the CNT and polymer and using straight CNTs seems to be important if the full potential of CNT reinforcement must be realized. Also, an increase in the polymer nanocomposite thermal conductivity is observed with rising the CNT volume fraction. Here, a parametric study is performed to show the effect of the CNT alignment on the thermal conducting properties of polymer nanocomposites. Based on the available literature [58–62], the increase in effective properties of the polymer nanocomposites occurs predominantly in the direction of the CNT longitudinal axis. In general, electric fields have proven to be an effective method to create aligned arrays of CNTs. Several techniques, including polarized Raman spectroscopy and polarized absorbance spectroscopy were successfully employed to determine the degree of CNTs alignment [60–64]. Table 1 shows the thermal conductivities of randomly dispersed CNT-polycarbonate nanocomposite and aligned CNT-polycarbonate nanocomposite systems extracted by the proposed Hapin-Tsai method. It should be noted that in the case of aligned CNT-polycarbonate nanocomposite, the longitudinal (along the CNt) thermal conductivity is reported in the table. It can be clearly observed that the thermal conductivities of the aligned CNT-polycarbonate nanocomposite are significantly higher than those of the randomly dispersed CNT-polycarbonate nanocomposite presented in the third column of the table. Consequently, the aligned arrays of CNTs lead to an improvement in nanocomposite thermal properties which is approved by previously studies too [58–62]. It is worth noting that in addition to the CNTs alignment and interfacial bonding with matrix, the CNTs dispersion, damage (e.g., oxidation defects) can also affect the conductivity of nanocomposite materials [65].

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H-T model, With 3D random dispersion, waviness and interface factors

(a)

0.6

Thermal conductivity (W/mK)

H-T model, With 3D random dispersion, waviness factors H-T model, With 3D random dispersion factor 0.5

Experiment [26]

0.4

0.3

0.2

0

0.4

0.8

1.2

1.6

2

Volume fraction of long CNT (%)

H-T model, With 3D random dispersion, waviness and interface factors 0.6

(b)

H-T model, With 3D random dispersion, waviness factors Thermal conductivity (W/mK)

H-T model, With 3D random dispersion factor 0.5

Experiment [26]

0.4

0.3

0.2

0

0.4

0.8

1.2

1.6

2

Volume fraction of short CNT (%) Fig. 11. Comparison between the results of the presented H-T model and experiment [26] for polycarbonate nanocomposite reinforced with (a) long CNT and (b) short CNT.

Table 1 Comparison of the thermal conductivities (W/mK) of randomly dispersed CNTpolycarbonate nanocomposite and aligned CNT-polycarbonate nanocomposite systems. CNT volume fraction (%)

Randomly dispersed CNTs

Aligned CNTs

Improvement (%)

0 0.2000 0.4000 0.6000 0.8000 1.0000 1.2000 1.4000 1.6000 1.8000 2.0000

0.2100 0.2316 0.2533 0.2749 0.2966 0.3182 0.3399 0.3615 0.3832 0.4048 0.4265

0.2100 0.3370 0.4639 0.5909 0.7180 0.8450 0.9721 1.0992 1.2263 1.3534 1.4806

0 45 83 115 142 165.5 186 204 220 234 247

To assess the predictive capability of the developed unit cell micromechanical model for the thermal conductivity of fibrous composite systems, comparisons between the predictions of the ESUC model and other micromechanical methods as well as available experimental data are performed. Fig. 12a and b shows the

variation of axial and transverse thermal conductivities of the phenolic composite, respectively, versus CF volume fraction. The thermal conductivity of CF and phenolic matrix is equal to 500 and 0.2878 W/mK, respectively [19]. The experimental data [19] for the phenolic composite thermal conductivity along the axial direction is provided in Fig. 12a as well. Moreover, the results of the CCA [37] model are represented in Fig. 12. Using the CCA model, the effective axial (K C;L ) and transverse (K C;T ) thermal conductivities of the long fiber-reinforced composites can be expressed as follows [37]

K C;L ¼ f F K F þ ð1  f F ÞK m K C;T ¼ K m

g 1 ð1 þ f F Þ þ 1  f F ; g 1 ð1  f F Þ þ 1 þ f F

ð11Þ g 1 ¼ K F =K m :

ð12Þ

where f F is the fiber volume fraction and K F denotes the fiber thermal conductivity. It is seen from Fig. 12a that no discrepancy exists between the results of the ESUC, and CCA model as well as experiment [19]. The phenolic composite thermal conductivity along the axial direction linearly increases with rising the CF volume fraction. As can be seen in Fig. 12b, the predictions of the ESUC and CCA models are in close agreement with each other. The phenolic

M.K. Hassanzadeh-Aghdam et al. / International Journal of Heat and Mass Transfer 124 (2018) 190–200

Axial thermal conductivity (W/mK)

196

300

(a)

ESUC model

250 CCA model

200 Experiment [19]

150 100 50 0 0

10

20

30

40

50

60

50

60

Transverse thermal conductivity (W/mK)

Carbon fiber volume fraction (%) 1.2

(b)

1 ESUC model

0.8 CCA model

0.6 0.4 0.2 0 0

10

20

30

40

Carbon fiber volume fraction (%) Fig. 12. Variation of (a) axial and (b) transverse thermal conductivities of carbon fiber-reinforced phenolic composite versus carbon fiber volume fraction.

composite transverse thermal conductivity nonlinearly increases with increasing the CF volume fraction. Fig. 13a and b shows the variation of axial and transverse thermal conductivities of randomly oriented CNT-coated CF-reinforced polycarbonate hybrid nanocomposite, respectively, with the CF volume fraction. The results have been extracted for two different CNT volume fractions, including 1% and 2%. For a comparison purpose, the thermal conductivity of traditional CF-reinforced polycarbonate composite, i.e. CNT volume fraction is assumed to be 0, has been provided in Fig. 13. Note that the role of the CNT waviness, random dispersion and the CNT/polymer interfacial thermal resistance in the hybrid nanocomposite overall thermal response of is

incorporated. It can be concluded from Fig. 13a that as the CF volume fraction increases, the hybrid nanocomposite axial thermal conductivity linearly increases. The growing of CNTs on the CF surface does not affect the thermal conducting behavior of the hybrid nanocomposite along the axial direction. However, in the transverse direction, the contribution of CNTs to the thermal conducting response of hybrid nanocomposites seems to be very important, as indicated in Fig. 13b. The CNTs growth on the CF surfaces leads to a significant enhancement in the transverse thermal conductivity. For example, when the CF volume fraction is equal to 60%, the value of transverse thermal conductivity for traditional fibrous composite and hybrid nanocomposite containing 2% CNT is equal to 0.8387 W/mK and 2.194 W/mK, respectively, corresponding to a 161.6% increment. Since the transverse thermal conductivities are significantly enhanced, the hybrid nanocomposite would have better thermal management to avoid temperature buildup. Also, the hybrid nanocomposite transverse thermal conductivity nonlinearly increases as the CF volume fraction increases. The variation of longitudinal and transverse thermal conductivities of randomly oriented CNT-coated glass fiber-reinforced polycarbonate hybrid nanocomposite versus glass fiber volume fraction is illustrated in Fig. 14a and b, respectively. The thermal conductivity of glass fiber is 0.05 W/mK [35]. The significant contribution of CNTs in the thermal conducting behavior of the glass fiberreinforced hybrid nanocomposite is obvious in Fig. 14. It is found that the growth of CNTs on the glass fiber surfaces leads to an enhancement of axial and transverse thermal conductivities of glass fiber-reinforced hybrid nanocomposite. For example, when glass fiber volume fraction is equal to 60%, the value of axial thermal conductivity for traditional fibrous composite and hybrid nanocomposite containing 2% CNT is equal to 0.114 W/mK and 0.3305 W/mK, respectively, corresponding to a 190% increment. Also, the value of transverse thermal conductivity for glass fiberreinforced composite and glass fiber-reinforced hybrid nanocomposite containing 2% CNT is equal to 0.0967 W/mK and 0.2315 W/mK, respectively, corresponding to a 139.4% increment. Since the overall thermal conductivities are significantly enhanced, the hybrid nanocomposite would possess better thermal management to avoid temperature buildup. The effect of CNT length on the axial and transverse thermal conductivities of randomly dispersed CNT-coated CF-reinforced polycarbonate hybrid nanocomposite is shown in Fig. 15. The results of Fig. 15a clearly illustrate that the CNT length does not affect the hybrid nanocomposite axial thermal conductivity.

Axial thermal conductivity (W/mK)

(a)

250 200 150 100

CNT volume fraction=0 CNT volume fraction=1%

50

CNT volume fraction=2%

0

Transverse thermal conductivity (W/mK)

2.5 300

15

25

35

45

55

Carbon fiber volume fraction (%)

CNT volume fraction=0 CNT volume fraction=1%

2

CNT volume fraction=2%

1.5

1

0.5

0 5

(b)

5

15

25

35

45

55

Carbon fiber volume fraction (%)

Fig. 13. Effect of the CNT volume fraction on the (a) axial and (b) transverse thermal conductivity of CNT-coated carbon fiber-reinforced polycarbonate hybrid nanocomposite.

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However, increasing the CNT length leads to a rise in the transverse thermal conductivity shown in Fig. 15b. Also, a peculiar value for CNT length exists after which further increasing L negligibly affect the hybrid nanocomposite transverse thermal response.

0.5

0.5

(b)

(a) Transverse thermal conductivity (W/mK)

Axial thermal conductivity (W/mK)

Now, a parametric study is carried out to investigate the role of non-straight shape of CNTs in the hybrid nanocomposite overall thermal conducting behavior. Fig. 16a and b depicts the effect of CNT waviness on the axial and transverse thermal conductivities

0.4

0.3

0.2 CNT volume fraction=0

0.1 CNT volume fraction=1% CNT volume fraction=2%

15 25 35 45 Glass fiber volume fraction (%)

CNT volume fraction=1%

0.4

CNT volume fraction=2%

0.3

0.2

0.1

0

0 5

CNT volume fraction=0

5

55

15 25 35 45 Glass fiber volume fraction (%)

55

Fig. 14. Effect of the CNT volume fraction on (a) axial and (b) transverse thermal conductivity of CNT-coated glass fiber-reinforced polycarbonate hybrid nanocomposite.

2.5

(a)

(b)

CNT length=10 µm

250 200 150 CNT length=10 µm

100

CNT length=100 µm

50

CNT length=200 µm

Transverse thermal conductivity (W/mK)

Axial thermal conductivity (W/mK)

300

CNT length=100 µm

2

CNT length=200 µm

1.5 CNT length=400 µm

1

0.5

CNT length=400 µm

0

5

0

15 25 35 45 55 Carbon fiber volume fraction (%)

5

15 25 35 45 55 Carbon fiber volume fraction (%)

Fig. 15. Effect of the CNT length on the (a) axial and (b) transverse thermal conductivity of CNT-coated carbon fiber-reinforced polycarbonate hybrid nanocomposite.

300

Transverse thermal conductivity (W/mK)

Axial thermal conductivity (W/mK)

250

2.8

(a)

Kw=1 Kw=0.5 Kw=0.3

200 150 100 50 0

Kw=0.3

(b)

Kw=0.5

2.3

Kw=1

1.8

1.3

0.8

0.3 5

15

25

35

45

Carbon fiber volume fraction (%)

55

5

15

25

35

45

55

Carbon fiber volume fraction (%)

Fig. 16. Effect of the CNT waviness on the (a) axial and (b) transverse thermal conductivity of CNT-coated carbon fiber-reinforced polycarbonate hybrid nanocomposite.

M.K. Hassanzadeh-Aghdam et al. / International Journal of Heat and Mass Transfer 124 (2018) 190–200

Axial thermal conductivity (W/mK)

300

R=0.77×10^-8 (m^2.K/W)

2.3

(a)

R=0.77×10^-8 (m^2.K/W)

Transverse thermal conductivity (W/mK)

198

R=8.3×10^-8 (m^2.K/W)

250

R=20×10^-8 (m^2.K/W)

200 150 100 50 0

5

15 25 35 45 Carbon fiber volume fraction (%)

R=8.3×10^-8 (m^2.K/W)

1.8 R=20×10^-8 (m^2.K/W)

1.3

0.8

0.3

55

(b)

5

15 25 35 45 55 Carbon fiber volume fraction (%)

Fig. 17. Effect of the interfacial thermal resistance on the (a) axial and (b) transverse thermal conductivity of CNT-coated carbon fiber-reinforced polycarbonate hybrid nanocomposite.

(b)

Fig. 18. Schematic diagram of the carbon fiber cross-sectional shape (a) circular cross-section and (b) square cross-section.

of randomly oriented CNT-coated CF-reinforced polycarbonate hybrid nanocomposite, respectively. As expected, the nonstraight shape of CNT does not affect the hybrid nanocomposite axial thermal conducting response concluded from Fig. 16a. However, the transverse thermal conductivity of hybrid nanocomposites containing straight CNTs are higher than that of hybrid nanocomposites containing wavy CNTs, as clarified in Fig. 16b. The results of Fig. 17 illustrate the effect of interfacial thermal resistance on the overall thermal conductivities of randomly dispersed CNT-coated CF-reinforced polycarbonate hybrid nanocom-

Axial thermal conductivity (W/mK)

2.5

(a)

300 250 200 150 100

Square cross-section 50 Circular cross-section 0

5

posite. The results are extracted for three different values of R, including 0.77  108 m2 K/W, 8.3  108 m2 K/W and 20  108 m2 K/W [37]. The axial thermal conducting response is not affected by change of R shown in Fig. 17a. However, it is observed from Fig. 17b that decreasing value of R can improve the hybrid nanocomposite transverse thermal conductivity. A parametric study is conducted to display the CF cross-section effect on the thermal conducting behavior of the hybrid nanocomposite. Two RVEs including circular and square cross-section shape of the CF shown in Fig. 18 are selected. A RVE with 50  50 subcells is made to simulate the CF circular cross-section, as mentioned above. Fig. 19a and b exhibits the hybrid nanocomposite axial and transverse thermal conductivities, respectively, versus CF volume fraction. It is found from Fig. 19a that the CF crosssection shape does not affect the hybrid nanocomposite axial thermal properties. Although, the predictions of the ESUC model with square cross-section is slightly higher than those of the model with circular cross-section. However, the CF cross-section shape influence on the transverse thermal conducting behavior can be neglected. So, for the simplicity of the simulations implemented by the unit cell models, the fiber square cross-section shape has been employed for general fibrous composite materials.

15

25

35

45

Carbon fiber volume fraction (%)

55

Transverse thermal conductivity (W/mK)

(a)

(b) Square cross-section

2 Circular cross-section 1.5

1

0.5

0

5

15

25

35

45

55

Carbon fiber volume fraction (%)

Fig. 19. Effect of the carbon fiber cross-sectional shape on the (a) axial and (b) transverse thermal conductivity of CNT-coated carbon fiber-reinforced polycarbonate hybrid nanocomposite.

M.K. Hassanzadeh-Aghdam et al. / International Journal of Heat and Mass Transfer 124 (2018) 190–200

6. Conclusions In this research, a multi-stage micromechanical analysis for thermal conducting response of unidirectional polymer hybrid nanocomposites was presented. In this type of hybrid nanocomposite, the unidirectional fibers are coated with randomly oriented CNTs. The novel micromechanical approach consisted of the ESUC method and a modified version of semi-empirical H-T model. The thermal conductivities of both CNT-polymer nanocomposite and conventional fibrous composite systems predicted by the present model were found to be in excellent agreement with that directly measured by experimental method. The results implied that for an accurate prediction of the CNT-polymer nanocomposites thermal properties, consideration of the CNT random orientation, waviness and, the CNT/polymer interfacial thermal resistance is critically essential. Specifically studies were conducted on a hybrid nanocomposite comprised of multi-walled CNT and CF as reinforcements embedded in a polycarbonate resin. It is found that the CF-reinforced hybrid nanocomposite axial thermal behavior does not affected by the CNTs coating on the CFs. However, it was observed that the transverse thermal conductivities of the hybrid nanocomposites is significantly improved in comparison with that of traditional fibrous composites. Since the transverse thermal conductivities are significantly increased, the hybrid nanocomposite would have better thermal management to avoid temperature buildup. Also, the transverse thermal conducting behavior of hybrid nanocomposites can be enhanced with (i) increasing the CNTs volume fraction and length (ii) using straight CNTs and (iii) forming a perfect bonding interface. Moreover, the effects of the CFs volume fraction and cross-sectional shape on the thermal properties of the CNT-coated CF-reinforced polymer hybrid nanocomposites were investigated. The presented efficient model with reported results could be actually useful in manufacturing the optimum materials to meet specific design requirements in many aerospace applications.

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