Applying a micromechanics approach for predicting thermal conducting properties of carbon nanotube-metal nanocomposites

Journal of Alloys and Compounds 789 (2019) 528e536

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Journal of Alloys and Compounds journal homepage: http://www.elsevier.com/locate/jalcom

Applying a micromechanics approach for predicting thermal conducting properties of carbon nanotube-metal nanocomposites Xiaolong Shi a, Ehsan Gholamalizadeh b, **, Rasoul Moheimani c, * a

Institute of Computing Science and Technology, Guangzhou University, Guangzhou, 510006, China Department of Mechanical Engineering, Sejong University, Seoul, Republic of Korea c Department of Mechanical and Civil Engineering, Washington State University, Pullman, WA, United States b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 1 February 2019 Received in revised form 2 March 2019 Accepted 4 March 2019 Available online 5 March 2019

Heat dissipation is a very important issue in the harsh thermal loads affecting the reliability of structures. In this study, the thermal conducting behavior of carbon nanotube (CNT)-reinforced metal matrix nanocomposites (MMNCs) has been analyzed using a micromechanical model based on the method of cell (MOC) approach. The critical role of interfacial thermal resistance between the CNT and metal matrix in the MMNC thermal conducting response has been extensively explored. Also, the effects of volume fraction, length, diameter and curvature of CNTs have been investigated. It has been found that forming a perfect CNT/matrix interface contact would enhance remarkably the overall thermal conductivities. Generally, the axial thermal conductivity of the CNT-reinforced MMNCs can be increased with (i) rising CNT length and (ii) using straight CNTs. Moreover, the transverse thermal conductivity of these nanocomposite materials are improved with (i) rising CNT diameter and (ii) using wavy CNTs. Besides, increasing CNT volume fraction leads to a significant increment in the effective thermal conductivities in the presence of a perfect bonding at the CNT/matrix interface. Effective thermal conductivities of the MMNCs estimated by the MOC approach have been compared with those predicted by the effective medium and Halpin-Tsai models. A quite good agreement has been observed between the predictions of the MOC approach and experimental data. © 2019 Elsevier B.V. All rights reserved.

Keywords: Metal matrix nanocomposite Carbon nanotube Thermal conductivity Interfacial thermal resistance Micromechanics

1. Introduction Due to their superior mechanical properties, including specific stiffness and strength, good wear and fatigue resistance as well as good thermal properties, metal matrix composites (MMCs) are broadly investigated and extensively utilized in the aerospace, automobile, defence, marine, sports and recreation industries [1e4]. Nowadays, current engineering applications in high advanced industries increasingly need on structural materials that have a better performance under more severe loads and environmental conditions. The development in nanotechnology opens new possibilities to manufacture nanoscale reinforcements which can be used in metal matrixes. Hence, a more attention has been drawn to the metal

* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (E. Gholamalizadeh), mohemanir@gmail. com (R. Moheimani). https://doi.org/10.1016/j.jallcom.2019.03.064 0925-8388/© 2019 Elsevier B.V. All rights reserved.

matrix reinforced with nanoparticles called as metal matrix nanocomposites (MMNCs). In general, MMNCs are a new generation of MMC materials that have the potentials of satisfying the recent demands of advanced engineering applications [5e8]. It has been reported that the reinforcement of carbon nanotubes (CNTs) in metal matrix can provide us a new class of materials [9,10]. CNTs have excellent mechanical properties (elastic modulus ~1 TPa and strength ~30 GPa) [11,12], low thermal expansion [13,14], very high thermal conductivity (up to 6000 W/mK) [15,16] together with ultrahigh aspect ratio and low density [17,18]. There are lots of experimental and theoretical studies for evaluating the overall thermo-mechanical behavior of MMNCs containing CNTs [19e23]. For instance, Cha et al. [24] experimentally found that a small amount of CNTs can effectively reinforce copper's strength. Choi et al. [25] reported that the aluminum (Al) MMNC containing 4.5 vol% multi-walled CNTs displays a yield strength of 620 MPa and fracture toughness of 61 MPa mm1/2, the values of which are nearly 15 and 7 times higher than those of the corresponding Al matrix, respectively. Esawi et al. [26] used the ball milling to disperse up to 5 wt% CNT in an Al matrix. Improvements

X. Shi et al. / Journal of Alloys and Compounds 789 (2019) 528e536

of up to 50% in tensile strength and 23% in stiffness for Al MMNC were observed in comparison to the pure Al [26]. Barai and Weng [27] developed a micromechanical model consisted of secant moduli and a field fluctuation technique to examine the effect of CNT/metal matrix interface condition on the elastoplastic stressstrain curves of MMNCs. It was indicated that CNTs are indeed a very effective strengthening agent, but an imperfect interface condition can decrease the stiffness and elastoplastic strength of CNT-reinforced MMNCs [27]. Long et al. [28] investigated the strengthening mechanisms of CNT-reinforced copper (Cu) MMNCs employing the 2-dimensional axial symmetric unit cell finite element method (FEM). The effects of volume fraction and aspect ratio of CNTs, size of hardening zone were studied on the elastoplastic behavior of Cu MMNCs [28]. Sharma and Sharma [29] experimentally measured the hardness, elastic modulus and the coefficient of thermal expansion (CTE) of the Al MMNCs at different CNT volume fractions. They observed that the addition of CNTs into the Al matrix can improve the hardness and elastic modulus [29]. Also, the CTE of Al MMNC containing 5 wt% CNTs can be decreased to 70% compared with that of Al matrix [29]. Unit cell micromechanical models such as method of cell (MOC), generalized method of cell (GMC) and simplified unit cell (SUC) have been extensively used to investigate the effective properties of CNT-reinforced nanocomposite materials. For example, Kundalwal and Ray [30] predicted the elastic stiffness tensor of a novel fuzzy fiber-reinforced composites containing CNTs by employing the MOC approach. In another study, the effect of CNT waviness on the effective coefficient of thermal expansion (CTE) of CNT-reinforced hybrid nanocomposites has been explored [31]. For this purpose, analytical micromechanics model based on the (MOC) approach was derived. It was found that the amplitude, number of CNT waves and volume fraction have a significant role in the hybrid composite CTEs [31]. Dhala and Ray [32] utilized the GMC technique to investigate the effective piezoelectric and elastic properties of CNTcoated fiber-reinforced composites. It was reported that the effective in-plane piezoelectric coefficient and elastic coefficients of this hybrid composite material are significantly enhanced over those of the existing 1e3 piezoelectric composite without reinforced with CNTs [32]. Haghgoo et al. [33] evaluated the elastoplastic behavior of Al nanocomposites reinforced with aligned CNTs by means of the SUC model. The influences of content and diameter of CNT, material properties and size of interphase on the elastoplastic stress-strain curve of Al nanocomposites during uniaxial tension were examined [33]. Too, Ansari et al. [34] developed the SUC micromechanics model to investigate the Young's modulus and initial yield envelope of CNT-Al nanocomposites considering Al4C3 interphase region. The Von-Mises yield criterion was employed to achieve the yielding response of the nanocomposites under biaxial transverse/transverse and transverse/longitudinal loads. In recent years, with the quick development of microelectronic technique, the electronic devices are becoming smaller and more integrated. In this condition, when designing and manufacturing the electronic devices, the heat dissipation capability must be well known. Overall, increasing the thermal conductivity of materials used for manufacturing the electronic devices leads to an enhancement in their performance, life cycle and reliability [35e38]. On the other hand, the thermal conductivity with a high value is very appropriate due to the reduction of the temperature gradient and therefore, the thermal induced stresses especially arisen from the manufacturing process [35e38]. So, the evaluation of thermal conducting behavior of the MMC materials is very necessary to know the amount of heat dissipation from their surfaces. Chu et al. [39] fabricated CNT-reinforced Cu MMNCs by means of a particles-compositing process followed by spark plasma sintering technique. Then, the MMNC thermal conductivities for

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various CNT volume fractions were measured by a laser flash technique [39]. According to the experimental observation, the Cu MMNC thermal conductivity slightly reduced after the addition of CNTs into the Cu matrix [39]. Also, Pal and Sharma [40] manufactured some specimens of CNT-reinforced silver (Ag) MMNCs and evaluated their thermal conductivities. The experimental outcomes showed that thermal conductivity decreased after incorporation of covalently functionalized multi-walled and single-walled CNTs into the Ag matrix [40]. Although the thermomechanical properties of CNT-reinforced MMNCs have been theoretically studied [27,28,30], however, to the best of authors' knowledge, there is no thermal conducting data currently available for the Cu MMNCs containing CNTs. In this work, a micromechanical model based on the MOC approach is developed to predict the thermal conductivities of the CNT-reinforced MMNCs. The influences of volume fraction, length, diameter and curvature of CNTs as well as the interfacial thermal resistance between the CNT and the metal matrix are analyzed. To illustrate the capability of the developed MOC approach, the predictions are compared with the experimental measurements available in the literature as well as other micromechanical methods. The herein simulation and reported results could be useful to guide the design of a wide range of CNT-reinforced MMNCs used as the structural materials for electronic and aerospace systems.

2. The MOC approach In this section, the development of a micromechanical model based on the MOC approach [41] is presented to predict the effective thermal conductivities of the straight CNT-reinforced MMNCs. The representative volume element (RVE) of the MMNC reinforced with CNTs is shown in Fig. 1. In this micromechanics model, the nanocomposite can be viewed to be composed of cells which form doubly periodic arrays along the x1 - and x2 -directions. It is assumed that the CNTs to be equivalent solid fibers [42e44] which are uniformly embedded in the matrix. Moreover, the CNTs are aligned along the x3 -direction. The CNTs are assumed to be equivalent solid fibers with a square cross-section which are uniformly embedded in the matrix. Note that the original attributes of cylindrical hollow structure of CNTs cannot satisfy the requirement of micromechanical methods; therefore, it is necessary to have an equivalent solid. In the unit cell-based micromechanical approaches, the square cross-section has been extensively employed

Fig. 1. RVE of the MOC for simulating CNT-MMNCs.

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for simulating the CNT-reinforced nanocomposites [30e34]. Each sub-cell is symbolized by bg, with b and g demonstrating the location of the sub-cell along the x1 - and x2 -directions, respectively. As shown in Fig. 1, the volume of each sub-cell (Vbg ) is calculated as

Using the effective thermal conductivity coefficients, the average heat flux components are connected to the temperature gradients

Vbg ¼ hb bg l

q1 ¼ K nc 1

(1)

where hb , bg and l are the height, the width and the length of the sub-cell bg, respectively. Furthermore, the total volume of the RVE is

V ¼ hbl

(2)

vT vT vT ; q ¼ K nc ; q ¼ K nc 2 3 vx1 2 vx2 3 vx3

where K nc denotes the thermal conductivity of the polymer i bg bg nanocomposite. Finally, eliminating the microvariables (x1 , x2 ) together with continuity conditions, the CNT-reinforced MMNC thermal conductivities are derived as

n o m CNT ½hðV11 þ V21 Þ þ h2 ðV12 þ V22 Þ þ K m h1 ðV12 þ V22 Þ 8 nc K K  K ¼ >  m > > 1 CNT > h þ K h hbl K > 1 2 > > > > n o < K m K CNT ½bðV11 þ V21 Þ þ b2 ðV12 þ V22 Þ þ K m b1 ðV12 þ V22 Þ nc >  > K2 ¼  > > hbl K m b1 þ K CNT b2 > > > > > : K CNT V11 þ K m ðV12 þ V21 þ V22 Þ K nc 3 ¼ hbl

where h and b are indicated in Fig. 1. In the MOC approach, the deviation of the temperature from a reference temperature TR (at which the material is stress free when its strain is zero), DQbg , is expanded as [41].

(9)

where K m and K CNT are the thermal conductivities of metal matrix and CNT, respectively. So, the thermal conductivity tensor of the MMNC reinforced with unidirectional straight CNTs is given as

2

nc

g bg DQbg ¼ DT þ xb1 xbg 1 þ x2 x2

(3)

(8)

K

K nc 1 ¼4 0 0

0 K nc 2 0

3 0 0 5 K nc 3

(10)

g

where xb1 and x2 are the local coordinate systems which have oribg gins that are located at the centroid of each sub-cell. Also, x1 and xbg are the linear dependence of the temperature on the local co2 ordinates [41]. The following relations can be given from the continuity conditions of the temperature at the interfaces of the sub-cells [41], as

8 vT > 1g 2g > > < h1 x1 þ h2 x1 ¼ ðh1 þ h2 Þ vx 1 > vT > b1 b2 > : b1 x2 þ b2 x2 ¼ ðb1 þ b2 Þ vx2

(4)

bg

For the average heat flux qi in the sub-cell bg, following relations are expressed based on the Fourier's law bg

bg vT

q1 ¼ K 1

vx1

bg

bg vT

; q2 ¼ K 2

vx2

bg

bg vT

; q3 ¼ K 3

vx3

(5)

bg

where K i signifies the thermal conductivity of the sub-cell bg. The average heat flux of the RVE of the nanocomposite can be calculated as

qi ¼

2 X 2 1 X bg Vbg qi V g¼1

3. Interfacial thermal resistance A large interfacial region between the CNT and metal matrix is generated in the CNT-reinforced MMNCs because of the high surface area to volume ratio of CNTs [27]. Therefore, the CNT/metal matrix interface plays a main role in computing the MMNC thermal conductivity. The interface influence is considered to be the interfacial thermal resistance between the metal matrix and the CNTs [15,39,45,46]. 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 [15,45,46]. Also, the interfacial thermal resistance is known as Kaptiza resistance [15,45,46]. The schematic of a CNT coated with a very thin interfacial thermal barrier layer is shown in Fig. 2. The CNT and surrounding interfacial thermal barrier layer can be integrated as an equivalent nanofiber. The effective thermal conductivities of the nanofiber along the CNT nf nf direction (axial) K A and normal to CNT direction (transverse) K T can be determined by the rule of mixture for a simple series model

(6)

b¼1

Also, the continuity conditions of the heat flux at the interfaces of the sub-cells lead to 1g 2g q1 ¼ q1 ; qb21 ¼ qb22

(7)

Fig. 2. Schematic diagram of a CNT coated with a very thin interfacial thermal barrier layer.

X. Shi et al. / Journal of Alloys and Compounds 789 (2019) 528e536

of the barrier layer/CNT/barrier layer, as follows

K CNT 8 nf KA ¼ > > > 2r K CNT > > 1þ K < L Km > > K CNT > nf > > : KT ¼ 2r K CNT 1þ K d Km

n p  z y ¼ A sin L

(11)

tan q ¼ (12)

where L and d are the CNT length and diameter, respectively. Also, rK and RK are the Kaptiza radius and resistance, respectively [15,45,46]. In other words, the interfacial thermal property is concentrated on a surface of zero thickness and characterized by Kaptiza radius. A perfect bonding interface means that CNT/matrix interfacial layer thickness is zero. The heat flux is well transmitted from metal matrix to the CNTs by constructing a perfect bonding at CNT/metal matrix interface. In this situation it is assumed that no interfacial thermal resistance exists between the matrix and CNT which leads to rK ¼ 0. However, due to interaction between the CNT and matrix which generates the large interfacial region between them, the heat flux cannot be well transmitted from the matrix to the CNTs. Note that in this condition that the bonding between the CNT and metal matrix is imperfect, the value of interfacial layer thickness is not zero and interfacial thermal resistance exists; i.e. rK s0. Note that K CNT in Eq. (9) is replaced with nanofiber thermal conductivity mainly to consider the effect of interfacial thermal resistance. 4. Simulating curvature of CNTs The wavy CNTs are modeled as sinusoidal solid CNT fibers [47,48]. Fig. 3 shows a schematic sketch of a MMNC containing the wavy CNT. The RVE of the nanocomposite is divided into infinitesimally thin slices of thickness dz. By averaging the thermal conductivities of these slices over the length (L) of the RVE, the homogenized thermal conductivities of the MMNC containing wavy CNTs is obtained. Each slice is treated as an off-axis unidirectional ply and its thermal conductivities can be calculated by transforming the thermal conductivities of the corresponding specially ply. Thus, the wavy CNT is described as [47,48].

n p  dy np ¼A z cos dz Ln L

(14)

The effective thermal conductivities of the MMNC at any point in the ply where the CNT is inclined at the angle q can be derived using the transformation law. So, the overall thermal conductivities at any point in the lamina is given as

8 <

K 1 ¼ K nc 1 2 K 2 ¼ K nc p2 þ K nc 2 3 q : nc 2 2 K 3 ¼ K nc q þ K p 2 3

(15)

where

8 > > > > > <

  npz2 12 npA cos p ¼ cos q ¼ 1 þ L L   >   npz2 12 > npA npz npA > > > 1þ cos : q ¼ sin q ¼ L cos L L L

(16)

Note that the effective thermal conductivities for the unidirecnc nc tional straight CNT-reinforced MMNC, i.e. K nc 1 ; K 2 and K 3 are attained by means of the MOC approach. The thermal conductivities of the ply with the wavy CNTs vary along the CNT length as the value of q changes over the length of the CNT. Thus, the thermal conductivities of the wavy CNT-reinforced MMNCs are determined as

8 9 NC > < K1 > = 1 ¼ K NC 2 > : NC > ; L K3

8 9 ðL < K 1 = dz K : 2; K 3 0

(17)

5. Results and discussion In this research, a main attention is paid to investigate the thermal conducting behavior of CNT-Cu MMNCs which are very attractive to meet the increasing demands for high performance thermal management materials used in heat sinks and electronic packages. The values of thermal conductivity for CNT and Cu are equal to 3000 W/mK and 331 W/mK, respectively [39]. Also, the average diameter and length of CNT are d ¼ 20 nm and L ¼ 10 mm, respectively [39]. Fig. 4a and b illustrate the variation of axial and transverse thermal conductivities of the straight CNT-Cu MMNC, respectively, with CNT volume fraction evaluated by the present MOC approach. Also, the results of the Halpin-Tsai (H-T) and effective medium (EM) models are represented in Fig. 4. Note that a perfect bonding at the CNT/matrix interface is considered in this comparison. By the use of H-T model, the effective thermal conductivity for a composite (KC ) can be given by Ref. [45].

KC ¼ K m Fig. 3. A schematic sketch of a MMNC containing wavy CNTs.

(13)

where A and L indicate the amplitude of the CNT wave and the linear distance between the CNT ends, respectively, and n signifies the number of waves of the CNT. Also, as realized in Fig. 3, the angle q is characterized by

in which

rK ¼ RK K m

531

  1 þ ABf 1  Bf g

where f is the CNT volume fraction and

(18)

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X. Shi et al. / Journal of Alloys and Compounds 789 (2019) 528e536

Fig. 4. Comparison between the results of the MOC, EM and H-T models for (a) axial and (b) transverse thermal conductivities of CNT-Cu MMNC.

 CNT m

 K K 1 1  4m ; g¼1þ f: B ¼  CNT m

42m K K þA

(19)

Also, 4m ¼ 0.82 [49]. A is equal to 2S (S denotes fiber aspect ratio L=d) and 0.5 for obtaining the axial and transverse thermal conductivities, respectively [49]. Using the EM model, the effective axial (Knc;A ) and transverse (Knc;T ) thermal conductivities can be expressed as follows [50].

Knc;A ¼ fK CNT þ ð1  f ÞKM Knc;T ¼ K m

 K CNT þ K m þ f K CNT  K m

: K CNT þ K m  f K CNT  K m

(20)

(21)

It is seen from Fig. 4a that no discrepancy exists between the results of the MOC, H-T and EM models for Cu MMNC axial thermal conductivity. The Cu MMNC thermal conductivity along the axial direction linearly increases with rising the CNT volume fraction. As seen in Fig. 4b, all predictions of the MOC, H-T and EM models are in close agreement with each other. The transverse thermal conductivity predicted by the MOC approach is slightly lower than that of the M-T and EM methods. The Cu MMNC thermal conductivity along the transverse direction enhances with increasing CNT volume fraction. Now, the proposed MOC micromechanical model is tested against experimental data of the CNT-Cu MMNC reported by Chu et al. [39]. In this comparison, the CNTs are not aligned into the Cu metal matrix [39]. A random dispersion of CNTs into the Cu matrix has been considered [39] which leads to an isotropic property for MMNCs. Hence, orientation averaging is performed to account the effect of randomly oriented CNTs on the calculated thermal conductivities. The transformation of the second-rank thermal conductivity tensor from global to local coordinates is given by the following relation g

K lij ¼ aim ajn K mn ; g

ði; j; m; n ¼ 1; 2; 3Þ l

(22)

where K and K denote the tensors of thermal conductivity expressed in terms of global and local coordinates, respectively. Note that the components of Kg are obtained using micromechanical modeling; i.e Eq. (10). The general form of the globalto-local transformation tensor for 3D randomly oriented CNTs is

Fig. 5. Comparison between the MOC predictions and the experimental measurements [39] for thermal conductivity of CNT-Cu MMNC.

expressed as follows [48].

2

cos q sin∅ a ¼ 4 cos∅cos q sin q

sin q sin∅ cos∅sin q cos q

3 cos∅ sin∅ 5 0

(23)

Consequently, the average thermal conductivity tensor for a MMNC containing 3D randomly oriented CNTs can be given as

KR ¼

1 2p

ðp ðp

Kl sin qdqd∅

(24)

00

Fig. 5 shows the variation of CNT-Cu MMNC thermal conductivity with CNT volume fraction. The values of RK , n and A=L are selected to be 0.013  108 m2k/W, 2 and 0.03 [48], respectively. For a comparison purpose, the results of the MOC approach are provided for the perfect bonding at CNT/Cu matrix interface with

X. Shi et al. / Journal of Alloys and Compounds 789 (2019) 528e536

RK ¼ 0. Fig. 5 proves that the thermal conducting behavior of Cu MMNC is very sensitive to the CNT/Cu interfacial thermal resistance. It can be concluded from the figure that even small interface thermal resistance seriously limits the heat transport in CNTreinforced Cu MMNCs and results in an exceptionally low value of overall thermal conductivity. Furthermore, owing to the large influence of interface thermal resistance, the MMNC thermal conductivity can be insensitive to the CNT content, where the difference in thermal conductivity of MMNC at various CNT volume fractions is very negligible. The formation of a perfect bonding at CNT/Cu matrix interface; i.e. RK ¼ 0 leads to maximum levels of effective thermal properties of CNT-Cu MMNCs. In this condition, the rise of CNT content highly enhances the MMNC thermal conductivity. Including the CNT interfacial thermal resistance into the MOC approach yields the predictions to be in a good agreement with those of experimental data [39] indicated in Fig. 5. It has been observed that the interface plays an important role in heat conduction of CNT-Cu MMNCs. Hence, a parametric study is performed to investigate the CNT/Cu interfacial thermal resistance on the thermal conducting response of unidirectional MMNCs along the axial and transverse directions. The predictions of the MOC approach with the perfect (RK ¼ 0) and imperfect (RK ¼ 0.1  108 m2k/W) bonding conditions at interface are shown in Fig. 6. As expected, both axial and transverse thermal conductivities are found to be significantly sensitive to interfacial thermal resistance illustrated in Fig. 6a and b, respectively. The effective thermal conductivities highly decrease with an imperfect bonding at interface. It can be observed from Fig. 6b that in the presence of interfacial thermal resistance, the CNT-Cu MMNC transverse thermal conductivity prominently decreases with the rise of CNT volume fraction. For example, the transverse thermal conductivity of the Cu MMNC containing 5 vol% CNT in the absence and the presence of interfacial thermal resistance is about 349.4 W/mK and 258 W/mK, respectively, corresponding to a 26% reduction. This reduction along the axial direction is about 38% revealed in Fig. 6a. Now, the effect of change in CNT diameter on the Cu MMNC thermal conducting characteristics is evaluated. The variation in CNT diameter is taken in a range from 5 nm to 1000 nm. The volume fraction of CNT is chosen to be 5% in this analysis. The axial and transverse thermal conductivities of Cu MMNC versus CNT diameter are represented in Fig. 7a and b, respectively. The predictions of the MOC approach are provided for different values of RK . The change of CNT diameter does not affect the axial thermal

533

conductivity of unidirectional CNT-reinforced MMNCs, as can be seen in Fig. 7a. However, it is observed from Fig. 7b that the contribution of CNT diameter to the transverse thermal conducting behavior seems to be very important in the presence of interfacial thermal resistance. Overall, the Cu MMNC transverse thermal conductivity increases with the increase in CNT diameter. This trend is more prominent at a low value of RK . Note that increasing CNT diameter decreases the impact of CNT/matrix interfacial thermal resistance on the transverse thermal conductivity based on Eq. (11). In another parametric study using the MOC approach, the influence of change in CNT length on the Cu MMNCs thermal conducting properties is examined. It is assumed that the change of CNT length to be in the range of 1e1000 mm. The variation of axial and transverse thermal conductivities with CNT length is depicted in Fig. 8a and b, respectively. The results are obtained for different values of RK . It can be seen from Fig. 8a that the axial thermal conductivity significantly depends on CNT length in the presence of interfacial thermal resistance. A nonlinear increase in Cu MMNC axial thermal conductivity has been observed by increasing CNT length. At a low value of RK , there is a peculiar value for CNT length after which further increasing CNT length does not affect the axial thermal conductivity. The transverse thermal conductivity is found to be independent of the CNT length, as indicated in Fig. 8b. The MOC micromechanical model is employed to examine the role of CNT curvature in the thermal conducting behavior of Cu MMNCs. The trend of variation in CNT-reinforced Cu MMNC thermal conductivity along the axial and transverse directions with A=L is shown in Fig. 9a and b, respectively. Also, the results are extracted for three different values of n, including 1, 2 and 3. Also, numerical data given in Fig. 9 is presented in Table 1. It is observed that the CNT curvature strongly influences the thermal properties of CNT-Cu MMNCs. As the value of A=L increases, a significant decreasing pattern of axial thermal conductivity is observed in Fig. 9a. Also, increasing the number of CNT waves yields a reduction in axial thermal conductivity. It can found from Fig. 9b that CNT curvature improves the thermal conductivity in transverse direction. Since the transverse thermal conductivities are significantly enhanced, the Cu MMNCs will have better thermal management to avoid temperature buildup. Overall, increasing the values of A=L and n can rise the transverse thermal properties. Generally, the radius of curvature of CNT can affect the interfacial bonding [51]. It was proved that the CNT curvature along with the nature of the

Fig. 6. Effect of CNT/Cu interface conditions on the (a) axial and (b) transverse thermal conductivities of CNT-Cu MMNC.

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Fig. 7. Variation of the (a) axial and (b) transverse thermal conductivities of CNT-Cu MMNC with CNT diameter.

Fig. 8. Variation of the (a) axial and (b) transverse thermal conductivities of CNT-Cu MMNC with CNT length.

Fig. 9. Variation of the (a) axial and (b) transverse thermal conductivities of CNT-Cu MMNC with A=L.

X. Shi et al. / Journal of Alloys and Compounds 789 (2019) 528e536 Table 1 Numerical data of axial and transverse thermal conductivities (W/mK) of CNT-Cu MMNC for different values of A=L. A/L

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Axial direction

Transverse direction

n¼1

n¼2

n¼3

n¼1

n¼2

n¼3

464.45 446.59 429.96 415.39 403.23 393.37 385.52 379.32 374.41 370.50 367.38

464.45 429.96 403.23 385.52 374.41 367.38 362.79 359.70 357.53 355.96 354.80

464.45 415.39 385.52 370.50 362.79 358.52 355.96 354.32 353.22 352.45 351.88

349.38 367.23 383.86 398.43 410.59 420.45 428.30 434.50 439.42 443.32 446.44

349.38 383.86 410.59 428.30 439.42 446.44 451.03 454.12 456.29 457.86 459.02

349.38 398.43 428.30 443.32 451.03 455.30 457.86 459.50 460.60 461.37 461.94

interfacial bonding are critical factors in determining the efficacy of CNTs in providing enhanced mechanical or thermal properties to the nanocomposites. With the assumption of a perfect bonding at CNT/metal matrix interface, MMNCs reinforced with straight CNTs can take full advantage of their exceptional stiffness and thermal conductivity. However, In the case of poor (or imperfect) bonding, CNTs waviness has been shown to enhance the effective properties of nanocomposites [52]. 6. Conclusions In this research, micromechanical modeling of thermal conductivity of CNT-reinforced MMNCs with account of the interfacial thermal resistance between the CNT and the metal matrix was performed. A closed form model based on the MOC approach was developed, which incorporates the geometrical characteristics of CNTs. The results obtained using the MOC approach were compared with the analytical results obtained from H-T and EM methods and a quite good agreement was observed. Besides, the predictions were found to be in close proximity with the experimental data which validates the present model. The main findings from this research can be summarized below:  Even small interface thermal resistance seriously limits the heat transport in CNT-reinforced MMNCs and decreases the effective thermal conductivity.  In the absence of interface thermal resistance, increasing CNT volume fraction leads to an increment in the thermal conductivities.  The CNT curvature decreases the MMNCs axial thermal conductivities. However, a reverse trend was observed in the case of transverse thermal properties.  The transverse thermal conductivity increases with increasing CNT diameter. Moreover, the increase of CNT length can improve the axial thermal conductivity. Acknowledgments This work was supported by National Natural Science Foundation of China (61572213, 61772376 and 61672248). References [1] P. Vijayavel, V. Balasubramanian, Effect of pin profile volume ratio on microstructure and tensile properties of friction stir processed aluminum based metal matrix composites, J. Alloys Compd. 729 (2017) 828e842. [2] C. Lin, Y. Han, C. Guo, Y. Chang, X. Han, L. Lan, F. Jiang, Synthesis and mechanical properties of novel Ti-(SiCf/Al3Ti) ceramic-fiber-reinforced metalintermetallic-laminated (CFR-MIL) composites, J. Alloys Compd. 722 (2017)

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