- Email: [email protected]
Al composites containing pores: Numerical study and experimental validation
Composites: Part A 41 (2010) 161–167
Contents lists available at ScienceDirect
Composites: Part A journal homepage: www.elsevier.com/locate/compositesa
Thermal conductivity of spark plasma sintering consolidated SiCp/Al composites containing pores: Numerical study and experimental validation Ke Chu a,b,*, Chengchang Jia a, Wenhuai Tian b, Xuebing Liang a, Hui Chen a, Hong Guo c a
Institute of Powder Metallurgy, University of Science and Technology Beijing, Beijing 100083, China Department of Materials Physics and Chemistry, University of Science and Technology Beijing, Beijing 100083, China c Beijing General Research Institute for Nonferrous Metals, Beijing 100088, China b
a r t i c l e
i n f o
Article history: Received 17 March 2009 Received in revised form 26 September 2009 Accepted 3 October 2009
Keywords: A. Metal matrix composites B. Thermal properties C. Analytical modeling E. Sintering
a b s t r a c t A simple model was introduced for describing the effect of porosity on the effective thermal conductivity of spark plasma sintered (SPS) consolidated SiCp/Al composites in terms of an effective medium approximation (EMA) scheme. Numerical results of the present model were compared to an existing model of two-step Hasselman–Johnson approach and experimental data. Both models yielded very close predictions, which provided a satisfactory agreement to the experimental data, especially for the composites with porosity below 10%. At high levels of porosity the model predictions were slightly higher than the experimental values. These two models were further extended to account for the thermal conduction properties of porous composites with a multimodal size distribution or multiphase reinforced mixtures. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Metal matrix composites (MMCs) with high reinforcement volume fractions are used for thermal management applications due to their excellent thermo-physical properties, tailorable thermal expansion and low density. Particularly, Al matrix composites containing high volume fractions of SiC particles (SiCp/Al) composites are receiving the most attention as potential candidates for a variety of uses in advanced electronic packaging and are currently competing with established materials such as Cu/W or Cu/Mo in the electronic packaging industry [1]. As has been reported so far, SiCp/Al composites with high SiCp volume fraction are mainly prepared by infiltration of liquid metal into the ceramic preforms [2–4]. This process often uses Al-alloys with an addition of Si and Mg to avoid formation of interfacial reaction products such as Al4C3, which has a detrimental effect on the thermo-mechanical properties of the composites. Nevertheless, the dissolved Si and Mg decrease the matrix thermal conductivity leading to a low thermal conductivity of the composites. Spark plasma sintering (SPS) applied as a new method for preparing high performance of SiCp/Al composites has been presented recently [5]. The main advantage of the SPS process is that it allows fabrication
* Corresponding author. Address: Institute of Powder Metallurgy, University of Science and Technology Beijing, Beijing 100083, China. Tel.: +86 10 62334271; fax: +86 01 62334271. E-mail address: [email protected] (K. Chu). 1359-835X/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesa.2009.10.001
of bulk materials using relatively short sintering times at low temperatures [6], which has been known to be very beneficial to the prevention of Al4C3 at the interface. However, a small portion of pores can always appear as an unavoidable phase in SiCp/Al composites during the SPS process, which is linked to the non-wetting nature of SiC and aluminum, resulting in weak ceramic–metal interfaces and incomplete sintering, especially at elevated particle volume fraction. Pores can severely degrade the thermal conductivity of the composites due to scattering of the heat flow [7]. Although the effective thermal conductivity of SiCp/Al composites has been extensively investigated considering different parameters such as particle shape, size, size distribution, volume fraction and interfacial thermal resistance (ITR) [8–12], very limited theoretical work [13] has been developed to quantitatively characterize the effect of porosity on the thermal conduction properties of these composites, especially in comparison with theoretical analysis and experimental results on the composites thermal conductivity associated with the porosity. We therefore focus in this contribution on the effect of porosity on the thermal conductivity of SiCp/Al composites with high volume fabrication. A reasonable model is proposed in the framework of the effective medium approximation scheme. For the purposes of assessing the predictive capacity, the present analytical model is then compared to an existing model of two-step Hasselman– Johnson approach [13] and experimental results for the thermal conductivity of SPS consolidated SiCp/Al composites with various levels of porosity. Various calculational methods are used to
162
K. Chu et al. / Composites: Part A 41 (2010) 161–167
determine the relevant parameters of the composites, i.e., the thermal conductivity of constituents and the interfacial thermal conductance.
2. Modeling A number of reported results [8–15] have proved that the effective medium approximation (EMA) is a simple and powerful approach in describing the essential thermal conduction properties of the composites with perfect or imperfect interface. Our idea is to develop the EMA formula to describe the thermal conductivity of particle-reinforced composites with the porosity content. First we briefly review the EMA theory following Hasselman and Johnson [14,15]. A schematic diagram of modeling process is shown in Fig. 1. Let us consider a two-phase composite medium consisting of n spherical inclusions of radius a and thermal conductivity Kp embedded in an effective matrix with thermal conductivity Km (Fig. 1a). We make the assumption that the n inclusions are enclosed by a spherical region of radius R and the interaction between the particles is negligible. The rigorous solutions for the temperature Tp within the dispersed particles and the temperature Tm in the surrounding matrix can be obtained, which have the following form
3 r cos h 2K p =ah þ K p =K m þ 2 " 3 # R K p =ah K p =K m þ 1 r cos h T m ¼ ðrTÞ 1 þ V p r 2K p =ah þ K p =K m þ 2
T p ¼ ðrTÞ
Now we consider a multi-phase composite medium with multiple reinforcement phases (i) (Fig. 1b). Each of reinforcement phases contains n spherical inclusions of radius, thermal conductivity, volume fraction and interfacial thermal conductance, corresponding to ai, K pi , V pi , hi, respectively. In such a scheme the Tm depends on the cumulative effect of N phases of reinforced inclusions within a region R, hence Eq. (1b) becomes
( T m ¼ ðrTÞ 1 þ cos h
ð1aÞ
"
3 # R Km Kc r cos h T m ¼ ðrTÞ 1 þ r 2K m þ K c
ð3Þ
Since the two expressions, Eqs. (2) and (3), are equivalent, it follows that N X
V pi
K pi =ai h K pi =K m þ 1 2K pi =ai h þ K pi =K m þ 2
¼
Km Kc 2K m þ K c
ð4Þ
For explicit expression of Kc, Eq. (4) can be rearranged as
1 þ 2A 1A
ð5aÞ
. 13 K eff Km 1 pi 4V p @ A5 . i K eff Km þ 2 pi
ð5bÞ
Kc ¼ Km ð1bÞ
where (rT) is the temperature gradient at a radial distance r (r R) from the centre of the spherical cluster region R; h is the angle between the radius vector r and the temperature gradient; h is the interfacial thermal conductance between the spherical dispersions and the matrix. Vp is the particle volume fraction defined as Vp = n(a/R)3.
ð2Þ
Now if all the inclusions are treated as an ‘‘effective homogeneous medium” of radius R and thermal conductivity Kc, suspended in a matrix of conductivity Km (Fig. 1b), the Tm at any radial location r (r R) is given as
i
3 X ) N K pi =ai h K pi =K m þ 1 R r V pi 2K pi =ai h þ K pi =K m þ 2 r i
with
A¼
N X i
K eff pi ¼
2
0
K pi 1 þ K pi =ai h
ð5cÞ
Fig. 1. Schematic illustration of the modeling process in terms of an effective medium approximation (EMA): (a) a two-phase composite medium consisting of unitary reinforcement phase; (b) a multi-phase composite medium consisting of multiple reinforcement phases; (c) effective homogeneous medium.
K. Chu et al. / Composites: Part A 41 (2010) 161–167
where K eff p stands for the effective thermal conductivity of the particles due to the existing finite interface thermal conductance between the two solid phases. The volume fractions of mixed particles of N phases inclusions fulfill the relationship: P V p ¼ Ni V pi . For convenience, Eq. (5) is named as multiple effective medium approximation (MEMA) model. Typically, for composites with unitary particles of single size, Eq. (5) is reduced to
h
Kc ¼
K m 2K m þ K eff p
i þ 2 K eff p Km Vp
eff 2K m þ K eff p ðK p K m ÞV p
ð6Þ
Eq. (6) is identical to Hasselman and Johnson (H–J) model [14] and Benveniste’s [16] self-consistent scheme. We now consider the porosity effect. To quantitatively account for the thermal conductivity of the composites with ternary system consisting of metal matrix, ceramic particles and pores, a so-called two-step H–J approach proposed by Molina et al. [13] is to consider the particles to be embedded in a ‘‘effective matrix” made up of a composite of matrix with certain amount of pores. The effective fraction of pores, n0 ; in the effective matrix is given by
n0 ¼
n 1 Vp
ð7Þ
where n is the porosity, the effective matrix thermal conductivity, eff K eff m ; derived from the H–J model by taking K p ¼ 0, is
K eff m ¼
K m ð1 n0 Þ 1 þ 0:5n0
ð8Þ
The thermal conductivity of the overall composite can then be obtained using the H–J model again (Eq. (6)) by replacement of the intrinsic matrix conductivity (Km) with the effective matrix conductivity (K eff m ). In the present contribution, we use Eq. (5) to develop another simple model for evaluating the thermal conductivity of the composites with the porosity content. In light of the spirit of Eq. (5), for the limited cases of composites with binary reinforcements, we assume that two reinforced inclusions are treated as two types of given particles including conducting SiC particles and non-thermally
163
conducting particles, namely the residual pores. Accordingly, Eq. (5) can be simplified as
Kc ¼ Km
1 þ 2A 1A
ð9aÞ
with
0 eff . 1 Kp Km 1 A1n . A ¼ V p@ eff 2 Kp Km þ 2
ð9bÞ
From Eq. (9), the thermal conductivity of the composites containing pores can be also estimated for a given porosity. To verify the validity of present model, we will use Eq. (9) to confront the accepted model of two-step H–J approach (Eqs. (6)–(8)) and experimental data in the following.
3. Experimental 3.1. Preparation of composites Aluminum powder (99.9% in purity) that passed through 200 mesh sieve was used. The selected reinforcement particles were high-purity (>99%) green a-SiC with mean size of 40 lm and 100 lm, respectively. The starting powders were blended with different SiCp sizes and volume fractions of 55–60% by three-dimensional vibratory mill of 1400 rpm for 10 min. A SPS system (mod. 1050, Sumitomo Coal Mining Co. Ltd., Japan) was used to synthesize SiCp/Al composites. Fig. 2 shows a schematic diagram of the SPS apparatus. The mixed powders were put into a cylindrical graphite die with an inner diameter of 30 mm. With the aim to prevent powders from sticking to the inner wall of the die during the sintering, the die was covered with a 2 mm thick layer of graphite felt. In order to obtain composites with various porosities, not all the composites were prepared by the same procedure. The compact powders were sintered at a temperature in the range of 540–560 °C for 5 min in vacuum (less than 4 Pa). The heating rate was 50 °C/min and an axial pressure of 40, 45 and 50 MPa was applied throughout the sintering cycle. After sintering, the surfaces of
Fig. 2. Schematic illustration of SPS apparatus.
164
K. Chu et al. / Composites: Part A 41 (2010) 161–167
samples were ground to remove the graphite layer. The characteristics of the samples are shown in Table 1. 3.2. Actual volume fraction and porosity measurements The weight fractions of particles (Wp) and matrix (Wm) are related to the total composite weight, and these variables can typically be established prior to composite processing. The volume fractions of particles (Vp) and matrix (Vm) are related to the total composite volume; however, these variables cannot be established without knowledge of the amount of the third volume-based part in composites, i.e., the porosity (n). Therefore, the volume fractions of particles, matrix and porosity are mutually dependent variables, and the particle weight fraction will be used as the basic independent variable. The weight and volume fractions of the composites are:
Wm þ Wp ¼ 1
ð10Þ
Vm þ Vp þ n ¼ 1
ð11Þ
Based on the densities of particles (qp) and matrix (qm), expressions for the actual volumes of the particle and matrix parts are
Vp ¼
W p =qp qc ¼ Wp 1=qc qp
ð12Þ
Vm ¼
ð1 W p Þ=qm qc ¼ ð1 W p Þ 1=qc qm
ð13Þ
From Eqs. (11)–(13), the porosity can be then expressed as:
n ¼ 1 qc
Wp
qp
þ
1 Wp
! ð14Þ
qm
where qc is the density of the composite, which can be measured by Archimedes’ principle. The densities of Al and SiC were measured by He-pycnometry to be 2.69 g/cm3 and 3.23 g/cm3, respectively. Based on the measured values for densities and weight fractions, the actual volume fractions of particles and porosity were calculated by Eqs. (12) and (14), respectively. 3.3. Electronic conductivity and thermal conductivity measurements The room temperature static electric conductivity was measured with the four probe method for determining the matrix thermal conductivity. Only pore-free samples were considered in order to eliminate the influence of porosity. All specimens had a rectangular cross section Q of 1 1 mm2 and 25 mm length. Both ends of the specimens were connected to a constant current source, which
Table 1 The characteristics of all the samples investigated in this work: Vp is the actual particle volume fraction, d is the average particle size (lm), T is the sintering temperature (°C), P is the applied pressure (MPa), n is the porosity (%), Kc and rc are the thermal conductivity (W m1 K1) and electronic conductivity (106 X1 m1) of the composites, respectively. Samples/n #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12
d 40
100
Vp
T
P
n
Kc
rc
55 60 58 56 56 53 50 55 58 55 53 52
560 560 560 560 550 540 560 560 560 550 540 540
50 50 45 40 45 40 50 50 45 40 45 40
0 0 2.6 7.2 6.3 12.5 0 0 3.7 8.8 11.3 13.4
208 211 204 185 192 165 220 224 208 197 181 173
8.17 5.75 – – – – 8.58 7.24 – – – –
provided a current of magnitude I. The potential drop V was measured over the distance L with the aid of two knife-edge ridges pressed on the sample. The conductivity r ¼ ðI=VÞðL=Q Þ was calculated by taking the mean value of two voltage and current measurements for normal and reversed current direction in order to minimize the effect of contact voltages. The laser flash method was used to measure the thermal diffusivity of the composites at room temperature. The thermal diffusivity measurement was performed by a JR-3 thermal physical testing instrument, which involves exposing the front face of a small disc-shaped sample to a short laser burst and recording the temperature rise on the rear face. The samples were cut from assintered composites to disc-shaped with a diameter of 10 mm and a thickness of 3 mm. The uncertainty in the thermal measurements is ±2%. The specific heat was derived from the theoretical value calculated by rule of mixture (ROM). The thermal conductivity of the composites was then calculated as a product of the density, thermal diffusivity and specific heat. All measurement results are also shown in Table 1. 3.4. Microstructure characterization A thorough microstructural examination of samples was also carried out on an LEO1450 scanning electron microscopy (SEM). Samples were sectioned and polished using standard metallographic techniques. 4. Results and discussion 4.1. Microstructure Fig. 3 shows the representative microstructures of SiCp/Al composites prepared by SPS under different temperatures and pressures. It can be seen in Fig. 3a and d that the composites have a rather uniform distribution of the particles in the matrix and there is no evidence of pores or separated interface, indicating that high volume fraction SiCp/Al composites can be well prepared via the prescribed SPS procedures followed in this work. As expected, inappropriate selected sintering parameters (temperature and pressure) can lead to the quantity of matrix being not sufficient to fill all the space left between touching particles, resulting in various degrees of presence of pores (white phase in Fig. 3b, c, e, and f). 4.2. Determination of material parameters In order to accurately evaluate the models and describe the thermal conductive behavior associated with porosity, one needs exact values of material parameters (Km, h, Kp) used for calculations in Eqs. (6)–(9). However, according to the reported data, these parameters cover wide ranges of 175–237 W m1 K1 for Km, 0.5– 2 108 W m2 K1 for h and 90–270 W m1 K1 for Kp depending on their purity, as well as the fabrication route [8–13]; thus it is very necessary to determine their pertinent values relevant for the present materials. The matrix thermal conductivity can be derived from the electrical conductivity measurements [12] of the composites rc (see Table 1) and the volume fraction of SiC particles (Vp) according to the following expression:
rc ¼ rm ð1 V p Þn
ð15Þ
where n is a parameter reflecting the average particle shape. The data for the electrical conductivity of the composites are fitted with Eq. (15) using n and rm as parameters. Fig. 4 shows data fitting results by carrying out a plot of ln rc vs. ln(1 Vp), given by:
K. Chu et al. / Composites: Part A 41 (2010) 161–167
165
Fig. 3. SEM micrographs of SPS consolidated SiCp/Al composites under various sintering conditions: (a) 60 vol.% 40 lm SiCp under a pressure of 50 MPa sintered at 560 °C (50 MPa/560 °C) (free of pores); (b) 58 vol.% 40 lm SiCp, 45 MPa/560 °C (porosity n = 2.6%); (c) 53 vol.% 40 lm SiCp, 40 MPa/540 °C (n = 12.5%); (d) 55 vol.% 100 lm SiCp, 50 MPa/560 °C (free of pores); (e) 58 vol.% 100 lm SiCp, 45 MPa/560 °C, (n = 3.7%); (f) 55 vol.% 100 lm SiCp, 40 MPa/550 °C (n = 8.8%).
ln rc ¼ 1:82 lnð1 V p Þ þ 3:48
ð16Þ
The obtained values of the fitted parameters are n = 1.82 and rm = 32.46 106 X1 m1. The matrix thermal conductivity (Km) can be then derived from the matrix electrical conductivity (rm) by the following well-known relation for metal and alloys:
K m ¼ L rm T
ð17Þ
where T is the temperature and L is the Lorentz constant (equal to 2.1 108 V2 K2 at T = 300 K [18]). By using Eq. (17), Km is calculated to be 210 W m1 K1. The value estimated for Km in present SPSed composites are in a reasonable value range, although there is a discrepancy from that of 237 W m1 K1 for monocrystalline aluminum, probably because powder sintered material pertain to polycrystal which contains grain boundaries and defects in its structure [18]. The value of the intrinsic interfacial thermal conductance, h, can be estimated by a simple Debye model in terms of the acoustic mismatch theory [19,20]:
hc
qm qp v m v p 1 v3 qm c m2 2 v p ðqm v m þ qp v p Þ2
ð18Þ
where c is the specific heat of metal, q the density and v the phonon velocity. Subscripts m and p denote metal and reinforcement, respectively. Substituting qm = 2.7 103 kg m3, C = 880 J kg1 K1 and mm = 3.52 103 m s1 [17,19], qp = 3.27 103 kg m3, mp = 1.16 104 m s1 [21] for the SiC into Eq. (18), h can be calculated to be 6.65 107 W m2 K1. The inverse calculation method [22] is always used to determine the thermal conductivity of the small reinforcements, i.e., particles, short fibers, and whiskers. This is because the actual values of the thermal conductivity for these small reinforcements are very different from those for their bulk single crystals and even polycrystals, and no appropriate experimental technique appears to be available for direct measurement. From a series of data with respect to the measured composites conductivity and other known parameters, this remained unknown parameter can be reasonably estimated based on associated formulations. We then take into account all the pore-free composites, and substitute four sets of data for Kc (Table 1), and relate the other parameters of Km, h, d and Vp
166
K. Chu et al. / Composites: Part A 41 (2010) 161–167
2.4
lnσc
2.2
2.0
1.8
1.6 -1.0
-0.9
-0.8
-0.7
ln (1-Vp) Fig. 4. The matrix electronic conductivity (rm) obtained from a linear fitting of ln rc vs. ln(1 Vp) for the experimental data of composites electronic conductivity (rc) (see Table 1). The data can be satisfactorily fitted by the straight linear of ln rc = 1.82 ln(1 Vp) + 3.48 with determined n = 1.82 and rm = 32.46 106 X1 m1.
into Eqs. (5c) and (6), the average Kp value of 248 W m1 K1 can be ultimately back-calculated. This value represents the average intrinsic thermal conductivity of the SiC particles for the present composites, which is in a reasonable value range for fully dense polycrystalline structural SiC, but remains well below the value of up to 400 W m1 K1 for a high-purity single crystal at room temperature [23]. 4.3. Numerical results and comparison with experiments Fig. 5 shows the comparison between the predictions given by the present model (MEMA), the two-step H–J (TS-HJ) approach and experimental data for the effect of porosity on the thermal conductivity the SiCp/Al composites with two particle sizes of 40, 100 lm. It can be clearly seen that the increase of porosity causes a pronounced decrease in composites conductivity. For the considered porosity of 7.2%, the thermal conductivity decrement is about
450
MEMA
100 μm ( )
220
TS-HJ
Thermal conductivity (W/mK)
Thermal conductivity (W/mK)
230
12.3%, a value of 71% higher than its corresponding porosity confirms this great negative effect of pores. In the present cases, both models give very close predictions, which are in good agreement with the experiment data for the composites with low porosity (roughly below 10%). When exceeding this value, the predictions are slightly higher than experimental data. The possible reason is that the pores congregate preferentially to the aluminum–SiCp interfaces. When the porosity is so high, it is reasonable to assume that the large amount of pores can lead to a marked reduction of the interfacial bonding between the aluminum and SiCp, which introduces notably additional thermal barriers, rationalizing the measured composite conductivity being lower than those predicted. Since the high porosity in excess of 10% is typically avoided for composites made in practice, it can be considered that both models of MEMA and TS-HJ can be used for a sufficiently accurate prediction of the effective composites thermal conductivity as a function of porosity. Actually, the simple comparison of these two schemes for the limited cases of SiCp/Al composites does not mean that they can always lead to the nearly same estimations. If rising the phase contrast (conductivity ratio of particle-to-matrix), i.e., using higher thermal conductive particles such as diamonds, it is clearly seen in Fig. 6 that there is an markedly increasing deviation between two predictions with increasing porosity, in which the TS-HJ model presents a lower value all the while. This deviation may be correlated with the morphology and distribution of pores in the composites. However, for high phase contrast composites, it is not certain whether the two approaches are also available to their corresponding porous structures, or the TS-HJ model is the only one that can provide fully satisfactory predictions for both porous structures. We hence limit ourselves to the statement that, since in SiCp/Al composites phase contrasts are generally low, the use of both models is probably justified for porous composites having any morphology and distribution of pores as the difference between the predictions of two schemes is often lower than the experimental uncertainty. For illustration, Fig. 7 shows a comparison between MEMA model and experimental values of Ref. [13], it is apparent that the MEMA model also offers a good prediction for the thermal conductivity of their composites that have probably different porous structure from present composites. In this work we have only performed the simple calculations for porous composites reinforced by unitary SiC particles with a
210 200 40 μm ( )
190 180 170 160
MEMA TS-HJ
Kp/Km= 6
400
350 Kp/Km= 4
300
250 Kp/Km= 2
200
0
2
4
6
8
10
12
14
Porosity / % Fig. 5. Comparison between predictions given by the multiple effective medium approximation (MEMA), two-step Hasselman–Johnson (TS-HJ) models and experiment data for the thermal conductivity the composites as a function of porosity with two considered particle sizes of 40 lm (filled squares), 100 lm (empty squares).
0
3
6
9
12
15
Porosity / % Fig. 6. Comparison between predictions given by the MEMA, TS-HJ models for the thermal conductivity of the composites with various phase contrasts (conductivity ratio of particle-to-matrix) of reinforced particles to matrix (Kp/Km) as a function of porosity.
Caculated thermal conductivity (W/mK)
K. Chu et al. / Composites: Part A 41 (2010) 161–167
167
behavior for the porous composites with a multimodal size distribution or multiphase reinforced mixtures.
220
Acknowledgements 200
This study was financially supported by National Natural Science Fund of China (No. 50971020) and National 863 Plan Project of China (No. 2008AA03Z505). We thank the reviewers for helpful suggestions.
180
References 160
140 140
160
180
200
220
Experimental thermal conductivity (W/mK) Fig. 7. Comparison of the measured (Table 2 in Ref. [13]) and calculated values for the thermal conductivity of Al–12 wt.% Si/SiC composites containing various levels of porosity using a auto-coherent linear relation. Filled and empty circles correspond to pore-free and porous samples, respectively.
monomodal distribution. This work can, however, be extended further to porous composites with a multimodal size distribution or multiphase reinforced mixtures. For TS-HJ model, a considered composite is treated to be obtained from an initial matrix by successively adding small quantities of particles (pores are first added considered as the zero-conductive particles) to the system until the final volume fraction of the particles is reached. For the present MEMA method, pores can also be regarded as one type of non-thermally conducting particles present with other types of inclusions. Obviously, the stepwise method involves complicated calculation steps, in each of which one needs to calculate the effective volume fraction of the new embedded particles as well as the thermal conductivity of the new effective matrix, whereas the MEMA method can be expressed in a single equation that can be evaluated readily for such materials. However, the further experiments are required to verify the extension of the two schemes. 5. Conclusion A developed effective medium approximation (EMA) approach for estimating the effect of porosity on the thermal conductivity of SPS consolidated SiCp/Al composites was established. The present model gave predictions quite close to those obtained by an existing model of two-step Hasselman–Johnson approach. With the help of the pertinent materials parameters, both models gave a fairly well description to the experimental data, especially for the composites with porosity below 10%. At high levels of porosity the model predictions were slightly higher than the experimental values. The present work also gave the fast and efficient ways to determine the materials parameters that seem not to be measured directly via appropriate experimental techniques. This work further generalized two models to describe the thermal conductivity
[1] Zweben C. Advanced electronic packaging materials. Adv Mater Process 2005;163:33–7. [2] Molina JM, Piñero E, Narciso J, García-Cordovilla C, Louis E. Liquid metal infiltration into ceramic particle preforms with bimodal size distributions. Curr Opin Solid State Mater Sci 2005;9:202–10. [3] Chu K, Jia CC, Liang XB, Chen H, Guo H. The thermal conductivity of pressure infiltrated SiCp/Al composites with various size distributions: experimental study and modeling. J Mater Des 2009;30:3497–503. [4] Chu K, Jia CC, Liang XB, Chen H, Guo H, Yin FZ, et al. Experimental and modeling study of the thermal conductivity of SiCp/Al composites with bimodal size distribution. J Mater Sci 2009;44:4370–8. [5] Yang MJ, Zhang DM, Gu XF, Zhang LM. Effects of SiC particle size on CTEs of SiCp/Al composites by pulsed electric current sintering. Mater Chem Phys 2006;99:170–3. [6] Sáncheza JM, Alvarez M, Rodriguez N, Aristizabal M. Effect of Ni powder characteristics on the consolidation of ultrafine TiMoCN cermets by means of SPS and HIP technologies. Mater Sci Eng A 2009;500:225–32. [7] Kumari L, Zhang T, Du GH, Li WZ, Wang QW, Datye A, et al. Thermal properties of CNT–Alumina nanocomposites. Compos Sci Technol 2008;68:2178–83. [8] Nan CW, Birringer R, Clarke DR, Gleiter H. Effective thermal conductivity of particulate composites with interfacial thermal resistance. J Appl Phys 1997;81:6692–9. [9] Kawai C. Effect of interfacial reaction on the thermal conductivity of Al–SiC composites with SiC dispersions. J Am Ceram Soc 2001;84:896–8. [10] Gui MC, Kang SB, Euh K. Thermal conductivity of Al–SiCp composites by plasma spraying. Scripta Mater 2005;52:51–6. [11] Euh K, Kang SB. Effect of rolling on the thermo-physical properties of SiCp/Al composites fabricated by plasma spraying. Mater Sci Eng A 2005;395:47–52. [12] Molina JM, Narciso J, Weber L, Mortensen A, Louis E. Thermal conductivity of Al–SiC composites with monomodal and bimodal particle size distribution. Mater Sci Eng A 2008;58:393–6. [13] Molina JM, Prieto R, Narciso J, Louis E. The effect of porosity on the thermal conductivity of Al–12 wt.% Si/SiC composites. Scripta Mater 2009;60:582–5. [14] Hasselman DPH, Johnson LF. Effective thermal conductivity of composites with interfacial thermal barrier resistance. J Compos Mater 1987;21:508–15. [15] Torquato S. Random heterogeneous media. New York: Springer; 2001. p. 460– 2. [16] Benveniste Y. Effective thermal conductivity of composites with a thermal contact resistance between the constituents non-dilute case. J Appl Phys 1987;61:2840–3. [17] Dyck JS, Chen W, Uher C. Heat transport in Sb2xVxTe3 single crystals. Phys Rev B 2002;66:125–206. [18] Nan CW, Birringer R. Determining the Kapitza resistance and the thermal conductivity of polycrystals: a simple model. Phys Rev B 1998;57:8264–8. [19] Swartz ET, Pohl RO. Thermal boundary resistance. Rev Mod Phys 1989;61:605–68. [20] Chu K, Jia CC, Liang XB, Chen H, Gao WJ, Guo H. Modeling the thermal conductivity of diamond reinforced aluminium matrix composites with inhomogeneous interfacial conductance. J Mater Des 2009;30:4311–6. [21] Liu DM, Lin BW. Thermal conductivity in hot-pressed silicon carbide. Ceram Int 1996;22:407–14. [22] Nan CW, Li XP, Birringer R. Inverse problem for composites with imperfect interface: determination of interfacial thermal resistance, thermal conductivity of constituents, and microstructural parameters. J Am Ceram Soc 2000;83:848–54. [23] Hasselman DPH, Donaldson KY, Geiger AL. Effect of reinforcement particle size on the thermal conductivity of a particulate silicon carbide-reinforced aluminum–matrix composite. J Am Ceram Soc 1992;75:3137–40.
Contents lists available at ScienceDirect
Composites: Part A journal homepage: www.elsevier.com/locate/compositesa
Thermal conductivity of spark plasma sintering consolidated SiCp/Al composites containing pores: Numerical study and experimental validation Ke Chu a,b,*, Chengchang Jia a, Wenhuai Tian b, Xuebing Liang a, Hui Chen a, Hong Guo c a
Institute of Powder Metallurgy, University of Science and Technology Beijing, Beijing 100083, China Department of Materials Physics and Chemistry, University of Science and Technology Beijing, Beijing 100083, China c Beijing General Research Institute for Nonferrous Metals, Beijing 100088, China b
a r t i c l e
i n f o
Article history: Received 17 March 2009 Received in revised form 26 September 2009 Accepted 3 October 2009
Keywords: A. Metal matrix composites B. Thermal properties C. Analytical modeling E. Sintering
a b s t r a c t A simple model was introduced for describing the effect of porosity on the effective thermal conductivity of spark plasma sintered (SPS) consolidated SiCp/Al composites in terms of an effective medium approximation (EMA) scheme. Numerical results of the present model were compared to an existing model of two-step Hasselman–Johnson approach and experimental data. Both models yielded very close predictions, which provided a satisfactory agreement to the experimental data, especially for the composites with porosity below 10%. At high levels of porosity the model predictions were slightly higher than the experimental values. These two models were further extended to account for the thermal conduction properties of porous composites with a multimodal size distribution or multiphase reinforced mixtures. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Metal matrix composites (MMCs) with high reinforcement volume fractions are used for thermal management applications due to their excellent thermo-physical properties, tailorable thermal expansion and low density. Particularly, Al matrix composites containing high volume fractions of SiC particles (SiCp/Al) composites are receiving the most attention as potential candidates for a variety of uses in advanced electronic packaging and are currently competing with established materials such as Cu/W or Cu/Mo in the electronic packaging industry [1]. As has been reported so far, SiCp/Al composites with high SiCp volume fraction are mainly prepared by infiltration of liquid metal into the ceramic preforms [2–4]. This process often uses Al-alloys with an addition of Si and Mg to avoid formation of interfacial reaction products such as Al4C3, which has a detrimental effect on the thermo-mechanical properties of the composites. Nevertheless, the dissolved Si and Mg decrease the matrix thermal conductivity leading to a low thermal conductivity of the composites. Spark plasma sintering (SPS) applied as a new method for preparing high performance of SiCp/Al composites has been presented recently [5]. The main advantage of the SPS process is that it allows fabrication
* Corresponding author. Address: Institute of Powder Metallurgy, University of Science and Technology Beijing, Beijing 100083, China. Tel.: +86 10 62334271; fax: +86 01 62334271. E-mail address: [email protected] (K. Chu). 1359-835X/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesa.2009.10.001
of bulk materials using relatively short sintering times at low temperatures [6], which has been known to be very beneficial to the prevention of Al4C3 at the interface. However, a small portion of pores can always appear as an unavoidable phase in SiCp/Al composites during the SPS process, which is linked to the non-wetting nature of SiC and aluminum, resulting in weak ceramic–metal interfaces and incomplete sintering, especially at elevated particle volume fraction. Pores can severely degrade the thermal conductivity of the composites due to scattering of the heat flow [7]. Although the effective thermal conductivity of SiCp/Al composites has been extensively investigated considering different parameters such as particle shape, size, size distribution, volume fraction and interfacial thermal resistance (ITR) [8–12], very limited theoretical work [13] has been developed to quantitatively characterize the effect of porosity on the thermal conduction properties of these composites, especially in comparison with theoretical analysis and experimental results on the composites thermal conductivity associated with the porosity. We therefore focus in this contribution on the effect of porosity on the thermal conductivity of SiCp/Al composites with high volume fabrication. A reasonable model is proposed in the framework of the effective medium approximation scheme. For the purposes of assessing the predictive capacity, the present analytical model is then compared to an existing model of two-step Hasselman– Johnson approach [13] and experimental results for the thermal conductivity of SPS consolidated SiCp/Al composites with various levels of porosity. Various calculational methods are used to
162
K. Chu et al. / Composites: Part A 41 (2010) 161–167
determine the relevant parameters of the composites, i.e., the thermal conductivity of constituents and the interfacial thermal conductance.
2. Modeling A number of reported results [8–15] have proved that the effective medium approximation (EMA) is a simple and powerful approach in describing the essential thermal conduction properties of the composites with perfect or imperfect interface. Our idea is to develop the EMA formula to describe the thermal conductivity of particle-reinforced composites with the porosity content. First we briefly review the EMA theory following Hasselman and Johnson [14,15]. A schematic diagram of modeling process is shown in Fig. 1. Let us consider a two-phase composite medium consisting of n spherical inclusions of radius a and thermal conductivity Kp embedded in an effective matrix with thermal conductivity Km (Fig. 1a). We make the assumption that the n inclusions are enclosed by a spherical region of radius R and the interaction between the particles is negligible. The rigorous solutions for the temperature Tp within the dispersed particles and the temperature Tm in the surrounding matrix can be obtained, which have the following form
3 r cos h 2K p =ah þ K p =K m þ 2 " 3 # R K p =ah K p =K m þ 1 r cos h T m ¼ ðrTÞ 1 þ V p r 2K p =ah þ K p =K m þ 2
T p ¼ ðrTÞ
Now we consider a multi-phase composite medium with multiple reinforcement phases (i) (Fig. 1b). Each of reinforcement phases contains n spherical inclusions of radius, thermal conductivity, volume fraction and interfacial thermal conductance, corresponding to ai, K pi , V pi , hi, respectively. In such a scheme the Tm depends on the cumulative effect of N phases of reinforced inclusions within a region R, hence Eq. (1b) becomes
( T m ¼ ðrTÞ 1 þ cos h
ð1aÞ
"
3 # R Km Kc r cos h T m ¼ ðrTÞ 1 þ r 2K m þ K c
ð3Þ
Since the two expressions, Eqs. (2) and (3), are equivalent, it follows that N X
V pi
K pi =ai h K pi =K m þ 1 2K pi =ai h þ K pi =K m þ 2
¼
Km Kc 2K m þ K c
ð4Þ
For explicit expression of Kc, Eq. (4) can be rearranged as
1 þ 2A 1A
ð5aÞ
. 13 K eff Km 1 pi 4V p @ A5 . i K eff Km þ 2 pi
ð5bÞ
Kc ¼ Km ð1bÞ
where (rT) is the temperature gradient at a radial distance r (r R) from the centre of the spherical cluster region R; h is the angle between the radius vector r and the temperature gradient; h is the interfacial thermal conductance between the spherical dispersions and the matrix. Vp is the particle volume fraction defined as Vp = n(a/R)3.
ð2Þ
Now if all the inclusions are treated as an ‘‘effective homogeneous medium” of radius R and thermal conductivity Kc, suspended in a matrix of conductivity Km (Fig. 1b), the Tm at any radial location r (r R) is given as
i
3 X ) N K pi =ai h K pi =K m þ 1 R r V pi 2K pi =ai h þ K pi =K m þ 2 r i
with
A¼
N X i
K eff pi ¼
2
0
K pi 1 þ K pi =ai h
ð5cÞ
Fig. 1. Schematic illustration of the modeling process in terms of an effective medium approximation (EMA): (a) a two-phase composite medium consisting of unitary reinforcement phase; (b) a multi-phase composite medium consisting of multiple reinforcement phases; (c) effective homogeneous medium.
K. Chu et al. / Composites: Part A 41 (2010) 161–167
where K eff p stands for the effective thermal conductivity of the particles due to the existing finite interface thermal conductance between the two solid phases. The volume fractions of mixed particles of N phases inclusions fulfill the relationship: P V p ¼ Ni V pi . For convenience, Eq. (5) is named as multiple effective medium approximation (MEMA) model. Typically, for composites with unitary particles of single size, Eq. (5) is reduced to
h
Kc ¼
K m 2K m þ K eff p
i þ 2 K eff p Km Vp
eff 2K m þ K eff p ðK p K m ÞV p
ð6Þ
Eq. (6) is identical to Hasselman and Johnson (H–J) model [14] and Benveniste’s [16] self-consistent scheme. We now consider the porosity effect. To quantitatively account for the thermal conductivity of the composites with ternary system consisting of metal matrix, ceramic particles and pores, a so-called two-step H–J approach proposed by Molina et al. [13] is to consider the particles to be embedded in a ‘‘effective matrix” made up of a composite of matrix with certain amount of pores. The effective fraction of pores, n0 ; in the effective matrix is given by
n0 ¼
n 1 Vp
ð7Þ
where n is the porosity, the effective matrix thermal conductivity, eff K eff m ; derived from the H–J model by taking K p ¼ 0, is
K eff m ¼
K m ð1 n0 Þ 1 þ 0:5n0
ð8Þ
The thermal conductivity of the overall composite can then be obtained using the H–J model again (Eq. (6)) by replacement of the intrinsic matrix conductivity (Km) with the effective matrix conductivity (K eff m ). In the present contribution, we use Eq. (5) to develop another simple model for evaluating the thermal conductivity of the composites with the porosity content. In light of the spirit of Eq. (5), for the limited cases of composites with binary reinforcements, we assume that two reinforced inclusions are treated as two types of given particles including conducting SiC particles and non-thermally
163
conducting particles, namely the residual pores. Accordingly, Eq. (5) can be simplified as
Kc ¼ Km
1 þ 2A 1A
ð9aÞ
with
0 eff . 1 Kp Km 1 A1n . A ¼ V p@ eff 2 Kp Km þ 2
ð9bÞ
From Eq. (9), the thermal conductivity of the composites containing pores can be also estimated for a given porosity. To verify the validity of present model, we will use Eq. (9) to confront the accepted model of two-step H–J approach (Eqs. (6)–(8)) and experimental data in the following.
3. Experimental 3.1. Preparation of composites Aluminum powder (99.9% in purity) that passed through 200 mesh sieve was used. The selected reinforcement particles were high-purity (>99%) green a-SiC with mean size of 40 lm and 100 lm, respectively. The starting powders were blended with different SiCp sizes and volume fractions of 55–60% by three-dimensional vibratory mill of 1400 rpm for 10 min. A SPS system (mod. 1050, Sumitomo Coal Mining Co. Ltd., Japan) was used to synthesize SiCp/Al composites. Fig. 2 shows a schematic diagram of the SPS apparatus. The mixed powders were put into a cylindrical graphite die with an inner diameter of 30 mm. With the aim to prevent powders from sticking to the inner wall of the die during the sintering, the die was covered with a 2 mm thick layer of graphite felt. In order to obtain composites with various porosities, not all the composites were prepared by the same procedure. The compact powders were sintered at a temperature in the range of 540–560 °C for 5 min in vacuum (less than 4 Pa). The heating rate was 50 °C/min and an axial pressure of 40, 45 and 50 MPa was applied throughout the sintering cycle. After sintering, the surfaces of
Fig. 2. Schematic illustration of SPS apparatus.
164
K. Chu et al. / Composites: Part A 41 (2010) 161–167
samples were ground to remove the graphite layer. The characteristics of the samples are shown in Table 1. 3.2. Actual volume fraction and porosity measurements The weight fractions of particles (Wp) and matrix (Wm) are related to the total composite weight, and these variables can typically be established prior to composite processing. The volume fractions of particles (Vp) and matrix (Vm) are related to the total composite volume; however, these variables cannot be established without knowledge of the amount of the third volume-based part in composites, i.e., the porosity (n). Therefore, the volume fractions of particles, matrix and porosity are mutually dependent variables, and the particle weight fraction will be used as the basic independent variable. The weight and volume fractions of the composites are:
Wm þ Wp ¼ 1
ð10Þ
Vm þ Vp þ n ¼ 1
ð11Þ
Based on the densities of particles (qp) and matrix (qm), expressions for the actual volumes of the particle and matrix parts are
Vp ¼
W p =qp qc ¼ Wp 1=qc qp
ð12Þ
Vm ¼
ð1 W p Þ=qm qc ¼ ð1 W p Þ 1=qc qm
ð13Þ
From Eqs. (11)–(13), the porosity can be then expressed as:
n ¼ 1 qc
Wp
qp
þ
1 Wp
! ð14Þ
qm
where qc is the density of the composite, which can be measured by Archimedes’ principle. The densities of Al and SiC were measured by He-pycnometry to be 2.69 g/cm3 and 3.23 g/cm3, respectively. Based on the measured values for densities and weight fractions, the actual volume fractions of particles and porosity were calculated by Eqs. (12) and (14), respectively. 3.3. Electronic conductivity and thermal conductivity measurements The room temperature static electric conductivity was measured with the four probe method for determining the matrix thermal conductivity. Only pore-free samples were considered in order to eliminate the influence of porosity. All specimens had a rectangular cross section Q of 1 1 mm2 and 25 mm length. Both ends of the specimens were connected to a constant current source, which
Table 1 The characteristics of all the samples investigated in this work: Vp is the actual particle volume fraction, d is the average particle size (lm), T is the sintering temperature (°C), P is the applied pressure (MPa), n is the porosity (%), Kc and rc are the thermal conductivity (W m1 K1) and electronic conductivity (106 X1 m1) of the composites, respectively. Samples/n #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12
d 40
100
Vp
T
P
n
Kc
rc
55 60 58 56 56 53 50 55 58 55 53 52
560 560 560 560 550 540 560 560 560 550 540 540
50 50 45 40 45 40 50 50 45 40 45 40
0 0 2.6 7.2 6.3 12.5 0 0 3.7 8.8 11.3 13.4
208 211 204 185 192 165 220 224 208 197 181 173
8.17 5.75 – – – – 8.58 7.24 – – – –
provided a current of magnitude I. The potential drop V was measured over the distance L with the aid of two knife-edge ridges pressed on the sample. The conductivity r ¼ ðI=VÞðL=Q Þ was calculated by taking the mean value of two voltage and current measurements for normal and reversed current direction in order to minimize the effect of contact voltages. The laser flash method was used to measure the thermal diffusivity of the composites at room temperature. The thermal diffusivity measurement was performed by a JR-3 thermal physical testing instrument, which involves exposing the front face of a small disc-shaped sample to a short laser burst and recording the temperature rise on the rear face. The samples were cut from assintered composites to disc-shaped with a diameter of 10 mm and a thickness of 3 mm. The uncertainty in the thermal measurements is ±2%. The specific heat was derived from the theoretical value calculated by rule of mixture (ROM). The thermal conductivity of the composites was then calculated as a product of the density, thermal diffusivity and specific heat. All measurement results are also shown in Table 1. 3.4. Microstructure characterization A thorough microstructural examination of samples was also carried out on an LEO1450 scanning electron microscopy (SEM). Samples were sectioned and polished using standard metallographic techniques. 4. Results and discussion 4.1. Microstructure Fig. 3 shows the representative microstructures of SiCp/Al composites prepared by SPS under different temperatures and pressures. It can be seen in Fig. 3a and d that the composites have a rather uniform distribution of the particles in the matrix and there is no evidence of pores or separated interface, indicating that high volume fraction SiCp/Al composites can be well prepared via the prescribed SPS procedures followed in this work. As expected, inappropriate selected sintering parameters (temperature and pressure) can lead to the quantity of matrix being not sufficient to fill all the space left between touching particles, resulting in various degrees of presence of pores (white phase in Fig. 3b, c, e, and f). 4.2. Determination of material parameters In order to accurately evaluate the models and describe the thermal conductive behavior associated with porosity, one needs exact values of material parameters (Km, h, Kp) used for calculations in Eqs. (6)–(9). However, according to the reported data, these parameters cover wide ranges of 175–237 W m1 K1 for Km, 0.5– 2 108 W m2 K1 for h and 90–270 W m1 K1 for Kp depending on their purity, as well as the fabrication route [8–13]; thus it is very necessary to determine their pertinent values relevant for the present materials. The matrix thermal conductivity can be derived from the electrical conductivity measurements [12] of the composites rc (see Table 1) and the volume fraction of SiC particles (Vp) according to the following expression:
rc ¼ rm ð1 V p Þn
ð15Þ
where n is a parameter reflecting the average particle shape. The data for the electrical conductivity of the composites are fitted with Eq. (15) using n and rm as parameters. Fig. 4 shows data fitting results by carrying out a plot of ln rc vs. ln(1 Vp), given by:
K. Chu et al. / Composites: Part A 41 (2010) 161–167
165
Fig. 3. SEM micrographs of SPS consolidated SiCp/Al composites under various sintering conditions: (a) 60 vol.% 40 lm SiCp under a pressure of 50 MPa sintered at 560 °C (50 MPa/560 °C) (free of pores); (b) 58 vol.% 40 lm SiCp, 45 MPa/560 °C (porosity n = 2.6%); (c) 53 vol.% 40 lm SiCp, 40 MPa/540 °C (n = 12.5%); (d) 55 vol.% 100 lm SiCp, 50 MPa/560 °C (free of pores); (e) 58 vol.% 100 lm SiCp, 45 MPa/560 °C, (n = 3.7%); (f) 55 vol.% 100 lm SiCp, 40 MPa/550 °C (n = 8.8%).
ln rc ¼ 1:82 lnð1 V p Þ þ 3:48
ð16Þ
The obtained values of the fitted parameters are n = 1.82 and rm = 32.46 106 X1 m1. The matrix thermal conductivity (Km) can be then derived from the matrix electrical conductivity (rm) by the following well-known relation for metal and alloys:
K m ¼ L rm T
ð17Þ
where T is the temperature and L is the Lorentz constant (equal to 2.1 108 V2 K2 at T = 300 K [18]). By using Eq. (17), Km is calculated to be 210 W m1 K1. The value estimated for Km in present SPSed composites are in a reasonable value range, although there is a discrepancy from that of 237 W m1 K1 for monocrystalline aluminum, probably because powder sintered material pertain to polycrystal which contains grain boundaries and defects in its structure [18]. The value of the intrinsic interfacial thermal conductance, h, can be estimated by a simple Debye model in terms of the acoustic mismatch theory [19,20]:
hc
qm qp v m v p 1 v3 qm c m2 2 v p ðqm v m þ qp v p Þ2
ð18Þ
where c is the specific heat of metal, q the density and v the phonon velocity. Subscripts m and p denote metal and reinforcement, respectively. Substituting qm = 2.7 103 kg m3, C = 880 J kg1 K1 and mm = 3.52 103 m s1 [17,19], qp = 3.27 103 kg m3, mp = 1.16 104 m s1 [21] for the SiC into Eq. (18), h can be calculated to be 6.65 107 W m2 K1. The inverse calculation method [22] is always used to determine the thermal conductivity of the small reinforcements, i.e., particles, short fibers, and whiskers. This is because the actual values of the thermal conductivity for these small reinforcements are very different from those for their bulk single crystals and even polycrystals, and no appropriate experimental technique appears to be available for direct measurement. From a series of data with respect to the measured composites conductivity and other known parameters, this remained unknown parameter can be reasonably estimated based on associated formulations. We then take into account all the pore-free composites, and substitute four sets of data for Kc (Table 1), and relate the other parameters of Km, h, d and Vp
166
K. Chu et al. / Composites: Part A 41 (2010) 161–167
2.4
lnσc
2.2
2.0
1.8
1.6 -1.0
-0.9
-0.8
-0.7
ln (1-Vp) Fig. 4. The matrix electronic conductivity (rm) obtained from a linear fitting of ln rc vs. ln(1 Vp) for the experimental data of composites electronic conductivity (rc) (see Table 1). The data can be satisfactorily fitted by the straight linear of ln rc = 1.82 ln(1 Vp) + 3.48 with determined n = 1.82 and rm = 32.46 106 X1 m1.
into Eqs. (5c) and (6), the average Kp value of 248 W m1 K1 can be ultimately back-calculated. This value represents the average intrinsic thermal conductivity of the SiC particles for the present composites, which is in a reasonable value range for fully dense polycrystalline structural SiC, but remains well below the value of up to 400 W m1 K1 for a high-purity single crystal at room temperature [23]. 4.3. Numerical results and comparison with experiments Fig. 5 shows the comparison between the predictions given by the present model (MEMA), the two-step H–J (TS-HJ) approach and experimental data for the effect of porosity on the thermal conductivity the SiCp/Al composites with two particle sizes of 40, 100 lm. It can be clearly seen that the increase of porosity causes a pronounced decrease in composites conductivity. For the considered porosity of 7.2%, the thermal conductivity decrement is about
450
MEMA
100 μm ( )
220
TS-HJ
Thermal conductivity (W/mK)
Thermal conductivity (W/mK)
230
12.3%, a value of 71% higher than its corresponding porosity confirms this great negative effect of pores. In the present cases, both models give very close predictions, which are in good agreement with the experiment data for the composites with low porosity (roughly below 10%). When exceeding this value, the predictions are slightly higher than experimental data. The possible reason is that the pores congregate preferentially to the aluminum–SiCp interfaces. When the porosity is so high, it is reasonable to assume that the large amount of pores can lead to a marked reduction of the interfacial bonding between the aluminum and SiCp, which introduces notably additional thermal barriers, rationalizing the measured composite conductivity being lower than those predicted. Since the high porosity in excess of 10% is typically avoided for composites made in practice, it can be considered that both models of MEMA and TS-HJ can be used for a sufficiently accurate prediction of the effective composites thermal conductivity as a function of porosity. Actually, the simple comparison of these two schemes for the limited cases of SiCp/Al composites does not mean that they can always lead to the nearly same estimations. If rising the phase contrast (conductivity ratio of particle-to-matrix), i.e., using higher thermal conductive particles such as diamonds, it is clearly seen in Fig. 6 that there is an markedly increasing deviation between two predictions with increasing porosity, in which the TS-HJ model presents a lower value all the while. This deviation may be correlated with the morphology and distribution of pores in the composites. However, for high phase contrast composites, it is not certain whether the two approaches are also available to their corresponding porous structures, or the TS-HJ model is the only one that can provide fully satisfactory predictions for both porous structures. We hence limit ourselves to the statement that, since in SiCp/Al composites phase contrasts are generally low, the use of both models is probably justified for porous composites having any morphology and distribution of pores as the difference between the predictions of two schemes is often lower than the experimental uncertainty. For illustration, Fig. 7 shows a comparison between MEMA model and experimental values of Ref. [13], it is apparent that the MEMA model also offers a good prediction for the thermal conductivity of their composites that have probably different porous structure from present composites. In this work we have only performed the simple calculations for porous composites reinforced by unitary SiC particles with a
210 200 40 μm ( )
190 180 170 160
MEMA TS-HJ
Kp/Km= 6
400
350 Kp/Km= 4
300
250 Kp/Km= 2
200
0
2
4
6
8
10
12
14
Porosity / % Fig. 5. Comparison between predictions given by the multiple effective medium approximation (MEMA), two-step Hasselman–Johnson (TS-HJ) models and experiment data for the thermal conductivity the composites as a function of porosity with two considered particle sizes of 40 lm (filled squares), 100 lm (empty squares).
0
3
6
9
12
15
Porosity / % Fig. 6. Comparison between predictions given by the MEMA, TS-HJ models for the thermal conductivity of the composites with various phase contrasts (conductivity ratio of particle-to-matrix) of reinforced particles to matrix (Kp/Km) as a function of porosity.
Caculated thermal conductivity (W/mK)
K. Chu et al. / Composites: Part A 41 (2010) 161–167
167
behavior for the porous composites with a multimodal size distribution or multiphase reinforced mixtures.
220
Acknowledgements 200
This study was financially supported by National Natural Science Fund of China (No. 50971020) and National 863 Plan Project of China (No. 2008AA03Z505). We thank the reviewers for helpful suggestions.
180
References 160
140 140
160
180
200
220
Experimental thermal conductivity (W/mK) Fig. 7. Comparison of the measured (Table 2 in Ref. [13]) and calculated values for the thermal conductivity of Al–12 wt.% Si/SiC composites containing various levels of porosity using a auto-coherent linear relation. Filled and empty circles correspond to pore-free and porous samples, respectively.
monomodal distribution. This work can, however, be extended further to porous composites with a multimodal size distribution or multiphase reinforced mixtures. For TS-HJ model, a considered composite is treated to be obtained from an initial matrix by successively adding small quantities of particles (pores are first added considered as the zero-conductive particles) to the system until the final volume fraction of the particles is reached. For the present MEMA method, pores can also be regarded as one type of non-thermally conducting particles present with other types of inclusions. Obviously, the stepwise method involves complicated calculation steps, in each of which one needs to calculate the effective volume fraction of the new embedded particles as well as the thermal conductivity of the new effective matrix, whereas the MEMA method can be expressed in a single equation that can be evaluated readily for such materials. However, the further experiments are required to verify the extension of the two schemes. 5. Conclusion A developed effective medium approximation (EMA) approach for estimating the effect of porosity on the thermal conductivity of SPS consolidated SiCp/Al composites was established. The present model gave predictions quite close to those obtained by an existing model of two-step Hasselman–Johnson approach. With the help of the pertinent materials parameters, both models gave a fairly well description to the experimental data, especially for the composites with porosity below 10%. At high levels of porosity the model predictions were slightly higher than the experimental values. The present work also gave the fast and efficient ways to determine the materials parameters that seem not to be measured directly via appropriate experimental techniques. This work further generalized two models to describe the thermal conductivity
[1] Zweben C. Advanced electronic packaging materials. Adv Mater Process 2005;163:33–7. [2] Molina JM, Piñero E, Narciso J, García-Cordovilla C, Louis E. Liquid metal infiltration into ceramic particle preforms with bimodal size distributions. Curr Opin Solid State Mater Sci 2005;9:202–10. [3] Chu K, Jia CC, Liang XB, Chen H, Guo H. The thermal conductivity of pressure infiltrated SiCp/Al composites with various size distributions: experimental study and modeling. J Mater Des 2009;30:3497–503. [4] Chu K, Jia CC, Liang XB, Chen H, Guo H, Yin FZ, et al. Experimental and modeling study of the thermal conductivity of SiCp/Al composites with bimodal size distribution. J Mater Sci 2009;44:4370–8. [5] Yang MJ, Zhang DM, Gu XF, Zhang LM. Effects of SiC particle size on CTEs of SiCp/Al composites by pulsed electric current sintering. Mater Chem Phys 2006;99:170–3. [6] Sáncheza JM, Alvarez M, Rodriguez N, Aristizabal M. Effect of Ni powder characteristics on the consolidation of ultrafine TiMoCN cermets by means of SPS and HIP technologies. Mater Sci Eng A 2009;500:225–32. [7] Kumari L, Zhang T, Du GH, Li WZ, Wang QW, Datye A, et al. Thermal properties of CNT–Alumina nanocomposites. Compos Sci Technol 2008;68:2178–83. [8] Nan CW, Birringer R, Clarke DR, Gleiter H. Effective thermal conductivity of particulate composites with interfacial thermal resistance. J Appl Phys 1997;81:6692–9. [9] Kawai C. Effect of interfacial reaction on the thermal conductivity of Al–SiC composites with SiC dispersions. J Am Ceram Soc 2001;84:896–8. [10] Gui MC, Kang SB, Euh K. Thermal conductivity of Al–SiCp composites by plasma spraying. Scripta Mater 2005;52:51–6. [11] Euh K, Kang SB. Effect of rolling on the thermo-physical properties of SiCp/Al composites fabricated by plasma spraying. Mater Sci Eng A 2005;395:47–52. [12] Molina JM, Narciso J, Weber L, Mortensen A, Louis E. Thermal conductivity of Al–SiC composites with monomodal and bimodal particle size distribution. Mater Sci Eng A 2008;58:393–6. [13] Molina JM, Prieto R, Narciso J, Louis E. The effect of porosity on the thermal conductivity of Al–12 wt.% Si/SiC composites. Scripta Mater 2009;60:582–5. [14] Hasselman DPH, Johnson LF. Effective thermal conductivity of composites with interfacial thermal barrier resistance. J Compos Mater 1987;21:508–15. [15] Torquato S. Random heterogeneous media. New York: Springer; 2001. p. 460– 2. [16] Benveniste Y. Effective thermal conductivity of composites with a thermal contact resistance between the constituents non-dilute case. J Appl Phys 1987;61:2840–3. [17] Dyck JS, Chen W, Uher C. Heat transport in Sb2xVxTe3 single crystals. Phys Rev B 2002;66:125–206. [18] Nan CW, Birringer R. Determining the Kapitza resistance and the thermal conductivity of polycrystals: a simple model. Phys Rev B 1998;57:8264–8. [19] Swartz ET, Pohl RO. Thermal boundary resistance. Rev Mod Phys 1989;61:605–68. [20] Chu K, Jia CC, Liang XB, Chen H, Gao WJ, Guo H. Modeling the thermal conductivity of diamond reinforced aluminium matrix composites with inhomogeneous interfacial conductance. J Mater Des 2009;30:4311–6. [21] Liu DM, Lin BW. Thermal conductivity in hot-pressed silicon carbide. Ceram Int 1996;22:407–14. [22] Nan CW, Li XP, Birringer R. Inverse problem for composites with imperfect interface: determination of interfacial thermal resistance, thermal conductivity of constituents, and microstructural parameters. J Am Ceram Soc 2000;83:848–54. [23] Hasselman DPH, Donaldson KY, Geiger AL. Effect of reinforcement particle size on the thermal conductivity of a particulate silicon carbide-reinforced aluminum–matrix composite. J Am Ceram Soc 1992;75:3137–40.