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Thermal conductivity of Al-SiC compositeswith monomodal and bimodal particle size

distribution

 ARTICLE  in  MATERIALS SCIENCE AND ENGINEERING A · MAY 2008

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Materials Science and Engineering A 480 (2008) 483–488

Thermal conductivity of Al–SiC composites with monomodaland bimodal particle size distribution

J.M. Molina a,b,c,∗, J. Narciso b,d, L. Weber c, A. Mortensen c, E. Louis a,b

a Departamento de Fısica Aplicada, Universidad de Alicante, Apartado 99, E-03080 Alicante, Spainb Instituto Universitario de Materiales de Alicante (IUMA), Universidad de Alicante, Apartado 99, E-03080 Alicante, Spain

c  ´  Ecole Polytechnique F´ ed´ erale de Lausanne, EPFL, Laboratory of Mechanical Metallurgy, CH-1015 Lausanne, Switzerland d Departamento de Quımica Inorg´ anica, Universidad de Alicante, Apartado 99, E-03080 Alicante, Spain

Received 22 January 2007; received in revised form 11 July 2007; accepted 16 July 2007

Abstract

The thermal conductivity of aluminum matrix composites having a high volume fraction of SiC particles is investigated by comparing data

for composites fabricated by infiltrating liquid aluminum into preforms made either from a single particle size, or by mixing and packing SiC

particles of two largely different average sizes (170 and 16m). For composites based on powders with a monomodal size distribution, the thermal

conductivity increases steadily from 151W/mK forparticlesof averagediameter8 mto216W/mKfor170m particles. For thebimodal particle

mixtures the thermal conductivity increases with increasing volume fraction of coarse particles and reaches a roughly constant value of 220 W/m K

for mixtures with 40 or more vol.% of coarse particles. It is shown that all present data can be accounted for by the differential effective medium

(DEM) scheme taking into account a finite interfacial thermal resistance.

© 2007 Elsevier B.V. All rights reserved.

Keywords:  Thermal conductivity; Size-effect; Al–SiC; Bimodal mixtures; Modeling

1. Introduction

The use of high volume fraction metal matrix composites

(MMCs) in multifunctional electronic packaging is steadily

increasing [1–5]. Thekey requirements for this application are a

high thermal conductivity anda coefficient of thermal expansion

(CTE) similar to that of materials commonly used in micro-

electronic systems. Among the different composites that are

nowadays being considered for this application, Al/SiC com-

posites with a high volume fraction of particulate reinforcement

are a suitable and versatile option. Due to the large difference

in CTE between the Al matrix and the SiC particles, the overall

CTE of the composite depends strongly on the particulate con-tent. By adjusting the overall SiC content, mainly by varying the

ratio between coarseandsmall particles, theCTE canbe tailored

in a certain range.

∗ Corresponding author at: Departamento de Fısica Aplicada, Universidad de

Alicante, Apartado 99, E-03080 Alicante, Spain. Tel.: +34 96 590 3400x2055;

fax: +34 96 590 3464.

 E-mail address: jmmj@ua.es (J.M. Molina).

In previous contributions [6,7], high volume fraction com-

posites were produced by infiltration of liquid aluminum into

preforms made by mixing and packing SiC particles with aver-

age diameters of 170 and 16m. Themaximum particle volume

fraction, near 0.74, was attained for a mixture containing 67%

coarse particles. As shown in Ref. [7], these composites have

a reasonably low coefficient of thermal expansion, in the range

of 7.8–10.8ppm/K, and are hence suitable for application as

substrates in microelectronics. Best among these are compos-

ites with the maximum particle volume fraction, attained using

a bimodal particle size distribution.

The thermal conductivity of these composites is mainly gov-

erned by the conductivity of the individual phases, their volumefraction and shape, and also by the size of the inclusion phase

due to a finite metal/ceramic interface thermal resistance. There

are quite a few analytical models in the literature [8–15] that

can account for a finite interface thermal conductance, h  (the

inverse of the interface thermal resistance). Many of these are

quite straightforward extensions of results obtained for compos-

ites with a matrix/inclusion topology having an ideal interface,

in which one replaces the intrinsic inclusion conductivity κinr   by

an effective inclusionthermal conductivity, κeff r   that incorporates

0921-5093/$ – see front matter © 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.msea.2007.07.026

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484   J.M. Molina et al. / Materials Science and Engineering A 480 (2008) 483–488

the effect of inclusion size. κeff r   is defined knowing the radius a

of the equivalent sphere and the interface thermal conductance,

h, according to

κeff r   =

κinr

1+ κinr   /ha

or   κeff r   =

κinr

1+ 1/Bi=

κinr

1+ B,   (1)

where Bi  is the relevant Biot number characterizing the rein-forcement and B is its inverse as introduced by Clyne [16].

One of the most widely used models is that proposed by Has-

selman and Johnson [8]. This model has often been applied to

compositeswith a monomodalparticle sizedistribution [17–21].

For composites with a bimodal particle size distribution, the

application of the Hasselman and Johnson (HJ) model is not

straightforward, because of the need to choose a single charac-

teristic length scale. A way of tackling the problem is to define

the characteristic particle size via the overall specific surface

area of the preform, e.g. [22].

We propose in this contribution that for bimodal mixtures,

the two populations of particles with distinct sizes should be

treated as two different particles with distinct effective thermal

conductivities according to the respective Biot number of the

two sizes used. We develop the leading equations for this case

in the framework of the differential effective medium (DEM)

scheme, which has recently been shown to be more robust than

the Hasselman and Johnson equation [23]. We then predict how

the composite thermal conductivity should vary in a composite

with a bimodal particle size distribution and a variable ratio of 

small-to-coarse particles. We confront this proposition to two

series of experiments. From a first series on composites made of 

aluminum and SiC with a monomodal particle size distribution

but variable particle size, we determine the relevant parameters

of the system, i.e. κinr   and h for the Al/SiC couple. The thermalconductivity of the matrix is inferred from measurements of its

electricalconductivity. The second series of composites features

a bimodal particle size distribution, with various ratios of small-

to-coarseparticles. Data from this secondseriesare then directly

confronted to the prediction of the DEM scheme for two distinct

inclusion phases.

2. Analysis

As has been shown previously, the differential effective

medium approach is well-suited to model conduction properties

of two-phase composites [23–25]. We limit our analysis to thechanges in the standard DEM theory needed to account for two

or several types of spherical inclusions having distinct effective

conductivities. The leading differential equation is

dκ

dV ∗p= κ

i

f i−κ− κeff 

ri

κ− κeff 

ri

S − κ

,   (2)

with f i the fraction of type i  of reinforcing phase in the blend

used to reinforce the composite (and, hence,i

f i  = 1), and S 

the polarization factor of a sphere, i.e. 1/3.  V ∗p   is the volume

fraction integration variable related to the effective final volume

fraction of reinforcement, V p, by  V ∗

p   = −ln(1− V p). We note

in passing that the extension to randomly oriented ellipsoidal

inclusions is not as simple as for the case of “perfectly bonded”

inclusions having no interfacial thermal barrier. We limit our-

selves to the observation that at relatively low phase contrast,

the aspect ratio of the inclusion is of minor importance. We

also note that a possible influence of particle to particle contact

points, which may ease heat conduction somewhat by creating

bridges between particles or slow conduction when partly infil-

trated, is not taken into account here other than by the choice of 

mean-field model. Separation of variables leads to the integral

equation

κc κm

dκ

κi

f i−κ− κeff 

ri

/κ− κeff 

ri

S − κ

= − ln1− V p

.

(3)

In general, the integral on the left-hand side has no analytical

solution; however, Eq. (3) can be solved numerically for κc.

For the special case of a bimodal particle mixture, one could

alternatively proceed by first introducing the small particles in

thematrix yielding a new “effective” matrix, in which thecoarse

particles are introduced in a second step. Let V p be again the

overall volume fraction of inclusion phase and  f s and f 1 be the

fraction of small and coarse particles, respectively. The fraction

V mps of small particles in the new “effective” matrix is given by

V mps=

f sV p

(1− f lV p)  (4)

and its thermal conductivity κ∗m according to the DEM scheme

is

1− V mps

=

(κs/κm)− (κ∗m/κm)

(κs/κm)− 1

κ∗m

κm

−1/3

,   (5)

with κs the effective thermal conductivity of the small particles.

The thermal conductivity of the composite is then given by

1− f lV p

=

(κl/κ∗m)− (κc/κ

∗m)

(κl/κ∗m)− 1

 κc

κ∗m

−1/3

,   (6)

where   κl  stands for the effective thermal conductivity of the

coarse particles. This second approach has the advantage of yielding an implicit analytical solution for  κc whereas the first

approach requires numerical integration. For the low phase con-

trast case considered here, it turns out that both approaches yield

the same result, i.e. the numerical values of the predicted com-

posite conductivity differ by less than 0.01 W/m K. We note in

passing that the two-step procedure would also be a way around

the need for choosing a single characteristic length scale in

the Maxwell approximation-based Hasselman and Johnson for-

mula. For low phase contrast this gives indeedvirtually thesame

result as the two DEM-based solutionspresented above. At high

phase contrast, however, the use of a Maxwell mean field based

approach may lead to erroneous results [23,24].

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Table 1

The main characteristics of monomodal composites investigated in this work: V p is the particle volume fraction, D is the average size ( D(4,3)) in m determined by

laser scattering, P0 is the threshold pressure for infiltration (kPa) taken from [6] or estimated (marked with *), TC is the thermal conductivity (W/mK), and EC is the

electrical conductivity (106 −1 m−1)

Mesh of SiC in MMC   V p (−)   D (m)   P0 (kPa) TC (W/m K) EC (106 −1 m−1)

F100 0.58 167 278 221 5.9

F180 0.58 86.4 340* 209 5.97

F240 0.6 56.8 400* 203 5.81F320 0.59 37.1 500* 204 5.74

F400 0.58 23.4 600* 194 6.57

F500 0.55 16.9 778 193 8.14

F800 0.53 8.9 2170 154 8.62

3. Experimental procedures

3.1. Processing of composites

Composites were made from 99.99% pure aluminum from

Norsk Hydro, Grevenhorst,Germany, andgreen-SiC(>99.5%)

supplied by Navarro SiC S.A, Cuenca, Spain, in nominal meshsizes F100, 180, 240, 320, 400, 500, and800. Thecorresponding

mean diametersof theSiCparticulate aregiven in Table1. Parti-

cles arerounded(asclassified by thesupplier) with a (low)aspect

ratio of 1.9 as obtained by image analysis. To clean the powders

from milling residues, the powder was repeatedly washed in

pure ethanol, separating the fine part of the powder by sedimen-

tation of the coarse particles. Particle mixing was carried out

in ethanol by means of the method described in Ref. [6]. Parti-

cles were packed into quartz tubes 16.5mm of inner diameter by

repeatedly addinga small amountof powder that was compacted

by tapping with well-defined impacts. To this end a plunger was

placed on the preform and was hammered by a weight falling

freely over a distance of 35 mm. Vibrations were intermittently

applied. The number of applied strokes was increased roughly

linearly with the height of the already packed preform. Not all

compacts were prepared with the same procedure: in the case

of coarse particulates (mesh 180 and 100) only vibrations were

used to avoid particle breaking, while in bimodal mixtures only

strokes were applied to reduce segregation. The particle volume

fraction was calculated by measuring the preform weight and

dimensions (height and diameter).

The composites were manufactured by gas pressure assisted

liquid metal infiltration. A metal ingot was placed on top of 

the particle compact. In order to avoid reaction between melt

and preform prior to infiltration, an alumina fabric was placedbetween the preform and the metal ingot.

Thequartz crucible containing thepreform and themetal was

introduced in an induction-heated cold-wall infiltration appara-

tus. Homogeneous heating was ensured by means of a graphite

susceptorplacedaround thecrucible.Afterhavingreacheda vac-

uum of about 0.3 mbar in the infiltration vessel, heating was set

to approximately 250 ◦C to allow for slow desorption of humid-

ity and gas adsorbed on the particles while continuously pulling

the vacuum. Subsequently, heating was continued up to 750 ◦C

at a rate of approximately 5 ◦C/min. The maximum tempera-

ture was maintained for 30min followed by argon injection to

reach a set pressure in the range of 8–50 bars. More precisely,

the infiltration pressure was chosen for all preforms to be about

five times the respective threshold pressure reported in Table 1,

based on previously published data [6].

Porosity was estimated by calculating the difference between

the theoretical and the actual composite density measured by

Archimedes principle. The porosity level was below 1%, in all

composites,whichwaswithin uncertainty in thepreform densityestimated as described above.

3.2. Measurement of electrical and thermal conductivity

The electrical conductivity of the composites was measured

using anEddy-current apparatus. Their thermalconductivitywas

measured by means of a relative steady-state technique, in an

experimental set up designed and assembled at EPFL. In this, a

sample of diameter around 16mm and length 40mm is clamped

between a water-cooled block and a brass reference sample

which is in its turn connected to a thermally stabilized hot water

bath. Losses through radiation and convection are neglected;

thus, we take the heat flow through the reference and the sample

to be equal. The system was calibrated against pure aluminum

(99.99 wt.%) and pure copper (99.9998%), having a thermal

conductivity at 40 ◦C of 236 and 400 W/m K, respectively. The

temperaturegradient in thereferenceand in thesamplewas mea-

sured by three, respectively two thermocouples. Thus, linearity

of thetemperaturegradient in thereference could bemeasuredas

well; this was typically within±1%. Accounting also for uncer-

tainty on the geometry of the sample (diameter after grinding,

precise position of the thermocouples, thermal conductivity of 

the reference) we evaluate the overall uncertainty of the mea-

sured thermal conductivities to within  ±5% of the indicated

value.

4. Results

After infiltration the composite typically stuck to the surface

of the quartz tube, such that quartz pieces had to be ground

off. This indicates that there might have been some reaction

between the quartz crucible and aluminum. Metallographic cuts

performed on the remaining metal on top of the composite after

infiltration revealed indeed the presence of minor amounts of 

silicon-rich phase in the matrix, cf. Fig. 1a. Since no aluminum

carbide was observed around the SiC particles by light opti-

cal microscopy, reaction with the crucible rather than reaction

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486   J.M. Molina et al. / Materials Science and Engineering A 480 (2008) 483–488

Fig. 1. Optical micrographs of cross sections of different samples: remaining metal on top of the infiltrated preform after infiltration (a) and composites containing

67% (b), 100% (c) and 75% (d) of coarse particles.

with the SiC is most likely the origin of the observed Si in the

matrix.

Metallography shows a homogeneous structure for both the

monomodal and the bimodal composites, and also the effective-

ness of the sedimentation cleaning procedure, cf. Fig. 1b and c.

The only composite where inhomogeneity at the mesoscale was

observed is that with the 3:1 mixture of coarse to small particles.

In this composite, the quantity of small particles was not suffi-

cient to fill all the space left between touching coarse particles,

resulting in variable degree of filling of these spaces with small

particles, as shown in Fig. 1d.

Table1 reportsexperimentaldata forparticle volumefraction,

average SiC particle diameter, and electrical and thermal con-

ductivity of the composites based on monomodal particle size

distributions used in this work. It is seen that all preforms have

similar volume fractions, with a slight tendency towards largervalues at larger particle size. The results for the particle volume

fractionincompactsofmixturesof100and 500mesh particlesin

differentrelative amountsare shownin Table2. Thevolume frac-

tion is maximal for 67% of coarse particles. Table 2 also gives

the experimental values of thermal and electrical conductivity

in composites based on bimodal mixtures.

5. Discussion

The microstructures shown in Fig. 1b–d indicate that the

present composite materials have indeed a regular and well-

defined microstructure; as such these can serve as model

materials suitable for confrontation with theory. We first ana-

lyze the data of the series based on monomodal particle size

distribution.

In order to reducethenumberof “free”parameters to account

for the experiments, the thermal conductivity of the matrix was

independently deduced from the electrical conductivity of the

composite. Indeed, knowing that in metal–insulator composites

the electrical conductivity scales directly with the matrix con-

ductivity, one can back-calculate the electrical conductivity of 

the matrix using the differential effective medium scheme to

account for the non-dilute volume fraction of the electrically

insulating disperse phase. The DEM was indeed shown to be

well-suited for the prediction of composite electrical conduc-

tivity based on matrix conductivity at large volume fractions of 

insulatingdispersephase[24]. In theDEMscheme,with insulat-

Table 2

The main characteristics of bimodal composites materials (obtained from par-

ticles F100 and F500 of  Table 1):  V p   is the total particle volume fraction,

TC is the thermal conductivity (W/mK), and EC is the electrical conductivity

(106 −1 m−1)

Coarse (%)   V p (−) TC (W/m K) EC (106 −1 m−1)

0 0.55 193 8.14

25 0.62 215 5.4

50 0.69 220 4

67 0.74 228 2.72

75 0.72 225 2.82

100 0.58 221 5.9

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 J.M. Molina et al. / Materials Science and Engineering A 480 (2008) 483–488   487

Fig. 2. Linear regression of log(σ c) vs. log(1−V p) of composite electrical con-ductivity in order to determine   σ m   and n   to 2.84× 107 −1 m−1 and 1.70,

respectively.

ing inclusions, the composite conductivity, σ c, is simply linked

to the electrical conductivity of the matrix, σ m, by

σ c  = σ m(1− V p)n,   (7)

where n is a parameter reflecting the average particle shape, cf.

[24,25]. For typical angular particles produced by grinding,  n

takes a value of 1.6–1.75. For the specific case of an aspect ratio

of 1.9 measured in 2D micrographs one expects n≈ 1.62 [25].

The experimental results for the electrical conductivity of both monomodal and bimodal composites have been fitted with

Eq. (7) using  n  and   σ m  as parameters. The fitting was car-

ried out by linear regression in a plot of log(1 −V p) versus

log(σ c), given in Fig. 2. The values of the fitted parameters are

n = 1.70 and σ m≈ 2.84× 107 −1 m−1, to be compared with

n = 1.62 expected from metallography and the conductivity of 

pure aluminum, namely, 3.7× 107 −1 m−1 (had we taken the

Maxwellmean-fieldmodelinsteadof theDEMto back-calculate

the matrix electrical conductivity, the back-calculated matrix

conductivity would have been lower by a further 10–20%).

Compared to pure Al the electrical conductivity of thematrix

is thus approximately 20% too low. This can be rationalized by

the presence of Si in solid solution due to the reaction with thequartz crucible as mentioned above. Using data compiled by

Bass [26], this would correspond to a content of 1.0 at.% of Si

in solid solution in the Al matrix; this is somewhat high but not

implausible if one considers that some of the Si is present in

the form of precipitates. Using the Lorentz number for Al at

T = 300 K, namely, κ /(σ T )≈ 2.16× 10−8 W /K2 [27], we then

estimate thematrix thermal conductivity to beκm≈ 185 W/m K,

also about 20% below that of pure Al.

As shown in Fig. 3, the thermal conductivity of the compos-

iteswith a monomodalparticlesizedistributioncanbeaccounted

for by a single pair of parameters κinr   and h taking the values of 

253 W/m K and 7.5× 107

W/m2

K, respectively. These values

Fig.3. Experimentalthermalconductivityof theSiC/Alcompositesproduced byinfiltration of aluminum into preforms of monomodal SiC particles of variable

average size compared to predicted values based on the differential effective

medium scheme including correction to find effective inclusion conductivities

according to Eq. (1). The values taken for the intrinsic thermal conductivity of 

the inclusion, κinr  , and the thermal interface conductivity, h, are 253 W/mK and

7.5 107 W/m2 K, respectively.

are in agreement with data found in the literature [19,22,28], yet

– at least for the interfacial thermal conductance – at the lower

end of the reported spectrum.

We cannow confront thepredictionsbasedon Eq.(3) withthe

experimental data gathered on the bimodal composites. As can

beseen in Fig. 4, the parametersetdetermined beforegivequali-

Fig. 4. Thermal conductivity of the SiC/Al composites produced by infiltra-

tion of aluminum into preforms of bimodal SiC mixtures (SiC-500/SiC-100) vs.

the percentage of coarse particles. The circles are the experimental results; the

dashed line is the calculation using the values of  κinr   and h derived to fit all com-

posites with monomodal particle size distribution, while the full line represents

the prediction based on the deterministic pair κinr   and h that fits the conductivity

of monomodal composites with particle sizes used in the bimodal mixture.

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488   J.M. Molina et al. / Materials Science and Engineering A 480 (2008) 483–488

tatively the experimentally observed evolution as the fraction of 

coarseandsmall particlesvaries, yet theexperimentalvalues are

systematically larger thanthe calculated onesincluding thosefor

the twocomposites based on monomodalparticles. If we instead

use the – deterministic – values of  κinr   and h to fit the two com-

posites with monomodal particle size distribution of F100 and

F500, namely 259 W/m K and 1.05× 108 W/m2 K, respectively,

the predictions come in better agreement with the experimental

data, Fig.4. Thedifference between thetwosets of κinr   and h may

be due to the fact that the smallest monomodal powder (F800)

was included into first above analysis: this particular composite

hasa surprisingly lowthermalconductivity(seeFig.3). Onemay

argue that for low particle sizes the silicon content in the matrix

might behighersincetherewas more SiC-surface fordissolution

available and, hence, the matrix conductivity might be lowered

further; however, the electrical conductivity measurements do

not confirm this assumption.

Withthedataathandwemightspeculateaboutwhattheupper

limit of achievable thermal conductivity of Al–SiC at room tem-

perature might be. We first note that the size of the employedparticles is presumably at its upper technical limit. Hence, an

increase in the particle effective conductivity might be difficult

to achieve. Furthermore, this parameter is limited to the intrin-

sic conductivity of technical SiC powder, i.e. 250–260W/m K.

Assuming therefore that the thermal conductivity of the SiC

filler cannot be further improved and that the interface ther-

mal conductance between Al and SiC is indeed of the order

108 W/m2 K (in agreement with values calculated using the

acoustic mismatch model [29]), the largest potential is obvi-

ously in improving the matrix conductivity to the value of pure

aluminum. This could be done by avoiding alloying with sili-

con both by using appropriate crucible or mold material and byreducing the contact time between melt and particles. Both can

be achieved by squeeze casting such composites. If we assume

that thematrix conductivity canbe maintained at thelevel ofpure

aluminum we may then expect the thermal conductivity of the

Al–SiC material to reach 250W/m K. For higher performance,

either better conducting matrices (e.g. copper or silver) or filler

materials (e.g. cBN, graphite or diamond) must be employed.

6. Conclusions

(1) The thermal conductivity of Al/SiC composites based on

bimodal powder mixtures canreasonably well be accounted

for by using the differential effective medium scheme inwhich thetwo particle populations are treated as twodistinct

types of inclusion.

(2) The finite interface thermal conductance at the parti-

cle/matrix interface is then taken into account by attributing

to theinclusionsofdifferentsizesa distincteffective thermal

conductivity.

(3) Based on this analysis the peak for thermal conductivity of 

Al/SiC composites with bimodal particle sizes distributions

can be estimated as 250W/m K.

Acknowledgements

The authors acknowledge partial financial support from the

Spanish “Ministerio de Educacion y Ciencia” (grant MAT2004-

03139) the “Generalitat Valenciana” (grants GRUPOS03/092

and GRUPOS03/212) and the Universidad de Alicante. Work of 

LW and AM was funded by Laboratory core funding at EPFL.

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