Bimodal and Monomodal Diamond Particle Effect on the Thermal Properties of Diamond‐Particle‐Dispers

Journal of Metallurgical Engineering (ME) Volume 4, 2015 www.me‐journal.org doi: 10.14355/me.2015.04.001

Bimodal and Monomodal Diamond Particle Effect on the Thermal Properties of Diamond‐Particle‐Dispersed Silver Matrix Composite Fabricated by SPS Kiyoshi Mizuuchi*1, Kanryu Inoue2, Yasuyuki Agari1, Masami Sugioka1, Motohiro Tanaka1, Takashi Takeuchi1, Jun‐ichi Tani1, Masakazu Kawahara3, Yukio Makino4, Mikio Ito5 Materials Science & Processing, Osaka Municipal Technical Research Institute, Joto‐ku, Osaka, 536‐8553, Japan

1

Materials Science & Engineering, University of Washington, Box 352120, Seattle, WA 98195‐2120, USA

2

Kawahara SPS Technical Office, 2‐4‐25, Ohyabe, Yokosuka, 238‐0024, Japan

3

Graduate School of Science, Kyoto University, Kitashirakawaoiwake‐cho, Sakyo‐ku, Kyoto, 606‐8502, Japan

4

Center for Atomic and Molecular Technologies, Osaka University, 2‐1, Yamadaoka, Suita, 565‐0871, Japan

5

mizuuchi@omtri.or.jp; 2inoueryu@msn.com; 3kawahara@bk2.so‐net.ne.jp; 4cool‐sarada@leto.eonet.ne.jp ; 5ito@mat.eng.osaka‐u.ac.jp *1

Abstract Diamond‐particle‐dispersed silver (Ag) matrix composites consisting of monomodal and bimodal diamond particles were fabricated in spark plasma sintering (SPS) process, where the mixture of diamond, pure Ag and pure Si powders were consolidated in liquid and solid co‐existent state. Microstructures and thermal properties of the composites fabricated in such a way were investigated and the bimodal and monomodal diamond particle effect was evaluated on the thermal properties of the composites. The composites can be well consolidated in a temperature range between 1113 K and 1188 K and scanning electron microscopy detects no reaction product at the interface between the diamond particle and the Ag matrix. Relative packing density of the composite containing monomodal diamond particles decreased from 97.4% to 93.4% with increasing volume fraction of diamond between 50% and 60%, whereas that of the composite containing bimodal diamond particles was as high as 99~97% in a volume fraction of diamond up to 65%. The composite containing bimodal diamond particles revealed the thermal conductivity around 720 W/mk at 60 and 65% of the volume fraction of the diamond particles and it was higher than that of the composite containing monomodal diamond particles in a volume fraction of diamond higher than 55%. The coefficients of thermal expansion (CTEs) of the diamond‐particle‐dispersed Ag‐matrix composites fall in the upper line of Kerner model, indicating good bonding between the diamond particle and the Ag matrix in the composite. The CTEs of Ag‐matrix composites containing bimodal diamond particles are 6.76×10‐6/K at 62 vol.% diamond and 6.54×10‐6/K at 65 vol.% diamond.

Keywords Metal‐matrix Composites (MMCs); Particle‐reinforcement; Thermal Properties; Sintering; Diamond; Silver; SPS

Introduction High‐performance thermal management materials should have high thermal conductivities [1, 2] and low coefficients of thermal expansion (CTEs) for maximizing heat dissipation [3] and minimizing thermal stress and distortion, which are critical issues in packaging of microprocessors, power semiconductors, high‐power laser diodes, light‐emitting diodes (LEDs), and micro‐electro‐mechanical systems (MEMS) [4‐9].

Traditional packaging base materials are Cu‐W [10], AlN [11, 12], BeO [13] and Al/SiC [14‐16] composites and these all have thermal conductivities, λ, near 200 W/mK. The highest thermal conductivity material is good quality diamond, which is available as a naturally occurring material and contains nitrogen less than 100 ppm, and its thermal conductivity is around 2000 W/mK [17]. In metals, silver (Ag) has the highest thermal conductivity which

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is 419 W/mK [18].

Recently, high thermal conductive composite materials were fabricated by one research group [19, 20], which used a metal infiltration technique to fabricate a diamond‐particle‐dispersed Ag matrix composite. In their studies, the composite was fabricated at temperatures higher than the melting point of Ag matrix (whose melting point being Tm=1233 K and thermal conductivity being λ=419 W/mK) [19]. High processing temperature in these infiltration techniques is considered to have drawbacks, such as, reductions in thermal conductivity caused by structural alterations occurring to diamond particles by direct contact with molten Ag. In addition, it is practically difficult to disperse diamond particles uniformly in the composite when such techniques are used. This situation is especially true when the composite contains a lower volume fraction of diamond, say, less than 50 vol.%, due to big differences in mass density between diamond and Ag. To fabricate high‐performance thermal management materials with ultra‐high thermal conductivities and low CTEs, we have recently initiated a series of investigations, where diamond‐particle‐dispersed metal‐matrix composites [21‐23] were fabricated by spark plasma sintering (SPS) process [15, 24‐32].

In our previous work, diamond‐particle‐dispersed Ag‐matrix composites were fabricated in solid‐liquid co‐existent state by SPS process, and the composites were shown to have high relative packing density and good bonding between the Ag matrix and the diamond particles. It was shown that the thermal conductivity was 717 W/mK at a diamond volume fraction of 50vol.% [32]. It should be mentioned that, however, the thermal conductivity of Ag‐ matrix composite containing monomodal diamond particles decreases with increasing volume fraction of diamond particles when the volume fraction is higher than 50 vol.%. This decrease of thermal conductivity is mainly caused by drastic declines of the relative packing density of the composite for the diamond fraction higher than 50vol.%.

In order to make Ag‐matrix composites having not only high thermal conductivity but also low CTEs in a diamond fraction range higher than 50 vol.%, the composites containing bimodal diamond particles (hereafter called bimodal composites) were fabricated in the present work along with those composites containing monomodal diamond particles (hereafter called monomodal composites). In the present fabrication, a solid–liquid co‐existent state [32] was created during SPS process. Process temperatures employed were lower than the melting point of Ag and SPS process parameters were optimized. Experimental results of thermal properties obtained from the monomodal and bimodal Ag‐matrix composites are reported in the present study along with microstructure observations of the composites fabricated under various process parameters. The packing density, the thermal conductivity and the CTEs of the composites are measured and the bimodal and monomodal diamond particle effect on the thermal property of the composites is evaluated.

Experimental Procedure According to three‐dimensional Euclidean space, the maximum volume fraction of diamond particles in Ag‐matrix composite can ideally be 74 vol.% if diamond particles are spherical and of uniform size. Due to some experimental constraints, Ag‐matrix composites are commonly produced with around 50 vol.% of diamond when roughly mono‐sized diamond particles are used. In the present study, two sizes of Ib type synthetic diamond powders (IMS‐15 diamonds with a mean particle diameter of 310μm and IMM diamonds with a mean particle diameter of 34.8μm, Tomei Diamond Co., Ltd) were employed as a filler material and their thermal conductivity of the synthetic diamonds are estimated to be about 2000 W/mK. As a matrix material, two kinds of powders, pure‐ Ag powder (99.9 % purity, 2μm in size, Koujundo Chemical Laboratory Co., Ltd.) and pure‐Si powder (99 %, 200 mesh, Koujundo Chemical Laboratory Co., Ltd.), were prepared. Scanning electron micrographs of large diamond, small diamond, pure‐Ag and pure‐Si powders used in the present study is shown in Fig.1.

Pure‐Si powder is important to use in the present study to create a solid‐liquid co‐existent state during SPS processing. We have developed unique SPS fabrication techniques to make metal‐matrix composites with or without dispersed diamond particles. In this developed method, specially designed powder mixtures were prepared to create solid‐liquid co‐existent state during SPS processing. That is, in the present study, Si powder is mixed with a mixture of Ag and diamond powders to decrease the melting point of Ag in an attempt to create a solid‐liquid co‐existent state during SPS processing. A binary phase diagram of Ag and Si shows that their eutectic point is 1113K at a chemical composition of Ag‐10.6 at.% Si [20]. To keep the solidus ratio to the matrix more than

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80 % during SPS, a Ag‐based powder mixture composed of 98.94 at.% (i.e., 98.76 v/o) of pure Ag powder and 1.06 at.% (i.e., 1.24 v/o) of pure Si powder was prepared as a matrix material in the present study. We believe that keeping the solidus ratio to the matrix more than 80% is important to avoid the invasion of molten Ag‐Si alloy into gaps between the die and the punch.

(a)

(b)

(c)

(d)

FIG. 1 SEM MICROGRAPHS OF AS‐RECEIVED (a) DIAMOND (310μm), (b) DIAMOND (34.8μm), (c) PURE Ag AND (d) PURE Si POWDERS, SHOWING THE MORPHOLOGY OF THE POWDERS USED IN THE STUDY. 40

Porosity, ε(%)

30

20

13.72% 10

76.15vol.% 0 0

20

40

60

80

100

Volume fraction of large particle. Sv1(%)

FIG. 2 CHANGE IN THE POROSITY OF THE DIAMOND POWDER MIXUTURE COMPOSED OF LARGE PARTICLES (310μm IN DIAMETER) AND SMALL DIAMOND PARTICLES (34.8μm IN DIAMETER) AS A FUNCTION OF THE VOLUME FRACTION OF LARGE DIAMOND PARTICLES IN THE DIAMOND POWDER MIXTURE. THE MIXTURE WAS TAPPED ONE HUNDRED TIMES PRIOR TO POROSITY MEASUREMENT.

The mixing ratio of large and small diamond particles is important in bimodal diamond particle mixture to achieve the high packing density of Ag‐matrix composites containing diamond particles. In the present study the optimum mixing ratio was determined as follows. When one puts a bimodal powder mixture into a container and tap it, the porosity of the bimodal powder mixture can be evaluated using Furnas model [33], which is described below.

Assuming (a) large particles are much larger in diameter than small particles, (b) the mass density of large particles is equal to that of small particles, and (c) small particles fill up the gaps between large particles, the porosityε of the bimodal powder mixture can be calculated in the following equation (1)

  1

1  1 (1) v1

where ε1 is the porosity when large particles are tapped into a container and v1 the volume fraction of large

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particles in a bimodal powder mixture composed of large and small particles.

Also assuming (d) larger particles are intermingles with small ones, the porosity of the bimodal powder mixture can be calculated in the following equation (2)

  1

1 2 (2) 1  v1   2

where  2 is the porosity when small particles are tapped into a container.

In Furnas model [33] every v1 value is proposed to choose a larger value of porosity between ones calculated in equation (1) and in equation (2). In the present study, experimentally obtained ε1 and ε2 values were 34.3% and 40.0%, respectively, after a hundred times of tapping. Using these ε1 and ε2 values, the porosity (ε) of a bimodal diamond powder was calculated and the result obtained is shown in Fig. 2 as a function of v1. As seen, when the volume fraction of large diamond particles in the bimodal diamond powder is 76.15%, the porosity of the bimodal diamond powder becomes minimum and it is 13.72%. From this result, in the present study, a bimodal diamond powder mixture composed of 76.15 vol.% of large diamond particles and 23.85 vol.% of small diamond particles was used to try to obtain the high relative packing density of Ag‐matrix composites containing diamond. Relative packing density is defined to be the ratio of the density of a material measured experimentally to the density of the material assumed to have no porosity.

Powder mixtures composed of four kinds of powders (large diamond, small diamond, pure‐Ag and pure‐Si powders) were prepared as starting materials, and these powder mixtures with various volume fractions of diamond were processed to fabricate Ag‐matrix composites using an SPS equipment, Model SPS SYNTEX DR SINTER 1020. The equipment consists of an electric power source and a pressure system, and the fabrication of composites is carried out immediately after powder mixtures are placed in a graphite die‐set. Schematic illustration showing processing of diamond particle dispersed Ag‐matrix composite by spark plasma sintering (SPS) is depicted in Fig. 3.

Pressure Graphite punch

Pulsed current

Graphite die

Graphite die

Diamond particle K-t ype the rmo couple

Si particle

Ag particle Graphite punch Pressure

Si particle dispersed Ag matrix

Diamond particle dispersed Ag matrix composite

K-t ype the rmo couple

Diamond particle

FIG. 3 SCHEMATIC ILLUSTRATION SHOWING PROCESSING OF DIAMOND PARTICLE DISPERSED AG‐MATRIX COMPOSITE BY SPARK PLASMA SINTERING (SPS). OPEN HEXAGONS REPRESENT DIAMOND PARTICLES; GREH MUSHY AREAS Ag PARTICLES; DARK CIRCLES Si PARTICLES; AND DARK DOTED AREAS Si‐PARTICLE‐DISPERSED Ag MATRIX.

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For making one disk‐shaped composite specimen with gauge dimensions of 2.5 mm thick and 10 mm in diameter, about 1.5 g of the mixture of diamond, pure‐Ag and pure‐Si powders was stirred up with a small amount of ethyl alcohol and placed carefully in the above described die‐set, and consolidation was carried out. The gauge dimensions of the graphite die used in the present study are φ35 mm in outer diameter, φ10mm in inner diameter and 40 mm high. During SPS process, die temperature was monitored by a thermo‐couple that was inserted in a hole located in a position of 5 mm distant from the inner surface of the die. This thermo‐couple hole was drilled on the outside surface of the die with dimensions of 2 mm in diameter and 7.5 mm deep.

The SPS process employed is described in detail as follows. First, the lower punch was inserted into the die from its one end, followed by pouring of 1.5 g of the powder mixture containing diamond particles. Then, the upper punch was placed onto the powder mixture. The powder mixture was pressed at a pressure of 80MPa through the upper and lower punches, and then spark plasma sintering was performed after evacuation to a vacuum of 2 Pa. Sintering was carried out first at a heating rate of 1.67 K/s from room temperature to 1113 K. Heating to 1188 K was then made at a rate of 0.17 K/s. To visualize a change in the relative packing density of each composite during the SPS process, longitudinal displacements between the punches were monitored during processing. The discharged pulsed current used was about 320 A and sparking was generated at a constant interval of approximately 2.78×10‐3 s. Disk‐shaped composites thus fabricated were, then, examined by scanning electron microscopy using a JEOL JSM 6460 LA microscope. Disks of the Ag‐matrix composite containing diamond were gold‐coated and their thermal diffusivities were measured by a laser flash technique using an Netzsch LFA‐457 thermal constants analyzer. The achieved packing density of composite disks fabricated was measured by Archimedes method for optimization of process parameters and for calculation of thermal conductivities.

Coefficients of the thermal expansion, CTEs, of the composites were investigated on square‐column‐shaped specimens, 4x4x15 mm, fabricated from the powder mixture of diamond, Ag and Si by SPS under a similar condition mentioned above. The thermal expansion of the composites was measured with a Rigaku TMA8310 dilatometer in a N2 flowing atmosphere with a heating ratio of 0.05 K/s in a temperature range between 243 and 483 K. The CTEs of the composites at room temperature were determined from the slope of the thermal expansion against temperature curve in a temperature range between 288 and 298 K. Results and Discussion Densification of Composites during Fabrication in Solid‐Liquid Co‐Existent State by SPS Specially designed powder mixtures prepared in the present study were composed of Ag powder and Si powder corresponding to Ag‐1.06 at% Si with various volume fractions of diamond, and these powder mixtures were processed to fabricate Ag‐matrix composite in SPS processing. We believe that the powder mixtures of Ag and Si particles will change its microstructure at various temperatures during SPS processing. To visualize such microstructural changes, a partial phase diagram of Ag‐Si alloy is illustrated in Fig. 4, along with the relevant microstructural changes, as depicted schematically in Fig. 5. In Fig. 4, grey arrows (1) and (2) indicate changes both in temperature and in alloy composition for molten Ag‐Si alloy during SPS processing. The microstructure of the powder mixture composed of diamond, Ag and Si at room temperature prior to SPS is drawn in Fig. 5(1). When the process temperature reaches 1113K, the eutectic point of Ag‐Si alloy, by heating during SPS, partial melting of the powder mixture occurs through alloying of all Si particles with Ag particles. As a result, the matrix phase in the powder mixture becomes solid‐liquid co‐existent state, which is composed of 90 v/o of solidus Ag particles and 10 v/o of molten Ag‐10.6 at.% Si eutectic alloy, as shown in Fig. 5(2). When the powder mixture is heated up from 1113K to 1188K, the Si content in the liquidus matrix phase and the ratio of the solidus matrix phase to the liquidus phase both decrease in a monotonic manner, as shown by the grey arrow (1) in Fig. 4. When the powder temperature reaches 1188K, the matrix phase in the powder mixture becomes solid‐liquid co‐existent state composed of 80 v/o of solidus Ag particles and 20 v/o of molten Ag‐5.3at.%Si alloy, as shown in Fig. 5(3). During the sintering stage at temperatures between 1113K and 1188K, such a liquidus phase infiltrates into pores between the diamond particles and Ag particles. As a result, those pores present in the powder mixture prior to SPS processing are filled up. When the temperature of the powder mixture reached 1188K, pulsed current is shut down and the powder mixture is furnace cooled. At a following cooling stage from 1188K to 1113K, both the ratio of the solidus versus the liquidus in the powder mixture and the Si content in the molten Ag‐Si alloy increase, as shown

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by grey arrow (2) in Fig. 4. When the powder temperature becomes lower than 1113K, the molten Ag‐Si alloy is solidified and becomes solidus Ag‐matrix containing Si precipitates, resulting in the formation of metal matrix composite composed of diamond particles and Ag matrix containing Si precipitates, Fig. 5(4).

1323

Temperature, T /K

1273 1223

Ag‐5.3at.%Si(T=1188K) ↓

1173

( 2) ( 1)

1123

1113K

(Ag-1.06at.%Si)

(Ag-10.6at.%Si)

1073 1023 973

Ag

0

5

10 at.% Si

15

20

FIG. 4 A PARTIAL PHASE DIAGRAM PARTIAL PHASE DIAGRAM OF Ag‐Si ALLOY. GRAY ARROWS INDICATE CHANGE IN TEMPERATURE AND ALLOY COMPOSITION OF LIQUID PHASE OF Ag‐Si ALLOY DURING SPS.

(1)T=R.T.

(2)T=1113K

(3)T=1188K

(4)T＜ 1113K

: Diamond particle

: Ag particle

: Si particle

: The mixture of 90vol.% solidus Ag and 10vol.% liquidus Ag-10.6at.%Si alloy : The mixture of 80vol.% solidus Ag and 20vol.% liquidus Ag-5.3at.%Si alloy : Ag matrix containing Si precipitates

FIG. 5 CHANGE IN THE STRUCTURE OF THE POWDER MIXTURE COMPOSED OF DIANOND, Ag AND Si PARTICLES AT VARIOUS TEMPERATURES DURING SPS.

Relative Packing Density of Composites Fabricated by SPS The relative packing density of the diamond‐particle‐dispersed Ag‐matrix composites fabricated was measured by Archimedes method, and results are shown in Fig. 6 as a function of diamond volume fractions. As seen, the relative packing density of the monomodal composite is higher than 98% in a range of diamond volume fraction lower than 45%. In a range of diamond volume fraction higher than 50%, on the other hand, the relative packing density drastically decreased with increasing volume fraction of diamond. That is, the relative packing density of

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Ag‐55vol.% and Ag‐60% diamond composite decreased to 95.5% and 93.4%, respectively. In the case of the bimodal composite maintained the relative packing density was shown to be as high as 99~97% in a range of diamond volume fraction up to 65%. The present results evidently indicate that the consolidation of the composite containing the bimodal diamond particles is very effective in filling up pores present in metal‐matric composite containing diamond particles.

Relative packing density, Dr (%)

100

95

90

85 Monomodal Bimodal

80 30

40

50

60

70

Volume fraction of diamond, Vf (%)

80

FIG. 6 CHANGES IN RELATIVE PACKING DENSITY OF AG‐MATRIX COMPOSITE WITH DISPERSED DIAMOND PARTICLES FABRICATED BY SPARK PLASMA SINTERING AS A FUNCTION OF THE VOLUME FRACTION OF DIAMOND IN THE COMPOSITE.

Microstructure The mixtures of powders composed of pure Ag, pure Si and diamond were compacted, consolidated and sintered in order to fabricate diamond‐particle‐dispersed Ag‐matrix composites by SPS process. The microstructure of as‐ sintered monomodal and bimodal composites containing 60 vol.% diamond was investigated by scanning electron microscopy. Figure 7 is scanning electron micrographs taken from cross sections of diamond‐Ag composites that were consolidated in solid‐liquid co‐existent state by SPS. As seen in Fig. 7(a) and (c), there are some cavities on interfacial boundaries between the diamond particle and the Ag matrix of the monomodal composite. On the other hand, cavities on the interfacial boundary between the diamond particle and the Ag matrix of the bimodal composite decrease considerably as shown in Fig. 7(b) and (d). It should be emphasized that the decrease of cavities or porosities in the latter composite is due probably to the effective filling up of gaps between large diamond particles by small diamond particles.

In the present study, no interfacial reaction layer on the boundary between the diamond particle and Ag matrix was detected in the diamond composites by scanning electron microscopy, as shown in Fig. 7(b) and (d). Further detailed microstructure investigation is underway by scanning transmission electron microscopy (STEM) with EDX analysis to clarify whether or not the interfacial reaction layer is formed on such interfaces in the composites fabricated in the present study. The results will be soon presented elsewhere [34]. (a)

(b)

(c)

(d)

FIG. 7 SEM MICROGRAPHS OF Ag‐60vol.% DIAMOND COMPOSITES FABRICATED BY SPARK PLASMA SINTERING (SPS).

MICROGRAPHS FOR MONOMODAL ARE (a) AND (c), AND FOR BIMODAL (b) AND (d).

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Thermal Conductivity Thermal conductivity has been evaluated for various materials including metal‐matrix composites [35], polymer composites [36], ceramics [37‐39] and foods, using the following Hasselman‐Johnson equation (3) [40] kd 2k d  kd k d   2( k  ah  1)Vd  k  ah  2  m c m c  (3) k  km  k d kd k d 2k d   (1 ) V 2      d   k ah k ah m c m c  

where, k is the thermal conductivity of the composite, k d the thermal conductivity of dispersions (in this case, diamond, 2000 W/mK), k m the thermal conductivity of the matrix (in this case, silver, 419 W/mK), Vd the volume fraction of dispersions, “a” the radius of dispersions (in this case, diamond, 155μm) and hc the boundary conductance.

Assuming perfect bonding between the diamond particle and the Ag matrix in the composite, the hc value become ∞ and equation (3) is simplified to Maxwell‐Eucken equation (4) proposed by Eucken [41]. kd  kd   2( k  1)Vd  k  2  m m  k  km   kd kd   V     (1 ) 2 d   k k m m  

(4)

Thermal conductivities were measured for diamond‐particle‐dispersed Ag‐matrix composites fabricated by SPS process and the results obtained were plotted in Fig. 8 as a function of the volume fraction of diamond particles. In the present study, equation (4) is also utilized to evaluate the effect of consolidation for the Ag‐matrix composite.

In Fig. 8 depicted are theoretically calculated values of thermal conductivity for the diamond‐dispersed Ag‐matrix composite using the Maxwell‐Eucken equation for cases with various percentages of conductivity as opposed to the 100% theoretical value. As seen, at 40, 45 and 50 vol.% of the volume fraction of diamond, the thermal conductivity of the monomodal composite was 648, 661 and 717 W/mK, respectively. The thermal conductivity of the composite increased with increasing volume fraction of diamond and fell in higher than 80% the theoretical value calculated using Maxwell‐Eucken equation in a diamond fraction range between 40% and 50%. It should be emphasized that this high value of the thermal conductivity indicates that the thermal conductance between diamond particles and the Ag matrix is high and that damages in the surface region of the diamond particles are small or negligible during consolidation under solid‐liquid co‐existent state in SPS processing. Similar results were also found in our previous works [24]. In the case for the present monomodal composite, however, the thermal conductivity decreased with increasing volume fraction of diamond in a fraction range higher than 50%. This decrease in the thermal conductivity is considered to be caused by reduction in the relative packing density of the composite with a diamond fraction higher than 50 vol.%.

Thermal conductivity, λ/(W/mK)

1500

Monomodal Bimodal 100% 90% 80% 70% 60% 50%

1200

900

600

300

0 35

45

55

65

Volume fraction of diamond, Vf (%)

75

FIG. 8 EEPERIMENTAL MEASUREMENTS AND THEORETICAL VALUES OF THE THERMAL CONDUCTIVITY DEPENDENCE ON THE VOLUME FRACTION OF DIAMOND FOR DIAMOND‐PARTICLE‐DISPERSED Ag‐MATRIX COMPOSITE. SOLID AND DOTTED LINES ARE VARIOUS PERCENTAGES OF THE THEORETICAL VALUES CALCULATED FROM MAXWELL‐EUCKEN EQUATION.

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The thermal conductivity of the bimodal composites at 55, 60, 65 and 70 vol.% of the volume fraction of diamond was obtained and they are 678, 723, 711 and 576 W/mK, respectively, which is shown in Fig. 8. As seen, the thermal conductivity of the bimodal composite increased with increasing volume fraction of diamond in a diamond fraction range between 55 and 60 vol.%. This high thermal conductivity of the composite in a diamond fraction range higher than 50 vol.% is primarily caused by the high relative packing density of the composite associated with a uniform bimodal diamond particle effect. It should be pointed out that the thermal conductivity higher than 650 W/mK is maintained in the Ag‐matrix composites containing both monomodal diamond particles with a diamond fraction from 40 to 50 vol.% and bimodal diamond particles with a diamond fraction range between 55 and 65 vol.%.

As shown above, the thermal conductivities for the bimodal composites are considerably high in a diamond fraction range between 55 and 65 vol.% when compared to those for the monomodal composite. These values are, however, far below the predictions calculated by Maxwell‐Eucken equation, as shown in Fig. 8. This low thermal conductivities of the bimodal composite would be caused by the following three reasons: 1) the increase of interfacial regions between diamond particles and the Ag matrix, which is caused by the increase of the volume fraction of small diamond particles in the composite; 2) the thermal conductivity of small diamond particles is lower than that of large diamond particles due to the lower purity in the small diamonds; and 3) lower thermal conductance hc between small diamond particles and the Ag matrix than that between large diamond particles and the Ag matrix. That is, in the present experimental setup, higher thermal conductivities would be obtained for diamond particle dispersed Ag matrix bimodal composite by using higher purity for small diamond powder. It should be emphasized here, however, that in the present study we have obtained the thermal conductivity of the bimodal composite close to or higher than 700 W/mK for a diamond fraction range between 55 and 65 vol.%.

Coefficients of Thermal Expansion Two models by Turner [42] and by Kerner [43] have been proposed to calculate theoretical coefficients of thermal expansion (CTEs) of a composite material [44‐48]. Kerner model takes shear effects into account at the boundaries between particles and the matrix, whereas Turner model does not take such effects into account. Consequently, Kerner model has widely been used for theoretical evaluations of the CTEs of composite material. Hence, we also utilized Kerner model in the present study to evaluate CTEs of the composites fabricated. Kerner model can be expressed in equation (5),

 c   m .Vm   d Vd  Vd Vm ( d   m ) 

Kd  Km (5) Vm K m  Vd K d  (3K d K m / 4Gm )

whereαc is the coefficient of thermal expansion of the composite, αd the coefficient of thermal expansion of the dispersions (in this case, diamond, 2.3×10‐6/K), αm the coefficient of thermal expansion of the matrix (in this case, silver, 19.6×10‐6/K), Vd the volume fraction of the dispersions in the composites, Vm the volume fraction of the matrix in the composites, Kd the bulk modulus of the dispersions (in this case diamond, 5.8×1011 Pa), Km the bulk modulus of the matrix (in this case, silver, 9.5×1010 Pa) and Gm the shear modulus of the matrix (in this case, silver, 2.78×1010 Pa), The comparison between theoretically predicted and experimentally obtained CTEs for the diamond‐particle‐ dispersed Ag matrix composites is shown in Fig. 9. In the figure, two lines are drown which were obtained by reversing the role of matrix and dispersions in Kerner model. As seen, experimentally obtained CTEs of the composites fall in the upper line of Kerner model. This is a good indication to show that the diamond particles have bonded well with the Ag matrix in the composite fabricated by SPS. The CTEs of the Ag‐matrix composites containing monomodal diamond particles obtained in the present study are 10.1×10‐6/K at 44 vol.% diamond and 8.97×10‐6/K at 51 vol.% diamond. On the other hand, the CTEs of Ag‐matrix composites containing bimodal diamond particles are 6.76×10‐6/K at 62 vol.% diamond and 6.54×10‐6/K at 65 vol.% diamond. It should be emphasized that the low CTEs obtained in the present bimodal composites are very promising as packaging base materials in terms of CTEs since the CTEs of currently used packing materials are 4~8×10‐6/K [10‐12]. Ag‐matrix composite with diamond particles higher than 57 vol.% are considered to be very promising to be used as a heat spreader.

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Coeeficient of thermal expansion, α/(K-1×10-6)

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20 Turner line Kerner lower line Kerner upper line

15

Experimental(Monomodal) Experimental(Bimodal)

10

5

0 0

20

40

60

80

Volume fraction of diamond, Vf (%)

100

FIG. 9 COMPARISON OF COEFFICIENTS OF THE THERMAL EXPANSION PREDICTED THEORETICALLY USING KERNER MODEL FOR Ag AND DIAMOND COMPOSITE TO THOSE OBTAINED EXPERIMENTALLY FOR Ag AND DIAMOND COMPOSITE CONTAINING 44, 51, 62 AND 65 vol.% DIAMOND PARTICLES. KERNERʹS UPPER LINE IS OBTAINED BY CONSIDERING Ag AS THE MATRIX OF THE COMPOSITE, WHEREAS ITS LOWER LINE IS FOR THE DISPERSED DIAMOND AS THE MATRIX. FOR COMPARISON, A THEORETICALLY‐OBTAINED CURVE USING TURNER MODEL IS ALSO SHOWN.

Summary Two kinds of diamond‐particle‐dispersed silver (Ag) matrix composites containing monomodal and bimodal diamond particles were fabricated in spark plasma sintering process, where the mixture of diamond, pure Ag and pure Si powders were consolidated in liquid and solid co‐existent state. Microstructures and thermal properties of the composites fabricated in such a way were investigated and the bimodal and monomodal diamond particle effect on the thermal properties of the composites was evaluated. The composites can be well consolidated in the temperature range between 1113 K and 1188 K and scanning electron microscopy detects no reaction product at the interface between the diamond particle and the Ag matrix. The relative packing density of the composite containing monomodal diamond particles decreased from 97.4% to 93.4% with increasing diamond volume fraction in a volume fraction range between 50 and 60 %. However, the relative packing density of the composite containing bimodal diamond particles was as high as 99~97% in a diamond volume fraction range up to 65%. The thermal conductivity of this composite was higher than that of the composite containing monomodal diamond particles in a diamond volume fraction higher than 55%. The coefficients of the thermal expansion (CTEs) of Ag‐ matrix composites containing diamond particles fell in the upper line of Kerner model, indicating good bonding between the diamond particle and the Ag matrix in the composite. The CTEs of Ag‐matrix composites containing bimodal diamond particles were 6.76×10‐6/K at 62 vol.% diamond and 6.54×10‐6/K at 65 vol.% diamond.

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