Preparation of W-Plated Diamond and Improvement of Thermal Conductivity of Diamond-WC-Cu Composite

Abstract

The tungsten (W)-plated diamond process was explored and optimized. A dense and uniform tungsten coating with a thickness of 900 nm was successfully prepared by the powder covering sintering method. The Diamond-WC-Cu composite with high density and high thermal conductivity were successfully prepared by cyclic vacuum pressure infiltration. The microstructure and composition of the W-plated diamond particles were analyzed. The effect of tungsten coating on the microstructure and thermal conductivity of the Diamond-WC-Cu composite was investigated. After calculation, the interface thermal resistance of the composite forming the tungsten carbide transition layer is 2.11 × 10⁻⁸ m²∙K∙W⁻¹. The thermal conductivity average value of the Diamond-WC-Cu composite with a diamond volume fraction of 60% reaches 874 W∙m⁻¹∙K⁻¹, which is close to the theoretical prediction value of Hasselman-Johnson (H-J) model and differential effective medium (DEM) model. Moreover, the Maxwell-Eucken (M-E) model, H-J model, and DEM model were used to evaluate the thermal conductivity of the Diamond-WC-Cu composite.

1. Introduction

With the rapid development of microelectronic technology and aerospace technology, the integration of semiconductor circuits is becoming higher and higher. Electronic components are developing in the direction of miniaturization, lightweight and integration, which greatly increases the power intensity. A large amount of heat will directly affect the operation stability, security, and reliability of electronic equipment [1,2,3,4]. Traditional heat dissipation materials, such as Al, Cu, Invar, Kovar, Mo/Cu, and W/Cu, can no longer meet the high-end heat dissipation requirements [5,6,7,8]. Therefore, it is very essential to research and develop new generation heat dissipation materials with higher thermal conductivity (TC).

As the third-generation thermal management material, metal matrix composites have become a research hotspot. Red copper is widely used in the field of thermal management due to its high thermal conductivity (400 W∙m⁻¹∙K⁻¹). However, the thermal conductivity of copper is not enough for high-power electronic components. Among natural materials, the thermal conductivity of diamond is the highest, which can reach 2000 W∙m⁻¹∙K⁻¹ at room temperature. The thermal expansion coefficient of diamond is as low as 1.2 ppm/K. It is very suitable for application in the field of thermal management. Nevertheless, the high cost and difficult processing hinder the independent application of diamond. However, as an ideal additive material, diamond is widely used in the research of composite. As a kind of “strong-strong bonding” effect, diamond particle reinforced copper matrix composites get more application opportunities, and diamond/copper composites are also considered as the representative of the new generation of thermal management materials [9,10]. However, there are two major issues in preparing diamond reinforced copper matrix composites with excellent properties. Firstly, for the non-wetting property of liquid copper to diamond, the contact angles is 122–129° [11]. Poor wettability leads to weak interface bonding, which seriously affects the comprehensive thermodynamic properties of composite materials. Secondly, due to the different density between diamond and copper, the distribution of diamond is not uniform.

In order to obtain high-performance diamond reinforced copper matrix composites and improve the interfacial bonding between diamond and copper, it is common to introduce strong carbide forming elements into the copper matrix or modify the surface of the diamond. A simple method is to uniformly disperse Ti [9,12], Cr [13], B [14,15], and Zr [16,17] powder into copper powder. In the process of sample preparation, alloying elements will diffuse in the liquid copper to the diamond surface and react with carbon atoms on the surface of diamond to form carbides, which enhances the interface bonding state and greatly improves the wettability of the copper matrix to the diamond. However, the amount of alloying elements is hard to control. If the alloying elements remain in the copper matrix, the thermal conductivity of the diamond/copper composites will decrease sharply. Another method is the surface modification of diamonds. Carbide is formed on the surface of diamond by a special process. The formation of the carbide transition layer can effectively improve the thermal properties of composites and avoid the negative effects of adding alloying elements. Ti [18,19], Cr [20], Mo [21], and W [22,23] have been coated on the diamond surface to effectively enhance the interface bonding of the composite. Among the above surface modification elements, WC has the highest thermal conductivity. The order of thermal conductivity is: WC (120 W∙m⁻¹∙K⁻¹) > Mo₂C (21 W∙m⁻¹∙K⁻¹) > Cr₇C₃ (20 W∙m⁻¹∙K⁻¹) > Cr₃C₂ (19 W∙m⁻¹∙K⁻¹) > TiC (17 W∙m⁻¹∙K⁻¹). Therefore, this article chooses tungsten as the metallization element on the diamond surface.

In this paper, tungsten alloying was carried out on the diamond surface by the powder covering sintering method and salt bath plating method. The effects of the two methods on the alloying of tungsten on the diamond surface were analyzed and compared. The formation mechanism of WC coating in composites was analyzed. It has been proved that the formation of WC transition layer can enhance the interface bonding between diamond and copper and reduce the interface thermal resistance. The dense Diamond-WC-Cu composites were successfully prepared by cyclic vacuum pressure infiltration. The thermophysical properties and microstructure of Diamond-WC-Cu composites were investigated and evaluated. The composite can be used as a thermal management material.

2. Experimental Procedures and Characterization

2.1. Raw Materials

Artificial single crystal diamond (140/170 mesh, average particle size 100 μm, purchased from Henan Huanghe Whirlwind Co., Ltd., Changge, China) was used as the reinforced phase. The WO₃ powders (T818832^#, 7.16 g/cm³ at 25 °C (lit.), 800 mesh, 99.99% metals basis, Shanghai Macklin Biochemical Co., Ltd., Shanghai, China) were used for surface modification of diamond particles. Pure copper powder (300 mesh, average particle size 50 μm, 99.5% purity) and tungsten powder (99.5% purity, average particle size 50 μm) were purchased from Beijing Xingrongyuan Technology Co., Ltd., Beijing, China. In addition, the aqueous solution of polyvinyl alcohol (PVA, model 17-99) was used as a binder and was purchased from Shandong Yousuo Chemical Technology Co., Ltd., Qingdao, China.

2.2. Composite Material Preparation

The whole preparation process of the composite material is shown in Figure 1. The diamond surface needed to be cleaned and roughened before modification. The purpose of cleaning was to remove organic impurities on the diamond surface. The roughening treatment facilitated the diamond surface to form a tungsten layer. The diamond particles were placed in a 10 wt% NaOH aqueous solution, continuously stirred and boiled for 15 min, and then rinsed repeatedly with deionized water. Then, the treated diamond particles were put into a 30 wt% dilute HNO₃ aqueous solution. After boiling and stirring for 30 min, the liquid was removed. The diamond particles were repeatedly cleaned with deionized water. Then, put into the drying box to dry at 100 °C for later use.

The process of the powder covering sintering method for surface modification of the diamond was as follows: According to the mass ratio, the diamond particles, WO₃ powder, and W powder were processed in a mixer for 3~5 h and then placed in a corundum burning boat, under a vacuum condition (4~6 Pa), heated up to 900~1100 °C temperature at a rate of 10 °C/min, and kept for 30~90 min. After the heating was stopped, the sample was cooled to room temperature in the furnace. Subsequently, the diamond in the mixture was separated with a 180-mesh sieve and washed repeatedly with deionized water to remove fine solid powder residues. Finally, diamond particles with a tungsten layer were obtained. Compared with the powder covered sintering method, the salt bath plating process included: (1) Mixed NaCl and KCl solid particles in a molar ratio of 1:1; (2) weight diamond and WO₃ powder at a molar ratio of 10:1, mixed evenly, and placed at the bottom of the corundum burn boat. The surface was covered with mixed salt particles of equal mass; (3) the heating temperature of the equipment was set to 900~1100 °C, the temperature was kept for 30~90 min, the heating rate was controlled within 10 °C/min, and high-purity Ar gas was used as the protective atmosphere; (4) after the holding time was over, the heating power supply was turned off, the crucible was taken out after cooling the sample to room temperature, the mixed salt was dissolved in hot water, and the residual molten salt was repeatedly washed and removed on the diamond surface to obtain W-plated diamond particles.

Then, the W-plated diamond/copper preform was prepared. The surface-modified diamond particles were weighed according to the calculated amount, and 1~2 wt% of the aqueous solution of polyvinyl alcohol was added in drops using a dropper. Subsequently, the mixture was placed in a mortar, grinded evenly, and then moved into a high purity graphite mold to be vibrated and paved. The pressing pressure was set to 8.5 MPa, and the pressure was maintained for 2 min. After prepressing, the graphite mold filled with diamond preform was dried for 4~5 h in a constant temperature (150 °C) drying oven for pre-degreasing to obtain a porous diamond preform with a certain strength. This method could avoid the problems that the prefabricated body was broken during the transfer process and was difficult to completely fit the mold. The experiment directly used compacted copper powder as the matrix infiltration material. During the pressing process, the pressure of the copper powder was 80 MPa, and the pressure was kept for 5 min. The copper body had good strength and was not easy to break. It could be directly used as the pressure infiltration matrix material.

As shown in Figure 1, the infiltration mold had a ladder structure. From top to bottom, there were Cu bodies and W-plated diamond particles. During the experiment, the W-plated diamond particles must be compacted and flattened before placing the Cu bodies. There were holes in the infiltration mould, which were convenient for gas circulation. The purpose of the experimental design was to maintain a certain pressure difference between the upper and lower liquid surfaces of the copper liquid layer during the infiltration stage. This could reduce the residual gas in the reinforced particles of the composite and eliminate the pore defects. Circulating the vacuum pressure had the same effect. The infiltration process was carried out in the SGL1700 vacuum tube furnace (Shanghai Daheng Optics and Fine Mechanics Co., Ltd., Shanghai, China). After the system was heated to the infiltration temperature (1200 °C), the vacuum pump was turned off, filled with argon gas to 0.5 MPa, the pressure was maintained for 10 min, and then vacuum infiltrated, after which the ultimate vacuum was maintained for 5 min, and the circulating vacuum pressure infiltration lasted for 1 h. The vacuum pumping during the infiltration is helpful to remove gas from the diamond body. The argon is used to make the upper surface pressure of the copper melt greater than the lower surface pressure and promote the penetration of the molten copper into the diamond body. Through the “vacuum-argon gas” process cycle, the final composites containing 50 to 70 vol% diamond were successfully prepared.

2.3. Characterization

The microstructure and morphology of original and W-plated diamond particles were observed by scanning electron microscope (SEM, LEO 1450, Baltimore, MD, USA). The qualitative analysis of the composition of diamond coating was characterized by X-ray diffraction (XRD, SmartLab, Akishima-shi, Japan). To measure the thickness of the coating and observe the cross-section, the Leica lapping machine and three ion beam cutting (EM TIC 3X, Wetzlar, Germany) processed the W-plated diamond. The cold field emission scanning electron microscope (FE-SEM, SU 8100, Hitachinaka, Japan) was used to observe the surface morphology and fracture microstructure of W-coated diamond particles and composites, and the local composition was analyzed by the energy dispersive spectrometer. The relative density and density of the composite material was measured using the Archimedes drainage method. The thermal conductivity (λ) of the Diamond-WC-Cu composites were calculated by the following formula: $\lambda =\rho \times \alpha \times C_p$, where $\rho$, $\alpha$, and $C_p$ are the density, thermal diffusivity coefficient, and specific heat capacity of the composite, respectively. The German Netzsch LFA 457 HyperFlash^® (NETZSCH-Gerätebau GmbH, Selb, Germany) was used to measure the thermal diffusivity coefficient of the sample and obtain the specific heat capacity. In order to ensure the validity of the data, the test data are the average of three measurements. The measurement error of the instrument gives an error bar result. The sample size for thermal diffusion measurement is Φ = 12.7 × 3 mm². The Langfang Supower Diamond Technology Co., Ltd., Langfang, China, was responsible for the mechanical processing of the experimental samples. Before testing, it was necessary to spray carbon on the surface of the sample to improve its absorption rate of laser pulses and reduce measurement errors.

3. Results and Discussion

3.1. Microstructure and Composition of W-Plated Diamond Particles

Figure 2 shows the surface morphology of W-plated diamond in a salt bath at different temperatures. As the plating temperature increases, the thickness of the tungsten coating gradually increases. After plating, the diamond kept a polyhedral shape and had no graphitization. At the test temperature of 900 and 1000 °C, the tungsten coating gradually crystallized on the diamond surface. The grains of tungsten coating in high temperature (1100 °C) were significantly larger than in the low temperature (900 and 1000 °C). Figure 2f shows that the tungsten coating on the diamond surface was rough, uneven in thickness, and poor in continuity. From the crystalline properties of diamond, the (111) crystal plane of single crystal diamond particles is hexagonal, and the (100) crystal plane is quadrilateral. It can be seen from Figure 2 that the (100) crystal plane of diamond was completely covered by the coating, while the (111) crystal plane is not completely covered by the tungsten coating. This indicates that the reaction product was easier to nucleate on the (100) crystal plane. This is related to the binding energy of C atoms on the crystal plane. The distance between adjacent carbon atoms on the (111) and (100) crystal planes of diamond is 0.252 and 0.356 nm, respectively. The binding energy of the diamond unit cell is 376.6 kJ/mol. According to the diamond lattice constant, the number of atoms per unit area of different crystal planes can be calculated. The total bond energy of the crystal plane can be obtained by the C-C bond energy and the number of atoms. The surface energy is half of the total bond energy. The calculation shows that the surface energy of the (111) crystal plane is 4.04 × 10⁻⁴ J/cm², and that of the (100) crystal plane is 6.23 × 10⁻⁴ J/cm² [24,25]. Therefore, carbon atoms on the (100) crystal plane are more active, and interface reactions are more likely to occur.

In addition, plating time is one of the important factors affecting tungsten coating by the salt bath. Figure 3 shows the surface morphology of W-plated diamond in a salt bath at 1100 °C with different holding times. When the plating time was 30 min, the plating layer on the diamond surface was discontinuously distributed. Figure 3c–f shows that the (100) crystal plane of diamond was covered by a thick tungsten layer when the plating holding time was 60 and 90 min. However, the (111) crystal plane of diamond still suffered from serious leakage.

The temperature of salt bath plating was too high, which led to the graphitization of diamond [26]. In addition, if the plating time was too long, the tungsten coating on the (100) crystal plane of diamond was too thick, and the tungsten coating still did not cover the (111) crystal plane. In the subsequent infiltration process, the thick tungsten coating of the (100) crystal plane, the (111) crystal plane without the tungsten coating, and the graphitization of diamond are not conducive to improving the wettability of diamond and the copper melt, thus adversely affecting the overall properties of the composite system.

In view of the results of salt bath plating, this paper tried the process of the powder covering sintering method for the surface modification of diamond. The surface morphology and area scanning analysis of W-plated diamond in the powder covering sintering method at different temperatures is shown in Figure 4. At 900 °C, the tungsten coating was thin, and the structure was fine, and there was a leakage of plating on some diamond crystal planes. As the plating temperature rised to 1000 °C, the vapor concentration of WO₃ increased and the reaction rate accelerated, and another layer of loose structure and coarser grains grew out of the thinner tungsten coating on the diamond surface. However, the bonds between the tungsten coatings were not tight. Part of the external diamond tungsten coating was damaged and peeled off. At 1100 °C, the outer tungsten coating was further thickened and dense, and the integrity of the tungsten coating was improved without missing plating. Figure 4g,h shows the result of the area scanning analysis of the red box areas 1 and 2 in Figure 4f, respectively. The elemental analysis shows that the main element of the coating was tungsten, and there were also trace amounts of carbon and oxygen.

To study the influence of time on the preparation of tungsten coating by powder covering sintering, this experiment was set to heat the mixed system of diamond and WO₃ for 30, 60, and 90 min under a vacuum condition at 1100 °C. The coating morphology is shown in Figure 5a–f. After a holding time of 30 min, small tungsten crystals were sporadically distributed on the diamond surface. Then, WO₃ reacts with C to form a coating layer. However, due to the diffusion rate of C atoms, the reaction was not completed in 30 min. Therefore, the diamond surface was not completely covered by the coating. As the plating time increased, the WO₃ adhesion-reaction continued in an infinite cycle, many W grains nucleated and grew, and the coating became dense, and the bonding strength with the diamond increased. The coating gradually thickened from loose to dense, covering all the crystal planes of diamond. Figure 5e shows that under the experimental conditions of 1100 °C and 90 min, the diamond surface coating prepared by the powder-covered sintering method was uniform and dense, and there was no leakage of plating.

Figure 6 is the element map of W-plated diamond in the powder covering sintering method at 1100 °C for 90 min. The surface of the diamond was completely covered with tungsten. The residual oxygen element contained reacted incompletely with WO_x and the oxygen adsorbed on the powder surface. Figure 7 shows the XRD diffraction patterns of W-plated diamond with the powder covering sintering method at 1100 °C for different plating times. The black, blue, and red curves in the figure represent the reaction time of 30, 60, and 90 min, respectively. The green vertical line is the characteristic peak position of the diamond. When the plating time was 30 min, the main components of the coating were WO_2.72 (W₁₈O₄₉) and element W, which indicated that the initial product of reducing WO₃ by the diamond surface C at this temperature was WO_2.72. As the plating time was extended to 60 min, another reduction product WO₂ appeared in the plating layer. Since the radius of the C atom (0.086 nm) is smaller than that of the W atom (0.141 nm), it might be caused by the continuous reduction of WO_2.72 by the C atoms diffused on the diamond side. When the plating time reached 90 min, the number of C atoms diffused to the outer layer increased, and the composition of the plating layer became more complicated. In addition to WO_2.72 and WO₂, WC was also produced [27]. It indicates that W will combine with C to form carbide (WC_x). The form of the product depends on the concentration of C atoms. The reduction of WO_x by C and the formation of WC_x proceed simultaneously. Therefore, the main component of the coating was elemental tungsten, which was formed from the reduction of WO₃ by C atoms.

The W-plated diamond prepared by the salt bath plating has a rough surface morphology and uneven thickness, and it was difficult to plate the (111) crystal plane, which had an adverse effect on improving the interface bonding and comprehensive properties of the composite. However, when the plating temperature was 1100 °C and the plating time was 90 min, a relatively uniform, complete, and compact coating could be prepared by the powder covering sintering method. Figure 8 shows the cross-section morphology and Energy Dispersive Spectroscopy (EDS) line scan analysis result of the W-plated diamond by argon ion beam cutting. The diamond was evenly wrapped by the tungsten coating, the coating structure was dense, and the interface between the coating and the diamond was firm. The average thickness of the W coating is about 900 nm. According to the results of the line scan energy spectrum analysis in Figure 8c, it was found that the coating is mainly composed of elemental W. The overall quality of the W coating is good. This has laid a foundation for the preparation of Diamond-WC-Cu composites.

3.2. Micro-Characterization of Diamond-WC-Cu Composites

In this experiment, Diamond-WC-Cu composites were prepared by the cyclic vacuum pressure infiltration. In the penetration experiment, under the set temperature (1100 °C) and time (90 min), the initial WO_x in the coating continues to react with the C atoms migrating in the diamond. Therefore, Cu, diamond, and tungsten carbide phases were present in the XRD diffraction peak analysis of the composite material (Figure 9). As shown in Figure 10, tungsten carbide existed at the interface between diamond and copper. The WC interface layer can improve the non-wetting interface between diamond and copper and reduce the interface thermal resistance [27].

3.3. Thermal Conductivity of Diamond-WC-Cu Composites

The compact and uniform W-plated diamond particle reinforced copper matrix composites were successfully prepared by the cyclic vacuum pressure infiltration method. The thermal conductivity of the composites first increases and then decreases with the increase of the content of the W-plated diamond. The thermal conductivity of the 60 vol% W-plated diamond and 900 nm WC layer composite reached 874 W∙m⁻¹∙K⁻¹. When the diamond content was over 60%, the thermal conductivity of the Diamond-WC-Cu composites decreased gradually. The tungsten carbide transition layer with low thermal conductivity hinders the thermal conductivity of the composites. The relative density of Diamond-WC-Cu composites had reached more than 98%. In contrast, the relative density and thermal conductivity of uncoated diamond/copper composites were significantly lower than that of Diamond-WC-Cu composites. The thermal conductivity of diamond/copper composites with different transition layers is shown in

Table 1. Thermal conductivity of diamond/copper composites prepared with different transition layers.

Interface Layer	Thickness (μm)	Diamond Content (vol%)	TC (W∙m−1∙K−1)	Ref
Mo2C	0.50	60	657	[21]
Cr7C3	1.00	65	562	[28]
TiC	0.22	60	811	[29]
WC	0.26	50	690	[27]
WC	0.11	63	900	[30]
B4C	2.11	90	731	[31]
Si	0.30	50	535	[32]
WC	0.90	60	874	This study

. The introduction of the transition layer can improve the thermal conductivity of the diamond/copper composite to varying degrees. The composite prepared in this paper has a high thermal conductivity.

To further analyze the thermal conductivity of Diamond-WC-Cu composites, we used the Maxwell-Eucken (M-E) model, Hasselman-Johnson (H-J) model, and differential effective medium (DEM) model to evaluate the thermal conductivity of the composites. Using these models to predict thermal conductivity, we consider the diamond particles as a nearly perfect sphere, while the copper matrix as a continuous phase.

The Maxwell-Eucken (M-E) model is Eucken’s improvement on the basis of the Maxwell model. The M-E model allows predicting the thermal conductivity of the continuous matrix reinforced by different particle phases. Equation (1) is the expression of the M-E model [33]:

$${\lambda }_{\mathrm{eff}}={\lambda }_m\frac{2{\lambda }_m+{\lambda }_D-2\left({\lambda }_m-{\lambda }_D\right)\varphi }{2{\lambda }_m+{\lambda }_D+\left({\lambda }_m-{\lambda }_D\right)\varphi }$$

(1)

where ${\lambda }_{\mathrm{eff}}$, ${\lambda }_D,$ and ${\lambda }_m$ represent the effective thermal conductivity of the composite, thermal conductivity of Diamond, and the Cu matrix, respectively. In addition, $\varphi$ is the volume fraction of diamond in the composite material.

Although the M-E model considers different particle reinforcement phases, the influence of the interface between the particles and the matrix is still ignored. The actual heat flow is conducted in composites, and the interface thermal resistance plays an important role. The interface thermal resistance ($R_{int}$) can be predicted by the following equation [34,35]:

$$R_{int}=\frac{2{\left({\rho }_m{\nu }_m+{\rho }_D{\nu }_D\right)}^2}{C_m{\rho }_m^2{\nu }_m^2{\rho }_D{\nu }_D}\cdot {\left(\frac{{\nu }_D}{{\nu }_m}\right)}^2$$

(2)

$$\nu =\sqrt{G/\rho }$$

(3)

where ρ and ν are the density and phonon velocity, respectively. C is the specific heat capacity. Subscripts “D” and “m” express the Diamond and matrix, respectively. The phonon velocity (ν) can be calculated by Equation (3). G is the shear modulus of the substance. Due to the introduction of tungsten carbide between copper and diamond, the calculation of the interface thermal resistance becomes complicated. To calculate the interface thermal resistance ($R_{Diamond/WC}$) between the diamond and tungsten carbide transition layer, we use the diamond as the reinforcing phase and the tungsten carbide transition layer as the matrix. In the same way, the tungsten carbide layer is used as the reinforcing phase and copper is used as the matrix to calculate the interface thermal resistance ($R_{WC/Cu}$) between the tungsten carbide layer and the copper matrix.

Table 2. Parameters for the interface thermal resistance calculation.

Material	Density ρ (kg·m−3)	Specific Heat C (J·kg−1·K−1)	Shear Modulus G (GPa)	Phonon Velocity ν (m·s−1)	TC Λ (W·m−1·K−1)	Ref
Diamond	3520	515	504	11,970	1500	[39]
Cu	8960	385	50	2362	400	[39,40]
WC	15,630	171	270	4156	120	[34]

shows the relevant parameters for calculating the interface thermal resistance. By substituting the physical values of the phases into formulas (2) and (3), $R_{Diamond/WC}$ and $R_{WC/Cu}$ were calculated as 6.25 × 10⁻⁹ and 4.13 × 10⁻⁹ m²∙K∙W⁻¹, respectively. In addition, the thermal resistance of the tungsten carbide interface layer can be calculated by $R_{WC}=L/{\lambda }_{WC}$, where L is the thickness of the WC layer. After the calculation, we obtained the thermal resistances of WC layers: $R_{WC}$ = 1.07 × 10⁻⁸ m²∙K∙W⁻¹. Therefore, the total interface thermal resistance ($R_{int}=R_{Diamond/WC}+R_{WC}+R_{WC/Cu}$) of the Diamond-WC-Cu composites is 2.11 × 10⁻⁸ m²∙K∙W⁻¹.

Hasselman and Johnson believed that the effective thermal conductivity of composite materials depends not only on the volume fraction of the reinforcing phase, but also on the size of the reinforcing phase. In addition, they also studied the effect of interface thermal resistance on the thermal conductivity of composites. Therefore, the novelty in the Hasselman and Johnson (H-J) model includes the particle radius ($r_D$) and interface thermal resistance ($R_{int}$). Equation (4) is the H-J model prediction of the spherical reinforced phase [36,37]:

$${\lambda }_{eff}={\lambda }_m\frac{\left[2\left(\frac{{\lambda }_D}{{\lambda }_m}-\frac{{\lambda }_D{R}_{int}}{r_D}-1\right)\cdot \varphi +\frac{{\lambda }_D}{{\lambda }_m}+\frac{2{\lambda }_D{R}_{int}}{r_D}+2\right]}{\left[\left(1-\frac{{\lambda }_D}{{\lambda }_m}+\frac{{\lambda }_D{R}_{int}}{r_D}\right)\cdot \varphi +\frac{{\lambda }_D}{{\lambda }_m}+\frac{2{\lambda }_D{R}_{int}}{r_D}+2\right]}$$

(4)

The H-J model belongs to the effective medium theory (EMT) or the effective medium approximation theory (EMA). It is the category of the mean field theory. Then, Tavangar et al. [38] used the differential effective medium theory or differential effective medium format (DEM) to obtain a new description of the effective thermal conductivity of composites. The DEM model is described as:

$$1-\varphi =\frac{{({\lambda }_m)}^{1/3}\left({\lambda }_D{\lambda }_{eff}{R}_{int}+r_D{\lambda }_{eff}-r_D{\lambda }_D\right)}{{({\lambda }_{eff})}^{1/3}\left({\lambda }_D{\lambda }_m{R}_{int}+r_D{\lambda }_{eff}-r_D{\lambda }_D\right)}$$

(5)

$$\gamma =\frac{{\lambda }_{\mathrm{eff}}}{\left(1+\frac{{\lambda }_D{R}_{int}}{r_D}\right){\lambda }_m}$$

(6)

Among them, $\gamma$ is the effective phase contrast. Tavnagar et al. believed that the H-J model fails under a high effective phase contrast between each phase of the composite. Therefore, when $\gamma$ is less than 4, the H-J model is effective. The effective phase contrast of different volume composite materials is shown in

Table 3. Effective phase contrast value(γ) of composite materials with different diamond volume fractions *.

${\mathit{\varphi }}_{\mathit{D}}(\mathbf{vol}\%)$	H-J Model/W·m−1·K−1	${\mathit{\gamma }}_{\mathit{H}-\mathit{J}}$	DEM Model/W·m−1·K−1	${\mathit{\gamma }}_{\mathit{D}\mathit{E}\mathit{M}}$
50	613	0.93	620	0.95
55	639	0.98	647	0.99
60	665	1.02	674	1.03
65	693	1.06	702	1.07
70	721	1.10	731	1.12

*: γ < 4, model prediction results are valid.

. The effective phase contrast (${\gamma }_{H-J}$ and ${\gamma }_{T-DEM}$) is less than 4. Therefore, the H-J and DEM models are effective in predicting thermal conductivity.

The thermal conductivity of Diamond-WC-Cu composites based on the H-J model, M-E model, and differential effective medium(DEM) model was shown in Figure 11. The red and green cones in the figure represent the thermal conductivity of un-plated diamond and W-plated diamond reinforced copper composites, respectively. The blue, cyan, and magenta cones are the thermal conductivity prediction values of the H-J model, the DEM model, and the M-E model, respectively. The experimental value of thermal conductivity of the Diamond-Cu composite material without a transition layer is significantly lower than the model predicted value. This shows that the interface thermal resistance is the main factor affecting the thermal conductivity of composites. The experimental value of thermal conductivity of Diamond-WC-Cu composites is obviously lower than the predicted value of the M-E model. This shows that the M-E model ignores the interfacial thermal resistance and has a high predicted value of thermal conductivity. The experimental values are close to the predicted values of thermal conductivity of the H-J and DEM models. This indicates that when $\gamma$ is less than 4, both models can be used to predict the thermal conductivity of composites. Due to the high thermal conductivity of Diamond-WC-Cu composites, it can satisfy the requirements of electronic packaging materials.

4. Conclusions

Through the powder covering sintering method, a dense and uniform tungsten coating was successfully prepared on the diamond at 1100 °C for 90 min. The thickness of the tungsten coating is approximately 900 nm. The Diamond-WC-Cu composites were prepared by the cyclic vacuum pressure infiltration. When the volume fraction of the diamond is 60%, the thermal conductivity of the Diamond-WC-Cu composites is 874 W·m⁻¹·K⁻¹. The microscopic morphology and composition analysis of the composites show that a tungsten carbide interface layer is formed at the interface between the diamond and copper matrix. The calculated value of interface thermal resistance of the composites is 2.11 × 10⁻⁸ m²∙K∙W⁻¹, due to the formation of the tungsten carbide interface layer. The wettability of copper and diamond is improved by the tungsten carbide interface layer, and the thermal conductivity of the composite is obviously improved. In this paper, the thermal conductivity of the prepared Diamond-WC-Cu composites is close to the theoretically predicted value (H-J and DEM models). The composites with an excellent thermal performance can meet the requirements of heat dissipation of thermal management materials.