**3. Results**

The microstructure of the porous Cu speciemns is shown in Fig. 2.

The average grain size of about 7-8μm indicates no coarsening effects during the SPS process. The pores are homogenously distributed within the metallic matrix.

**Figure 2.** Optical images of the SPS-processed specimens. The microstructure of the almost dense specimen (1vol% po‐ rosity) after etching was presented as the insert in the corresponded image.

Values of the electrical resistivity and thermal conductivity measured at room temperature as a function of porosity are shown in Fig. 3.

**Figure 3.** The electrical resistivity and thermal conductivity of porous Cu specimens.

As was expected, a general trend of increasing the electrical resistivity and decreasing the thermal conductivity values with increasing of the porosity amount was observed. Further‐ more, additional insights regarding the geometrical alignment and porous distribution can be obtained, upon applying the GEM and the Weidemann-Franz equations.

## **4. Discussion**

The thermal conductivity measurements were conducted under ~1atm argon, similarly to the

**Porosity, vol% Temperature, oC Holding time, min. Applied uniaxial pressure,**

0 700 15 97 1 600 15 97 10 400 2 88 15 400 2 70 20 400 2 53 30 400 2 35

**MPa**

The average grain size of about 7-8μm indicates no coarsening effects during the SPS process.

**Figure 2.** Optical images of the SPS-processed specimens. The microstructure of the almost dense specimen (1vol% po‐

The microstructure of the porous Cu speciemns is shown in Fig. 2.

**Table 1.** Spark Plasma Sintering conditions and porosity of the specimens.

The pores are homogenously distributed within the metallic matrix.

rosity) after etching was presented as the insert in the corresponded image.

SPS conditions.

160 Sintering Techniques of Materials

**3. Results**

In order to apply the GEM equation (1) and Weidemann-Franz relation (eq. 3) for analyzing the experimental results presented in Fig. 3, the thermal and electrical conductivities of pure Cu and argon, which filled the micrometric pores, have to be determined. The thermal and electrical conductivities of pure copper (κ1 and σ1, respectively, in eq. 1) are reported in [13, 14]. For argon, the electrical conductivity, σ2, is negligible, while the thermal properties has to be discussed. According to [15], at normal conditions, heat convection through a porous metallic structure becomes of practical importance only at temperatures above 1000K and for pore diameter larger than 5000μm. It was also reported in [15] that heat transfer by radiation in the pores at moderate and high temperatures might be neglected. Thus, for materials with relatively small pores, as in our case, conductive heat transfer is the dominant process at the investigated temperature.

Under normal pressure, the thermal conductivity of Ar [16, 17] may be calculated according to eq.10

$$\kappa\_{Ar} = 1.47 \cdot 10^{-5} \frac{T\_2^{\prime \prime}}{1 + 142} \Big/ \begin{bmatrix} \text{Watt } / \text{cmK} \end{bmatrix} \tag{10}$$

and is equal to ~1.73x10-2 W/mK at room temperature.

#### **4.1. General Effective Media (GEM) theory**

The aims of this paragraph is to confirm that the effective transport properties of the porous material may be calculated using one set of the geometrical parameters *A* and *t*. We started with the analysis of the measured values of the effective transport properties for the specimen showing a 20 vol.% porosity. By solving together two equations, based on the GEM approach, for the thermal and electrical conductivities, the values *A*=1.3 and *t*=1 were obtained. These values were used for calculation of the transport properties for the entire investigated specimens, showing various porosity amounts. A very good agreement was found between the calculated and experimental results, as can be seen in Figs. 4-5, while comparing the experimental data to various calculated curves with different *A* values. The experimental points are located between the curves corresponded to *A* values in the range of 1÷2. Thus, we may conclude that the pores are homogeneously dispersed and have nearly spherically shaped, in agreement with the micro-structural observations (Fig. 2).

**Figure 4.** Effective electrical resistivity (=σeff-1) values for the investigated porous Cu samples, calculated by the GEM equation substituting *A* and *t* parameters of 0.3,1; 0.6,1; 1.3,1; 2,1 and ∞,1 (parallel alignment); and by the Weidemann-Franz (W.F.) relation. The experimentally measured results are shown by the black points.

**Figure 5.** Effective thermal conductivity (κeff) values for the investigated porous Cu samples, calculated by the GEM equation substituting *A* and *t* parameters of 0.3,1; 0.6,1; 1.3,1; 2,1 and ∞,1 (parallel alignment); and by the Weidemann-Franz (W.F.) relation. The experimentally measured results are shown by the black points.

#### **4.2. Weidemann-Franz (W.F.) relation**

The calculated values of the transport properties as a function of the porosity levels using Weidemann-Franz relation (eq. 3) are also presented in Figs 4-5. The calculations took into account the Lorenz number for Cu at 0o C (2.23x10-8 [V/K]2 ) reported in [18]. This value is slightly differs from the Sommerfeld value of 2.47x10-8 [V/K]2 (derived from eq. 7), which is commonly used for highly degenerated materials. Nevertheless, there is a good agreement between the calculated and the experimental results for both the electrical resistivity and the thermal conductivity values.

#### **4.3. W.F. vs. GEM**

1 <sup>2</sup> <sup>5</sup> 1.47 10 / <sup>142</sup> <sup>1</sup> *Ar*


+

k

162 Sintering Techniques of Materials

**4.1. General Effective Media (GEM) theory**

and is equal to ~1.73x10-2 W/mK at room temperature.

shaped, in agreement with the micro-structural observations (Fig. 2).

*T*

The aims of this paragraph is to confirm that the effective transport properties of the porous material may be calculated using one set of the geometrical parameters *A* and *t*. We started with the analysis of the measured values of the effective transport properties for the specimen showing a 20 vol.% porosity. By solving together two equations, based on the GEM approach, for the thermal and electrical conductivities, the values *A*=1.3 and *t*=1 were obtained. These values were used for calculation of the transport properties for the entire investigated specimens, showing various porosity amounts. A very good agreement was found between the calculated and experimental results, as can be seen in Figs. 4-5, while comparing the experimental data to various calculated curves with different *A* values. The experimental points are located between the curves corresponded to *A* values in the range of 1÷2. Thus, we may conclude that the pores are homogeneously dispersed and have nearly spherically

**Figure 4.** Effective electrical resistivity (=σeff-1) values for the investigated porous Cu samples, calculated by the GEM equation substituting *A* and *t* parameters of 0.3,1; 0.6,1; 1.3,1; 2,1 and ∞,1 (parallel alignment); and by the Weidemann-

Franz (W.F.) relation. The experimentally measured results are shown by the black points.

*<sup>T</sup> Watt cmK*

ë û

(10)

While comparing the two suggested approaches for estimating the electrical and thermal transport properties of porous materials, several general guidelines can be obtained:


temperatures (e.g. room temperature) leads to inaccurate estimations due to inelastic scattering effects. Therefore, measurement of both κeff and σeff (or ρeff) for one known porosity level, for evaluation of *L*, should not be avoided. For less conductive matrix materials, phonons thermal conductivity effects are involved, which are expected to vary with the porosity amount. For semiconductors with reduced Fermi energy in the range of −0.4<η<1.2, *L* depends also on the scattering mechanism and the Fermi energy. For semiconductors or insulators with η<-0.4, *L* depends on the scattering mechanism as well. Such an approach can't supply any insight on the geometrical alignment and morphology of the pores.
