**1. Introduction**

Spark Plasma Sintering (SPS) is a consolidation approach, in which pulsed direct current passes through a graphite die and a compacted specimen, and enabling very high heating rates with simultaneously applying an external pressure. The combination of internal heating and external pressure provides the conditions for fast sintering. SPS parameters have to be determined based on the thermal and electrical properties of the consolidated material, which, evidently, depend on the compact porosity that decreases during the sintering process. In the present study, the effect of porosity of the SPS-processed Cu specimens on their thermal and electrical properties at room temperature was theoretically and experimentally investigated.

There are some theoretical approaches, which may be applied for analysis of transport properties of a porous material. One of the approaches, the "so-called" general effective media (GEM) method (eq. 1) was discussed in [1, 2].

$$\times\_{1} \cdot \frac{(\sigma\_{1}, \kappa\_{1})^{\mathsf{Y}\_{t}^{\mathsf{f}}} - (\sigma\_{\text{eff}'}, \kappa\_{\text{eff}})^{\mathsf{Y}\_{t}^{\mathsf{f}}}}{(\sigma\_{1}, \kappa\_{1})^{\mathsf{Y}\_{t}^{\mathsf{f}}} + A \cdot (\sigma\_{\text{eff}'}, \kappa\_{\text{eff}})^{\mathsf{Y}\_{t}^{\mathsf{f}}}} = (1 - \mathbf{x}\_{1}) \frac{(\sigma\_{\text{eff}'}, \kappa\_{\text{eff}})^{\mathsf{Y}\_{t}^{\mathsf{f}}} - (\sigma\_{\text{2}}, \kappa\_{2})^{\mathsf{Y}\_{t}^{\mathsf{f}}}}{(\sigma\_{2}, \kappa\_{2})^{\mathsf{Y}\_{t}^{\mathsf{f}}} + A \cdot (\sigma\_{\text{eff}'}, \kappa\_{\text{eff}})^{\mathsf{Y}\_{t}^{\mathsf{f}}}} \tag{1}$$

The GEM equation (eq. 1) is usually employed for calculating the effective electrical and thermal conductivities (σeff and κeff, respectively) for two-phase materials using the electrical (σ1 and σ2) and thermal (κ1 and κ2) properties of each phase. The morphological (or geometrical)

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parameters (*A* and *t*) may be derived from the equation of conductivity percolation [1] or by appropriate modeling of experimental results. The *x*1 and (1-*x*1) values are the volume fractures of the phases in a two-phase material.

Generally, *A* and *t* values depend on the phases distribution and their morphology within twophase materials. Let us consider a composite material, which consists of continuous matrix and homogenous distributed particles with high aspect ratio (for instance, fibers) of a second phase. For this kind of distribution *t* value is equal to 1 [1] and the so-called "parallel" and "series" alignment of the particles (relative to the electrical potential or temperature gradients) may be considered. The parameter *A* varies from ∞ for the parallel to 0 for series alignments. Recently [1], the GEM equation and measured transport properties were successfully used for estimating the fracture and distribution of Sn phase, which displays a fiber shape particles, in the SnxTe1-x inter-metallic matrix.

Ke-Feng Cai et al. [3] reported the results of prediction of effective thermal conductivity of porous Al-doped SiC ceramics using the Landauer's expression (eq. 2) derived for spherical and homogeneously dispersed second phase [4].

$$\kappa\_{\rm eff} = \kappa\_1 \left(\frac{3\chi\_1 - 1}{2}\right) \tag{2}$$

It may be shown that this equation may be derived from the GEM equation (1) if one takes into account that *A*=2 and *t*=1.

Another commonly used approach, yet not straightforward for multi-phased materials, to correlate effective transport properties (thermal and electrical conductivity) of porous materials is based on the Weidemann-Franz relation.

$$
\kappa\_{\rm eff} = L \cdot \sigma\_{\rm eff} \cdot T \tag{3}
$$

Where, the coefficient *L* is known as Lorenz number and *T* represents the absolute temperature. The Lorenz number is constant only in the case when the conduction electrons are scattered elastically. This condition requires that the temperature-independent electron scattering by impurities will dominate over the electron scattering by phonons [5]. These conditions may be achieved at high temperatures or at very low temperatures, where the residual resistivity is predominant. At intermediate temperatures, the condition of elasticity no longer holds and the Lorenz number decreases considerably from the Sommerfeld value. In order to get a better understanding of the *L* number, let us consider the general expression for *L,* derived from the Fermi-Dirac statistics (eq. 4) [6] and graphically presented in Fig. 1 for two different electron scattering mechanisms (or scattering parameters, *r*): by ionized impurities (using scattering parameter *r* of 3/2) and by acoustic phonons (*r*=-1/2).

Correlation between Thermal and Electrical Properties of Spark Plasma Sintered (SPS) Porous Copper http://dx.doi.org/10.5772/59010 157

$$L = \left(\frac{k}{e}\right)^2 \left[ \frac{\left(r + \frac{7}{2}\right)\left(r + \frac{3}{2}\right)F\_{r + \frac{5}{2}}\left(\eta\right)F\_{r + \frac{5}{2}}\left(\eta\right) - \left(r + \frac{5}{2}\right)^2F\_{r + \frac{3}{2}}^2\left(\eta\right)}{\left(r + \frac{3}{2}\right)^2F\_{r + \frac{5}{2}}^2\left(\eta\right)}\right] \tag{4}$$

Where, *e*, *k*, *F*r and η are the electrons charge, Boltzmann constant, Fermi integral (defined by eq. 5) and the reduced Fermi energy (*E*F/*kT*), respectively.

parameters (*A* and *t*) may be derived from the equation of conductivity percolation [1] or by appropriate modeling of experimental results. The *x*1 and (1-*x*1) values are the volume fractures

Generally, *A* and *t* values depend on the phases distribution and their morphology within twophase materials. Let us consider a composite material, which consists of continuous matrix and homogenous distributed particles with high aspect ratio (for instance, fibers) of a second phase. For this kind of distribution *t* value is equal to 1 [1] and the so-called "parallel" and "series" alignment of the particles (relative to the electrical potential or temperature gradients) may be considered. The parameter *A* varies from ∞ for the parallel to 0 for series alignments. Recently [1], the GEM equation and measured transport properties were successfully used for estimating the fracture and distribution of Sn phase, which displays a fiber shape particles, in

Ke-Feng Cai et al. [3] reported the results of prediction of effective thermal conductivity of porous Al-doped SiC ceramics using the Landauer's expression (eq. 2) derived for spherical

1

It may be shown that this equation may be derived from the GEM equation (1) if one takes into

Another commonly used approach, yet not straightforward for multi-phased materials, to correlate effective transport properties (thermal and electrical conductivity) of porous

*x*

æ ö - <sup>=</sup> ç ÷

3 1

è ø (2)

=× × *L T* (3)

1

k k

2 *eff*

*eff eff*

 s

Where, the coefficient *L* is known as Lorenz number and *T* represents the absolute temperature. The Lorenz number is constant only in the case when the conduction electrons are scattered elastically. This condition requires that the temperature-independent electron scattering by impurities will dominate over the electron scattering by phonons [5]. These conditions may be achieved at high temperatures or at very low temperatures, where the residual resistivity is predominant. At intermediate temperatures, the condition of elasticity no longer holds and the Lorenz number decreases considerably from the Sommerfeld value. In order to get a better understanding of the *L* number, let us consider the general expression for *L,* derived from the Fermi-Dirac statistics (eq. 4) [6] and graphically presented in Fig. 1 for two different electron scattering mechanisms (or scattering parameters, *r*): by ionized impurities (using scattering

k

of the phases in a two-phase material.

156 Sintering Techniques of Materials

the SnxTe1-x inter-metallic matrix.

account that *A*=2 and *t*=1.

and homogeneously dispersed second phase [4].

materials is based on the Weidemann-Franz relation.

parameter *r* of 3/2) and by acoustic phonons (*r*=-1/2).

$$F\_r = \bigcap\_{0}^{\circ} \xi^r f\_o \left(\eta \right) \cdot \partial \xi$$

$$f\_o(\eta) = \frac{1}{1 + \exp(\xi - \eta)}\tag{6}$$

In the expression for *F*<sup>r</sup> (eq. 5)-*f*o (defined in eq. 6) and ξ are the Fermi distribution function and the kinetic energy of a charge carrier, respectively.

**Figure 1.** Lorenz number variation with the reduced Fermi energy (η=*E*F/kT) for two different electron scattering mech‐ anisms, by ionized impurities (using scattering parameter *r* of 3/2) and by acoustic phonons (*r*=-1/2).

According to this analysis, for materials with low carrier concentrations (η<-0.4), *L* values reach the classical (Boltzmann statistics) values of 4(*k/e*)2 and 2(*k/e*) <sup>2</sup> for ionized impurities and acoustic phonons scattering, respectively. For degenerate materials (η>1.2) *L* reaches the Sommerfeld value, derived from eq. 7, regardless the scattering mechanism.

$$L = \frac{\pi^2}{3} \left(\frac{k}{e}\right)^2\tag{7}$$

For the intermediate η values, *L* exhibits a strong dependence on both Fermi energy and the scattering mechanism.

Since Cu is highly conductive metal, the value of η at room temperature may be calculated using eq. 8 and the physical properties of Cu (effective mass, *m*\*=1.01*m*<sup>o</sup> [7], and carrier concentration, *n*=8.4x1022 cm-3 [8]). The estimated η value is equal to 270 and corresponds to the energetic range where the constant Sommerfeld value valids.

$$m = \frac{8}{3\sqrt{\pi}} \left(\frac{2\pi m^\* kT}{h^2}\right)^{\frac{3d}{2}} \eta^{\frac{3d}{2}} \tag{8}$$

Nevertheless, the Lorenz number differs slightly from the Sommerfeld value due to the inelastic electrons scattering.

The simple consideration presented above shows that for the applying the Weidemann-Franz relation between the transport properties of materials, the specific Lorenz number has to be determined.

It was shown [9], for instance, that Lorenz number of copper alloys depends on their purity and thermo-mechanical treatments. Moreover, in [10] it was established that the Lorenz number for copper films depends on its thickness. This effect was partly attributed to scattering of electrons at the films surfaces and partly to scattering by frozen-in structural defects.

This explanation is based on a distinction between the free paths of electrons for the electrical conductivity process and those for the thermal conductivity process. Being more specific, in a metal, an electric field or a temperature gradient causes an electron drift, which is restricted only by the collisions of the electrons with lattice imperfections (static defects or lattice vibrations). When the electron distribution function is disturbed from its equilibrium value, the rate of return to equilibrium may be expressed by collision processes, which are usually expressed in terms of a relaxation time. Only in case that the relaxation time is the same for both electrical and thermal transport, eq. 7 can be used. This equation is based on the assump‐ tion that *L* is a constant independent of the band structure or the relaxation time. Regarding relaxation, it was pointed out in [11] that equilibrium can be reached in two ways: either by processes changing the direction of motion of an electron but not changing its energy signifi‐ cantly, or by processes changing its energy but not the direction – the so-called "horizontal" or "vertical" movements on the Fermi surface. Since the "vertical" movement was found as ineffective in producing electrical resistance, the relaxation times for electrical and thermal conduction are equal only in case that the "vertical" movement is absent. The effective scattering by phonons at high temperatures and by impurities at low temperatures is elastic, leading to similar relaxation time for the different transport properties. This is the reason for the validity of eq. 7 at very low or high temperatures.

Koh et al. [12] had used a modified Weidemann-Franz relation (eq. 9), which takes into account also the lattice component of the thermal conductivity, for analysis of porous stainless steel and Cu based alloys.

$$\kappa\_0 = L \frac{T}{\rho\_0} + b \tag{9}$$

Where, ρ0 and κo are the electrical resistivity and the thermal conductivity of the alloys, respectively, and *b* is the lattice component of the thermal conductivity.

It was established that for highly conductive materials, such as copper alloys, where the lattice component of the thermal conductivity is relatively small compared to the electronic one, the dependency of the thermal conductivity on *T*/ρ for various porosity levels was characterized by a straight curve with a slope *L* and intercept *b*. In this case the *L* and *b* values depend on the nature of metal only. For stainless steel, for which the lattice and the electronic components are comparable, *b* value depends on the porosity and specific curve for each porosity level was obtained.

It can be concluded that special care should be taken while using the Weidemann-Franz relation for porous or other multi-phased materials. On the other hand, applying the GEM approach is much more straightforward, giving additional information about the phases' alignment and distribution characteristics

In the presence study, the GEM and Weidemann-Franz relations were applied for investigation of the experimental transport properties results of porous SPS-processed Cu specimens.
