4.2. Models to Predict Sintered Metal Fibres Electrical Conductivity
In this section according to the rule of mixture, and considering the effect of tortuosity and critical porosity, two new models for predicting the electrical conductivity of porous materials are proposed. Based on the rule of mixture, the electrical conductivity of unidirectional metal fibres can be expressed as below [
44]:
Considering that the electrical conductivity of air is negligible compared to the conductivity of the solid metal phase, therefore the electrical conductivity can be expressed as:
Although this formula gives the correct value for long unidirectional fibres, this is not valid in sintered metal fibres as the length of the fibres can vary as well as their directions. Also, the connection of the fibres depends on the sintering conditions such as pressure, temperature, and duration [
40]. For instance, Zhou et al. [
40] have shown that for sintered copper fibres samples with a porosity of 90% porosity, the electrical conductivity at 1000 °C was almost 5.4 times as high as of the electrical conductivity at 700 °C. Therefore, in order to introduce the effect of manufacturing processes, inhomogeneity in the structure, and possible defects we can introduce a coefficient that accounts for these effects:
It is expected that the value of
K to be lower than one (i.e.,
K = 1 represents ideal conditions). Here the critical porosity,
εc, can be introduced to the previous equation:
One factor that contributes to the increase of the conductivity in porous metals is the increase in the path that electrons need to travel, i.e., the electrical tortuosity, due to the presence of nonconductive voids. This is illustrated in
Figure 6. It can be seen that the electrons have to travel a longer path in a porous metal in comparison to a bulk metal where they can travel in a straight line.
This can be expressed by the tortuosity concept. If we assume in the bulk material, the length travelled by electrons is
L0 and in the porous material it is
L, and consider
τ as tortuosity, then we can write:
This can also account for the fact that fibres are not unidirectional. The Bruggeman relation is a widely accepted estimation for tortuosity [
45,
46]:
However, this expression is hydraulic tortuosity in the void space in the porous media.
α is a coefficient, which depends on the structure of the porous media. For the electric tortuosity, we can rewrite the Bruggeman relation as below as the solid phase is important:
This relation satisfies the boundary conditions for tortuosity at ε = 0 and ε = 1. When porosity is equal to zero, the tortuosity must be 1 and when porosity is 1, tortuosity must be ∞.
Another accepted model for tortuosity is [
47]:
Here, P is a fitting parameter that depends on the porous media structure and can be estimated experimentally or numerically, such as using lattice Boltzmann simulation. Again, to satisfy the following conditions:
τ → ∞ when
ε = 1 and
τ → 1 when
ε = 0, we can change the formula to:
Now, by introducing the effect of tortuosity in Equation (10), we have:
By replacing tortuosity in Equation (17) by Equations (13) and (14) we have:
The results for the two new models are presented in
Table 3 for experimental data from Zhou et al. [
40] and in
Table 4, for data from Feng et al. [
30]. These models result in lower ARE compared to previous models. The two new models have almost similar results. The ARE for different structures and using different models can be seen in
Table 5. It is clear that for sintered metal fibres the accuracy of the models can be significantly improved by the critical porosity introduction; the improvement is less for closed-cell metal foam; for the open-cell foam the improvement is insignificant.
Therefore, by looking at the results from the modified models and the new proposed models, it can be concluded that for sintered metal fibres, the presumption of having zero conductivity at a porosity of 100% does not provide accurate results. By doing so, the errors for sintered copper fibres can reach up to 30% for the range of 70%–90%. Also, as it was expected, the critical porosity obtained by fitting the models to the experimental results for open-cell metal foam, was higher (close to unity) than that of obtained for sintered metal fibres and closed-cell metal foam. The suggested reason is that for having structural integrity for a similar structure, closed-cell metal foam requires more solid phase. The average ARE from using different models for different structures can be seen in
Table 5. As it can be observed, for the sintered metal fibres, significant improvement can be gained by the modified models and even more significantly by the new proposed models. For closed-cell the improvement is moderate; and finally, for the open cell metal foam, the reduction in error is marginal and average ARE is reduced by around 1% to 3%. As a result, it is suggested that for open cell metal foam Equation (3) can be used, which gives less error compared to Equations (2) and (4) and the inclusion of critical porosity does not bring significant improvement. For closed cell metal foams, modified models (i.e., Equations (5)–(7)) and new suggested models (i.e., Equations (18) and (19)) provide similar accuracy and outperform models without considering critical porosity. Finally, for sintered metal fibre porous media, the newly proposed models outperform the previous models and hence are recommended to be considered for this type of porous metal.
The critical porosity at which the electrical conductivity reaches zero (i.e., the maximum porosity that is achievable to have a connected cluster of solid phased) depends on the type of the porous metal (the method by which the porous metal is manufactured). The highest amount of porosity in porous metals can be achieved in metal foams, specifically open-cell metal foams. Metal foams typically have larger pore size compared to that of sintered metal fibres or powders. Larger pores mean higher void space ratio to solid phase. Also, in metal foams that are manufactured from molten metal, the solid phase is uniform and there is no contact resistance in the structure. Additionally, open-cell metal foams can provide higher porosities in comparison to the closed-cell metal foams as the walls (solid phase) are non-existent in open-cells and only struts are present to form a cell. On the other hand, sintered porous metals, are formed by means of pressure at high temperatures. This puts a limit on the maximum porosity achievable. Additionally, as the sintering depends on the temperature, pressure, and the duration, the electrical conductivity depends on these factors [
40,
48]. These factors affect the contact between the metal fibres or powders and hence the overall electrical conductivity. As a result, the critical porosity, has a more significant effect on the accuracy of the models for porous metals formed by sintering, followed by closed-cell metal foams, and finally less impact on the open-cell metal foams. This is also evident in the results by fitting the models and the critical porosity for the open-cell foam was obtained to be close to one, whereas for sintered metal fibres, this value was ~95%.
In order, to find the validity of the models for other cases of closed-cell metal foams the models are fitted to additional data from the literature. For metal foams, the data from Feng et al. [
30], Sevostianov et al. [
49], Kim et al. [
50] and Kovacik et al. [
51] are used. After fitting the data to these models, the coefficients are obtained as presented in
Table 5. The critical porosity and the tortuosity coefficient are similar for different datasets. The value of
K (manufacturing effects) is close for the three of the datasets ([
30,
50,
51], However,
K for the data from Sevostianov et al. [
49] is slightly lower. The value of
K is expected to vary between different datasets provided by different research groups as it depends on several factors and the complex interaction between them such as impurities and manufacturing methods implemented. If a
K value based on these data is applied to a new dataset, a higher error could potentially arise due to the effect of aforementioned parameters. Therefore, it is suggested for new porous metals, few measurements of the electrical conductivity of samples with different porosities to be experimentally measured (e.g., two different porosities). Subsequently, the value of
K can be determined based on those values to minimise the error and predict the electrical conductivity of the porous metal at different porosities by the model. This allows the prediction of the conductivity for a wide range of porosity that otherwise needs to be measured individually and samples with different porosities are required to be manufactured that are not practical in many circumstances. This will help in modelling of devices implementing such materials, where electrical conductivity is required such as fuel cells and electrolysers.
Also, if the value of
K for closed metal foams (
Table 6) is compared the value for sintered metal fibres (experimental data from Zhou et al. [
40]), ~0.03, it is evident that the value of
K is significantly higher for closed-cell metal foams as expected. This can be explained by the fact that in metal foams, the solid phase can be considered one connected piece, whereas in sintered porous metals, there are numerous connections points between the particles adding to the overall resistance (i.e., lower conductivity) resulting in a lower value of
K. In other words, the value of
K represents the deviation from ideal conditions.