## 3 Results and Discussion
### Isothermal Aging
#### 3.2.3 Stability of the Contact Resistivity and Service Life Prediction  
To study the high temperature stability of the electrical properties of these joints, the contact resistivity of the joints was measured after each different thermal aging time. Figure 8 exhibits the variation of the interfacial contact resistivity of the joints with the square root of the aging time between 550 \({}^{\circ}\)C and 650 \({}^{\circ}\)C. At the aging temperature of 550 \({}^{\circ}\)C, the contact resistivity of the joint increases very slowly and reaches about 4.5 \(\upmu\Omega\cdot\)cm\({}^{2}\) after being aged for 50 days. Increasing the aging temperature to 600 \({}^{\circ}\)C and 650 \({}^{\circ}\)C, the contact resistivity reaches 6.8 \(\upmu\Omega\cdot\)cm\({}^{2}\) and 10 \(\upmu\Omega\cdot\)cm\({}^{2}\) after 41 days and nine days, respectively. In contrast to the previous reports [26; 29; 30; 31], the stability of the contact resistivity of the Ti/Mo/Yb\({}_{0.3}\)Co\({}_{4}\)Sb\({}_{12}\) joints is significantly improved.  
The variation of the contact resistivity was then fitted with the square root of the aging time and a linear relationship between them was found (See Table 2). Based on Tables 1 and 2, it is clear that the increase in the contact resistivity of the Ti/Mo-Ti/Yb\({}_{0.3}\)Co\({}_{4}\)Sb\({}_{12}\) joints corresponds well to the increase in the thickness of the interfacial diffusion layer at the Mo-Ti/Yb\({}_{0.3}\)Co\({}_{4}\)Sb\({}_{12}\) interface. In another word, it is the growth of the interfacial diffusion layer at the Mo-Ti/Yb\({}_{0.3}\)Co\({}_{4}\)Sb\({}_{12}\) interface that dominates the stability of the interfacial contact resistivity of the Ti/Mo-Ti/Yb\({}_{0.3}\)Co\({}_{4}\)Sb\({}_{12}\) joints.  
The performance of the TE joints, as well as that of the TEGs, is sensitive to the contact resistivity, and therefore, the contact resistivity is a valuable criteria for the service life prediction of the TE joints. We assume that a Ti/Mo/Yb\({}_{0.3}\)Co\({}_{4}\)Sb\({}_{12}\) joint fails when the interfacial contact resistivity reaches 10 \(\upmu\Omega\cdot\)cm\({}^{2}\) or 20 \(\upmu\Omega\cdot\)cm\({}^{2}\), which are low values and satisfy most practical applications, and the service life of the joint is then predicted in Table 2. At the service temperature of 550 \({}^{\circ}\)C, the pretreated Ti/Mo/Yb\({}_{0.3}\)Co\({}_{4}\)Sb\({}_{12}\) joints can endure about 21 years before the contact resistivity reaches 10 \(\upmu\Omega\cdot\)cm\({}^{2}\). Even when increasing the service temperature to 600 \({}^{\circ}\)C, the joints can survive for 3.9 years before the contact resistivity reaches 20 \(\upmu\Omega\cdot\)cm\({}^{2}\).  
\begin{table}
\begin{tabular}{c c c} \hline
**Temperature/\({}^{\circ}\)C** & **D/10\({}^{-19}\) m\({}^{2}\)/s** & **Y/\(\upmu\)m** \\ \hline
550 & 0.21 & \(1.45\times 10^{-4}\cdot 4\cdot 0^{0.5}\) \\
600 & 9.53 & \(9.76\times 10^{-4}\cdot 4\cdot 0^{0.5}\) \\
650 & 251 & \(50.1\times 10^{-4}\cdot 4\cdot 0^{0.5}\) \\ \hline \end{tabular}
\end{table}
Table 1: Fitting results of the growth rate and the relationship between diffusion layer thickness and aging time (s).  
Figure 7: **(a)** Variation of the diffusion layer thickness of the pretreated joints with the square root of aging time between 550 \({}^{\circ}\)C and 650 \({}^{\circ}\)C. The inset is the enlargement of the result at 550 \({}^{\circ}\)C. **(b)** Variation of In\(D\) with 1000/\(T\) between 550 \({}^{\circ}\)C and 650 \({}^{\circ}\)C.