## 3 Results and Discussion
### Isothermal Aging
#### 3.2.2 Diffusion Kinetics of the Interface  
The pretreated joints were then isothermally aged at 550 \({}^{\circ}\)C and the inter-diffusion between Ti and Yb\({}_{0.3}\)Co\({}_{4}\)Sb\({}_{12}\) occured at the Mo-Ti/Yb\({}_{0.3}\)Co\({}_{4}\)Sb\({}_{12}\) interface. The BSE graphs of the interfacial cross section for the joints after 20-day and 50-day aging are shown in Figure 4a,b. After being aged for 50 days, the diffusion layer is about 0.3 \(\upmu\)m thick and consists of about 90 at % Ti and 10 at % Sb without any sign of Mo. The extremely slow growth-speed of the diffusion layer implies that the new Mo-Ti layer formed in the pretreatment effectively dampens the diffusion of Ti atoms before they reach the Yb\({}_{0.3}\)Co\({}_{4}\)Sb\({}_{12}\) matrix. Besides, Figure 4c,d show that there is no evident change in the distribution of Ti and Mo in the formed Mo-Ti layer, compared to the pretreated joints. The pretreated joints were also isothermally aged at 600 \({}^{\circ}\)C and 650 \({}^{\circ}\)C. It was found that the growth rate of the diffusion layer between Ti and Yb\({}_{0.3}\)Co\({}_{4}\)Sb\({}_{12}\) increases with an increasing aging temperature. According to Figure 5, after being aged at 600 \({}^{\circ}\)C for 41 days, the diffusion layer at the  
Figure 4: The interfacial evolution of the pretreated joints isothermally aged at 550 \({}^{\circ}\)C. (**a**) BSE graph, 20 days; (**b**) BSE graph, 50 days; (**c**) distribution of Mo, 50 days; (**d**) distribution of Ti, 50 days. Figure 3: The interfacial microstructure of the Ti/Mo/Yb\({}_{0.3}\)Co\({}_{4}\)Sb\({}_{12}\) joints pretreated at 550 \({}^{\circ}\)C for 16 days. (**a**) the BSE graph and spot testing, (**b**) line scan results, (**c**) the distribution of Ti. Mo-Ti/Yb\({}_{0.3}\)Co\({}_{4}\)Sb\({}_{12}\) interface reaches about 2 \(\upmu\)m and consists of two sub-layers of Ti\({}_{3}\)Sb and Ti\({}_{2}\)Sb. Figure 6 shows that after being aged for nine days at 650 \({}^{\circ}\)C, the total thickness of the diffusion layer at the Mo-Ti/Yb\({}_{0.3}\)Co\({}_{4}\)Sb\({}_{12}\) interface grows to about 4.5 \(\upmu\)m. Beside the sub-layers of Ti\({}_{3}\)Sb and Ti\({}_{2}\)Sb, there also appears a sign of TiCoSb in the front of the diffusion layer. To reveal the diffusion kinetics, we analyzed the relationship between the total diffusion layer thickness, aging time, and aging temperature. Figure 7a shows that at all the three aging temperatures, the total thickness of the diffusion layer is linear with the square root of the aging time, which indicates that the evolution of the Mo-Ti/Yb\({}_{0.3}\)Co\({}_{4}\)Sb\({}_{12}\) interface is a diffusion-controlling process between 550 \({}^{\circ}\)C and 650 \({}^{\circ}\)C. Therefore, the diffusion layer thickness can be expressed by the following parabolic equation [38,39]:  
\[Y-Y_{0}=(Dt)^{0.5} \tag{1}\]  
where \(t\) is the aging time, and \(Y_{0}\) and \(Y\) are the total diffusion layer thickness at the origin of aging and after being aged for a time period of \(t\), respectively. \(D\) is the growth rate of the diffusion layer, which is a function of the aging temperature. By fitting the data of the diffusion layer thickness as a function of \(t^{0.5}\), \(D^{0.5}\), which is the slope of the fitting result, can be obtained. To determine the activation energy for the interfacial diffusion, the Arrhenius relationship was used,  
\[D=D_{0}\exp(-\frac{Q}{RT}) \tag{2}\]  
where \(D_{0}\) is the growth constant, \(Q\) is the apparent activation energy for the growth of the total diffusion layer, \(T\) is the temperature in Kelvin, and \(R\) is the gas constant, 8.314 J/(mol\(\cdot\)K). Equation (2) can be transformed into:  
\[\ln D=-\frac{Q}{1000R}\cdot\frac{1000}{T}+D_{0} \tag{3}\]  
By plotting and fitting ln\(D\) against 1000/\(T\), \(Q\) can be obtained. Table 1 shows detailed \(D\) values and relations between the diffusion layer thickness and aging time.