## 3 Results and Discussion
### Effect of Heat Treatment on the Microstructure of Solder Powder  
To evaluate the effect of heating on microstructure evolution, the aging process was performed at elevated temperatures. Figure 5 presents SEM images of the eutectic Sn-Pb37 powder after heat treatment for 1 day in air at temperatures from 40 \({}^{\circ}\)C to 120 \({}^{\circ}\)C. Figure 6 shows the corresponding SEM micrographs of a cross-section through the powder. Microstructure coarsening similar to that caused by aging at room temperature occurs. However, the coarsening occurs more rapidly at elevated temperatures. The microstructure is coarsened to the same degree after aging at 120 \({}^{\circ}\)C for 1 day as in aging for more than 1 month at room temperature. The aging process in the alloy powder is the same at elevated temperatures as at room temperature; however, it occurs considerably more quickly at elevated temperatures. This indicates that the thermal diffusion coefficient between tin and lead is considerably higher for a higher aging temperature. Precipitation occurs with a further increase in aging temperature. Figure 7 shows SEM images of Sn-Pb37 powder after aging at 150 \({}^{\circ}\)C for 3 days, wherein several small particles were precipitated. The solubility of tin in lead at 150 \({}^{\circ}\)C is approximately 10 wt%, and thus some redissolution of the tin phase may be observed. After cooling to room temperature, the new precipitates are observed to have reduced tin solubility in lead. Figure 5: SEM images of rapidly solidified eutectic Sn–Pb37 powder after heat treatment for 1 day in air at (**a**) 40 \({}^{\circ}\)C, (**b**) 60 \({}^{\circ}\)C, (**c**) 90 \({}^{\circ}\)C, and (**d**) 120 \({}^{\circ}\)C.  
The coarsening kinetics of Pb-rich particles at different aging temperatures can be interpreted using the Arrhenius law. The Pb-rich phase size is measured using the intercept method [21]. Similar to that of conventional materials, the growth of Pb-rich particles can be described using the following equation:  
\[\mathrm{D}^{n}-D_{o}^{n}=k(T)t, \tag{1}\]  
where \(D\) and \(D_{o}\) are the average size of the Pb-rich phase and initial Pb-rich phase size before aging treatment, respectively; \(k\), \(T\), and \(t\) represent a constant, the temperature, and time, respectively. The grain growth constant is temperature dependent:  
\[k(T)\propto e^{-Q/k_{b}T}, \tag{2}\]  
where \(Q\) and \(k_{b}\) are the activation energy and Boltzmann constant, respectively. Figure 6: Cross-sectional microstructure of rapidly solidified eutectic Sn–Pb37 powder after heat treatment for 1 day in air at (a) 40 \({}^{\circ}\)C, (b) 60 \({}^{\circ}\)C, (c) 90 \({}^{\circ}\)C, and (d) 120 \({}^{\circ}\)C.  
Figure 7: SEM images show the precipitate particles in the powders after aging at 150 \({}^{\circ}\)C for 3 days. Therefore, the grain growth models for Pb-rich phase coarsening can be expressed using the following equation, which is in agreement with the models [22]:  
\[\mathrm{D}^{n}-D_{o}^{n}=K_{o}\exp(-Q/RT)t, \tag{3}\]  
where \(D\) and \(D_{o}\) are the phase size after annealing for different durations and the initial phase size, respectively. \(T\), \(n\), \(K_{o}\), and \(Q\) are the temperature, phase size exponent, pre-exponential factor, and activation energy, respectively. According to coarsening theory [23,24], the growth exponent \(n=4\) is used for the coarsening of Pb-rich particles, in which the coarsening process is controlled through grain boundary solution diffusion. The growth kinetics of Pb-rich phase particles are provided by  
\[\mathrm{D}^{4}-\mathrm{D}_{o}^{4}=K_{o}\exp(-Q/RT)t. \tag{4}\]  
Figure 8 plots \(\ln(\mathrm{D}^{4}-\mathrm{D_{o}}^{4})\) versus 1/T for the annealing of Sn-Pb37 powder based on the Pb-rich particle coarsening analysis. A straight line through the data points provides an activation energy of \(Q=45.53\pm 4.23\) kJ/mol, which is consistent with the activation energy of grain boundary diffusion in Sn and Pb [21].