## 2 Size Effects on Microstructure in Solder Joints
### Solidification and Recrystallization in the Solder Matrix  
Apart from the interfacial microstructure, the solder size also has a crucial effect on the microstructure, including the \(\beta\)-Sn phase, eutectic phase, and bulk primary solidification phase in the solder matrix. The mechanical strength of solder balls is commonly regarded to be dependent on the microstructure size, while both the dendrite arm size and the eutectic phase size depend on the solder ball dimension. Furthermore, the microstructures in solder balls and solder joints are directly dependent on the solidification behavior in the annealing or reflow procedures [37; 38; 39]. [PERSON] et al. [40] studied the undercooling behavior of Sn3.9 Ag0.6 Cu solder joints with diameters ranging between 200 to 2000 \(\upmu\)m by differential scanning calorimetry (DSC) reflow tests, and reported a linear relationship between the undercooling temperature and inverse nominal diameter of the solder balls. Additionally, the dendrite arm sizes, which were approximately 0.008 times of the solder ball diameter, were found to be strongly size-dependent. [PERSON] et al. [41] investigated the increasing undercooling behavior of Sn3.0 Ag0.5 Cu solder balls and solder joints with diameters decreasing from 760 to 300 \(\upmu\)m with DSC analysis. They found that the large primary solidification Ag\({}_{3}\)Sn phase had a large undercooling in the small solder balls, while the primary Cu\({}_{6}\)Sn\({}_{5}\) phase formed in the solder joints. In addition, the grain size of the \(\beta\)-Sn phase was reported to be smaller, and the contained eutectic structure was observed to be less in the sample with small undercooling. [PERSON] et al. [42] measured the size and substrate effects on the undercooling of solder alloys including Sn, Sn0.7 Cu, Sn3.5 Ag, Sn3.8 Ag0.7 Cu, and Sn0.1 Ni, and found that the primary solidification phases of those solder alloys were \(\beta\)-Sn, Cu\({}_{6}\)Sn, Ag\({}_{3}\)Sn, and Ni\({}_{3}\)Sn\({}_{4}\), respectively. The primary solidification phase was illustrated to be the principle factor for the degrees of undercooling, followed by the substrate factor and solder ball size factor [42]. Solder volume and substrate effects on the undercooling of different types of solder alloy are summarized in Figures 5 and 6, respectively. The undercooling of solder joints depends on the solidification behavior. The effect of solder joint size on the solidification is intuitively depicted in Figure 2. For Sn3.0 Ag0.5 Cu solder joints with a 200 \(\upmu\)m diameter, the \(\beta\)-Sn phase in Figure 2a is relatively small compared with that in  
Figure 5: Solder ball size effect on undercooling of several types of solder alloy. (Extracted data from [PERSON] et al. [42]). Figure 2b,c. As the solder joints' diameters increase to 500 and 760 \(\upmu\)m, the eutectic structures transform to pseudo-eutectic structures, which means that Ag3 Sn and Cu6 Sn5 phases precipitate during the solidification. Based on the classic solidification theory, the relationship between solder size and undercooling can be described as follows [41]:  
\[d^{3}=\frac{6r}{\pi K_{s}\Delta T}\exp\left[\frac{16\pi T_{sl}^{3}T_{m}^{2}f( \theta)}{3 kL_{v}^{2}T_{s}\Delta T^{2}}\right] \tag{2}\]  
where \(d\) is the diameter of the solder ball, \(\Delta T\) is the degree of undercooling, and \(T_{s}\) and \(T_{m}\) are the theoretical solidification and melting temperature, respectively. Additionally, \(r\) is the cooling rate, \(K_{s}\) depends on the atoms' vibration frequency, \(\gamma_{sl}\) is the crystal-liquid interfacial energy, \(L_{v}\) is the melting enthalpy per volume, and \(f(\theta)\) is dependent on the catalytic potency. For small solder bumps, the driving force for solidification is usually larger than big bumps, which leads to the stronger undercooling. Furthermore, smaller solder bumps are apt to bear more severe temperature gradients, which also promotes the rapid cooling and enhances the undercooling phenomenon.