## 2 Size Effects on Microstructure in Solder Joints
### IMC Formation and Growth
[28] investigated the bump size and reflow time effects on the IMC thickness and growth rate. They found that the IMC layers in smaller Sn1.0 Ag0.5 Cu/Cu solder joints were thicker than those in larger solder joints after the initial reflow, but the IMC growth rate in smaller joints was slower than in larger solder joints during the second and third reflows. In contrast to the findings by [PERSON] et al. [28], [PERSON] et al. [29] conducted thermal shock tests on Cu/Sn3.0 Ag0.5/Ni solder joints with 100 and 200 \(\upmu\)m pitches, and found that the IMC growth rates in the 100 \(\upmu\)m pitch solder joints were obviously higher than those in joints with a 200 \(\upmu\)m pitch. Studies of Cu/Sn/Cu under 160 \({}^{\circ}\)C solid-state thermal aging by [PERSON] et al. [30] indicated that the thickness effect of solder joints on the IMC growth rate was not that \"the thinner the solder layer was, the faster the IMC grew\" as usually believed. Instead, there was a threshold value of \(\delta\) = 30 \(\upmu\)m for solder thickness, and the initial growth rate of IMC in thinner solder joints (\(\delta\) < 30 \(\upmu\)m) was greater than that of thicker solder joints (\(\delta\) > 30 \(\upmu\)m), but the former decreased and the latter increased with an increasing aging time [30]. The size-dependent IMC growth rate was also reported by [PERSON] et al. [31]. Based on both solid-state and liquid-state diffusions, they found that the commonly reported size-dependent IMC growth rate may be only valid under a liquid-state diffusion, but invalid under a solid-state diffusion. To obtain a more general relationship between solder joint size and IMC growth, the IMC thicknesses corresponding to solder joints with different ratios of solder volume to substrate area (V/A) are presented in Figure 3. It is observed that a linear relationship, as shown in Equation (1), seems to exist between the logarithmic IMC thickness and the V/A ratio, even though the slope of the linear fitting line can be positive or negative. \[t=(c\cdot r)^{k}+t_{0} \tag{1}\]  
Figure 3: IMC thickness of solder joints with different substrate area/solder volume (V/A) ratio. where \(r\) is the V/A ratio, \(t\) is the thickness of the IMC, \(t_{0}\) is related to the IMC thickness of the initial state, \(c\) is a function related to the IMC growth rate, and the slope \(k\) depends on the specimen shape. Based on the data reviewed in Figure 3, a negative slope \(k\) usually occurs in line-type or bi-lateral solder joints, while unilateral solder joints always have a positive \(k\) value. Therefore, a constraint effect in line-type or bi-lateral solder joints, which slows down the IMC growth, is speculated to be the dominant mechanism for such a negative \(k\) value. The constraint effect on the IMC growth can be seen in Figure 4, where the solder joints with small gap sizes (50 and 150 \(\upmu\)m) only form thin lamellar-type IMCs instead of scallop-type IMCs, which have been observed in the other three types of solder joint [31]. From the interaction point of view, the IMC growth rate is dominated by the Cu diffusion. Therefore, the underlying mechanism for the size effects of solder joints is highly likely to be related to the Cu flux. [PERSON] et al. [32] established an analytical model to investigate the growth rate of Cu\({}_{3}\)Sn, which was found to be joint-size dependent and obey the parabolic relation with time. [PERSON] et al. [33] measured the IMC evolution of Sn3.0 Ag0.5 Cu solder joints on Cu or Ni-P pads, and the Cu flux variation was inferred to be the cause of solder volume size effects. [PERSON] et al. [34] performed experiments and simulations regarding the Cu concentration effect on different sizes of Sn3.0 Ag0.5C0.1 TiO\({}_{2}\)/Cu solder joints, and concluded that the IMC formation may not simply obey diffusion-controlled or interfacial reaction-controlled growth, but was instead a combination of both. Compared with bulk diffusion, surface diffusion and intergranular diffusion are fast diffusion channels.