## 4 Discussion
### Interface Mechanical Analysis of Liquid Solder CF Process  
Figure 12 displays the physical model of the liquid solder's front-end during the CF process. This model included surface tension values at the interfaces between liquid and solid, gas and solid, and gas and liquid, which were represented by symbols \(\sigma_{sl}\), \(\sigma_{sg}\), and \(\sigma_{lg}\), respectively. The surface tension caused the three-phase interface to balance at the front-end of the liquid solder before the EU wave was applied. When the EU wave was applied, the sound pressure difference was generated inside the liquid solder. When the sound pressure difference was greater than 0 and greater than the surface tension \(\sigma\) and friction \(\tau\), the liquid solder had a tendency to move forward. When the acoustic pressure difference was less than 0 and greater than the surface tension \(\sigma\) and friction \(\tau\), the liquid solder was hindered from moving forward. Due to the friction force and half-wave offset, the liquid solder could not be retracted to its original position. Over time, the liquid solder was able to fill the gap. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Micro-Point**} & **Sound Pressure** & **Sound Pressure** & **Initial Phase** & **Half-Wave** \\  & **Amplitude** & **Frequency** & **(μs)** & **Offset (MPa)** \\ \hline F & 0.121 & 20 & 13.501 & 0.092 \\ G & 0.07 & 20 & 13.025 & 0.031 \\ H & 0.039 & 20 & 13.115 & 0.018 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The sound pressure expression parameters of each micro-point. Figure 11: Variation curve of sound pressure of micro-point in liquid solder. To analyze the driving force of the liquid solder from the microscopic point of view, it is necessary to extract a micro-element in the liquid solder and establish a mathematical model, as shown in Figure 13. Before the EU wave was applied, the micro-element was subjected to gravity and the static pressure \(P_{A}\) with equal direction. After the EU wave was applied, it was subjected to the sound pressure \(p\) and the viscous resistance \(\tau\) when the micro-element moves. During the CF process of liquid solder, the micro-element motion in the gap of the substrate still satisfies [PERSON]'s law. The two-dimensional [PERSON]'s motion equation can be expressed as follows [36,37]:  
\[\rho\frac{du}{dt}=-\Delta P_{ux}-\eta\frac{du}{dy}-\Big{(}\sigma_{\rm lg}\cos \theta-\sigma_{sl}+\sigma_{sg}\Big{)}/S \tag{9}\]  
where \(P_{ux}\) is the sound pressure difference in the X direction, \(\eta\) is the viscosity of the liquid solder, and \(S\) is a cross-sectional area of the micro-element. Extending the analysis to a liquid layer consisting of continuous microelements with all cross-sectional areas \(S\), the overall equation of motion can be given by the following:  
\[-\rho\frac{du}{dt}=\Delta P_{ul}-P_{\sigma}-\int\limits_{0}^{d}\eta\frac{ \partial u}{\partial y}dx \tag{10}\]  
where \(P_{ul}\) is the sound pressure difference in the X direction of the liquid layer with a length of \(l\) and a thickness of \(d\). \(P_{\sigma}\) is the surface additional pressure formed by the surface tension pointing to the inside of the liquid solder, which can be expressed as follows:  
\[P_{\sigma}=\frac{\sigma_{\rm lg}}{l}=\frac{2\sigma_{\rm lg}\cos\theta}{d} \tag{11}\]  
According to Equation (8), the general expression of the sound pressure gradient on the liquid layer during the CF process of the liquid solder can also be expressed as follows:  
\[\Delta P_{u}=P_{0}\times\sin[2\pi f(t-t_{0})]+P_{s} \tag{12}\]  
Figure 12: Force model of liquid solder front-end in (**a**) the initial state, (**b**) moving forward state, and (**c**) retraction state. Figure 13: Stress analysis of a micro-element in liquid solder. According to Equation (12), it could be concluded that the sound pressure gradient changed with time, so it could explain why the liquid solder had an instantaneous process of repeated forward and contraction motion during the CF process.