# Research on Lifespan Prediction Methods Using Ultrasonic Microimaging for Electronic Packaging
## 4 Solder Joint Failure Modeling and Lifespan Prediction
### Validation of Solder Joint Failure Model Based on Positional Differences  
To validate the adaptability of the model when addressing different solder joint locations on flip chips, image data from four solder joints located in various positions, solder joint 4 (inner circle), solder joint 42 (outer circle), and solder joints 69 and 102 (the corners of the chip) were selected as samples for validation. The validation results are shown in Figure 13. The results indicate a high degree of fit between the failure curves of the solder joints and the cumulative failure model based on the two-parameter Weibull distribution. Although there are differences in the shape of the curves, primarily reflected in the failure rates and start and end times, all samples typically underwent a process of no defects, defect initiation, and final failure.  
Figure 12: Fitting result of failure data for solder joint 107 on flip chip 2 of experimental sample 1.  
To further analyze the differences in behavior at various stages, precise failure model parameters were calculated for the four solder joints and are summarized in Table 2. The combination of the failure curves in Figure 13 and the cumulative failure model parameters in Table 2 demonstrates the variability in the parameters of the two-parameter Weibull distribution model for solder joints at different locations. The failure shape parameter is consistently near 1.3 for all joints, indicating uniformity in the statistical distribution shape and suggesting that the failure mechanisms are fundamentally the same. However, the failure time parameter shows clear position dependence, reflecting the variance in aging rates due to differences in stress and thermal cycling at different locations. This finding is crucial for understanding the reliability of solder joints and directly relates to the performance stability and lifespan prediction of the chip under varying environmental conditions.  
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Solder Joint on Flip Chip 2 of Sample 1** & **Scale Parameter \(\mathbf{\lambda}\)** & **Shape Parameter \(\mathbf{k}\)** \\ \hline Solder joint 4 & 74.22 & 1.24 \\ Solder joint 42 & 81.18 & 1.26 \\ Solder joint 69 & 84.45 & 1.27 \\ Solder joint 102 & 87.65 & 1.34 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Fitting parameters of failure models for solder joints at different locations in Figure 13.  
Figure 13: Fitting results of failure models for solder joints at different locations on chip 2, sample 1. (a) Solder joint 4; (b) solder joint 42; (c) solder joint 69; (d) solder joint 102.  
After processing the entire life-cycle ultrasonic images of all solder joints as described above, the model's performance was evaluated using the maximum absolute value of the fitting residuals \(|e_{i}|\) as the metric. The formula for calculating the maximum absolute value of fitting residuals is:  
\[|e_{i}|=max(|y_{i}-F(t_{i})|) \tag{5}\]  
where \(y_{i}\) represents the actual observed value and \(F(t_{i})\) is the model's predicted value. The results computed for all chips on four experimental sample boards are shown in Table 3.  
The results in Table 3 demonstrate that when fitting the solder joint data of flip chips using the two-parameter Weibull distribution cumulative failure model, the absolute value of the residuals \(|e_{i}|\) does not exceed 0.24 on any of the experimental sample boards. This consistency in handling data from different samples underscores the model's reliability and universality in practical applications.