## 3 Finite Element Analysis  
### Modeling of the Cu-IMC-Resin Microstructure Based on 3D Image Reconstruction  
3D image reconstruction of the serial cross-sectional SEM images provided useful insights about the mechanical properties of the Cu-IMC-resin microstructure. Figure 1 shows the segmentation flow of these images. The resolution of the original SEM images was 0.06 \(\upmu\)m/pixel. These images were segmented into three regions, which were Cu, IMCs, and PI resin, based on their brightness using a commercial image processing software, Amira, as shown in Figure 1b,c. The subsequent steps were performed by an image processing and meshing software, Scan IP, Simpleware. These steps included resizing of the images into the resolution of 0.5 \(\upmu\)m/pixel and Gaussian smoothing, as shown in Figure 1d. The latter was done to reduce the number of FE mesh cells. Finally, the images were resegementated to remove the tiny cavities and islands, whose sizes were approximately less than 1 \(\upmu\)m, as shown in Figure 1e. Figures 2 and 3 show the reconstructed cubic 3D model with sizes of 15 and 50 \(\upmu\)m, respectively. These models were composed of quadratic hexahedral elements,where the number of nodes and elements in the 15- and 50-\(\upmu\)m-sized model were 118,305 and 65,919, and 1,535,038 and 833,105, respectively. The volume fractions of Cu, IMC, and PI resin in the latter model were 24.0, 32.7, and 43.3%, respectively. Figure 1: Segmentation flow of the copper- intermetallic compound-resin (Cu-IMC-resin) microstructure composed of Cu, Cu-Sn IMC, and polyimide (PI) resin. (**a**) Schematic of stacked SEM images, (**b**) preprocessed SEM image, (**c**) Otsu segmentation of the image, (**d**) image after Gaussian smoothing, and (**e**) resegementated image. Figure 2: Cu-IMC-resin model of size 15 \(\upmu\)m for finite element (FE) analysis, which is based on 3D image reconstruction. The model comprises Cu (green), IMC (pink), and resin (light blue). Mechanical FE simulations were performed to estimate the macro- and micro-deformation properties of the microstructure using the commercial software, ANSYS Workbench 18.2 [35]. The 15-\(\upmu\)m-sized model (Figure 2) was used to predict the effect of IMC and resin on the homogenized mechanical properties. The model was divided into regions A, B, and C, which represented the presence of Cu, IMC, and resin, respectively. The 1/8 model was used for FE analysis considering the symmetry conditions. Ideal contact was assumed between all the interfaces. An external displacement of 0.075 \(\upmu\)m along the x-axis, equivalent to the macroscopic strain of 0.5%, was applied to the yz-surface of the FE model. Ambient temperatures of -55, 22, 100, and 175 \({}^{\circ}\)C were considered in the simulations. The homogenized nominal stress-stain curves of the microstructure were calculated at each temperature from the displacement \(u\) and the averaged reaction force \(F\) on the surface subjected to external displacement, using the following expressions:  
\[\sigma=\frac{F}{L^{2}}, \tag{1}\]  
\[\varepsilon=\frac{u}{L}, \tag{2}\]  
where \(L\) is the size of the cubic model. The homogenized Young's modulus and the yield strength were obtained from these curves. A parameter study was performed based on the cases shown in Table 1, where the effects of materials properties in the regions B (IMC) and C (resin) on the microstructure were validated. The properties of PI resin were obtained using tensile tests of the cured PI resin film, and that of the epoxy resin (reference) were acquired from its commercial specifications. Further, air instead of resins was considered as the region C in the cases for comparison to clarify the effect of the mechanical properties of resins on the microstructure. Furthermore, the cases with SAC305 instead of IMC as the region B were examined to clarify the effect of the mechanical properties of IMC on the microstructure. The relevant parameters of the materials are summarized in Tables 2 and 3 [36; 37; 38; 39; 40; 41].