Fig. 4: Model validation by generating images for seen and unseen conditions with the developed C-DDPM.

a Physics-based C-DDPM validation workflow. Three microstructural images are generated for each experimentally available condition. The MS-SSIM is calculated in order to quantify the similarity between generated images and real FESEM-BSE micrographs, in terms of brightness, contrast and texture. Subsequently, the microstructural features, i.e. IMC-layers and Kirkendall pores, are qualitatively checked regarding their realism. In order to quantify how well the IMC-layer thicknesses and Kirkendall pore areas obey the underlying physical growth mechanisms, the microstructural features are manually labelled in both generated images and real FESEM-BSE micrographs. From the phase labels, microstructural features can be extracted and quantified. Based on the evolution of the depicted microstructural features, physics-based model validation is performed, for seen conditions and for unseen conditions. b Exemplary comparisons of real FESEM-BSE micrographs, marked ‘SEM’, with virtually generated ones, marked ‘genAI’ for seen datapoints and c unseen datapoints. Scalebars of 5 µm are valid for all images. The respective first rows show real FESEM-BSE micrographs for b exemplary seen conditions with ECD1-0 h, PVD-100 h, ECD2-1000 h and ECD3-3000 h and c the unseen conditions with PVD-300 h and ECD1-750 h. The second rows show virtually generated images for the respective conditions from the first row. d Phase labels for the micrographs shown in b and c. Cu is labelled in blue, Cu₃Sn in yellow, Cu₆Sn₅ in purple, the SAC305 bulk in magenta, pores in green and Ag₃Sn-precipitates near the interface in salmon. Scalebar of 5 µm is valid for all labelled images. e Deviations of the phase labels for pore areas (green), Cu₆Sn₅-thicknesses (purple) and Cu₃Sn-thicknesses (yellow) in generated images from the real mean values. Mean standard deviations of the respective features in real micrographs are shown as solid lines, whereas the deviations of generated images from the real mean values are shown as circles. For seen conditions, the mean deviations of all images that are generated for each of the four Cu-impurity contents are plotted, whereas for unseen conditions, the mean deviations of the three generated images for each of the two unseen conditions are plotted. The asterisk marks the deviation of the Cu₃Sn-layer thickness in the generated images for ECD1-750 h. Numerical values of all deviations are given in Supplementary Tables 3–5.