## 3 Data-Constrained Modelling for the Samples  
### Data-Constrained Modelling  
For each sample, the volume fractions of different compositions in each CT voxel of region c were obtained with data-constrained modelling [30] by minimizing the objective function T shown in Equation (1). \[\left\{ \begin{array}{l}\mathrm{T=\sum\limits_{n=1}^{N}\left[\sum\limits_{m=0}^ {M}\left(\mathrm{\mu^{(m)}V_{n}^{(m)}-\hat{\mathrm{\mu}}_{n}}\right)^{2}+\sum \limits_{m=0}^{M}V_{n}^{(m)}S^{(m)}+E_{n}\right]}\\ \sum\limits_{m=0}^{M}V_{n}^{(m)}=1\\ 0\leq V_{n}^{(m)}\leq 1\end{array} \right. \tag{1}\]  
where n (n = 1, 2, 3,..., N) represents the CT voxel number in region c of the sample, and N is the quantity of voxels in region c, as shown in Table 2. M denotes the total number of composition types in the sample. For a pure silica sample, M = 1, with m = 0, 1 representing pore and silica compositions, respectively. For samples 2-6 containing mixed compositions of iron powder and silica, M is 2, with m = 0, 1, 2 representing pore, silica, and iron compositions, respectively. \(V_{n}^{(m)}\) represents the volume fraction of composition m in voxel n. In (I) of Equation (1), \(\hat{\mathrm{\mu}}_{n}\) represents the reconstructed X-ray linear absorption coefficient value at voxel n in the CT slice, and \(\mathrm{\mu^{(m)}}\) represents the theoretical X-ray linear absorption coefficient value of composition m. Since continuous spectrum X-rays were used in this study, an equivalent energy method was employed to determine the theoretical absorption coefficient values of each composition. In order to determine the equivalent X-ray energy of the experiments, the average linear X-ray absorption coefficients of the standard aluminum samples imaged together with the region c of each sample were calculated. For each X-ray CT experiment, we selected one CT slice of the standard aluminum sample every 25 CT slices, and calculated the average linear X-ray absorption coefficient of these slices as the average linear X-ray absorption coefficient value in this experiment. Using the built-in X-ray linear absorption coefficient database in the DCM software [34], the corresponding  
Figure 5: Optical microscope images (**a**–**e**) and the threshold segmentation results (**a**–**e**) for samples 2–6. The pixel size of each image is 2560 \(\times\) 1920 pixels, and each pixel represents a physical size of 0.206 \(\times\) 0.206 \(\mathrm{\mu^{m}}\). (**a**,**a**) Sample 2; (**b**,**b**’) sample 3; (**c**,**c**’) sample 4; (**d**,**d**’) sample 5; (**e**,**e**’) sample 6. X-ray energy for the absorption coefficient values of the standard aluminum sample was determined. This energy was used as the equivalent X-ray energy for this experiment. The theoretical X-ray linear absorption coefficients of each composition of the samples were determined according to this equivalent energy. The equivalent energies of the samples are listed in Table 4. Table 4 shows that despite using the same imaging parameters and X-ray tube voltage for samples 2-6, the equivalent energies are different. Sample 3, in particular, shows a significant difference compared to the others. The reason may be related to the fact that the CT imaging experiments for these samples were conducted at different times. After system restarts, the stabilization time of the equipment or errors in the voltage control and feedback system may lead to differences in the equivalent energies even when the same settings are used before and after the restart. Additionally, varying degrees of image artifacts and differences in the relative positions of the aluminum samples during different experiments may result in slight differences in the statistical values of X-ray linear absorption coefficients at the same acceleration voltage. This may also contribute to variations in the equivalent energies across different experiments.