## 3 Results and Discussion
### Microstructural Analyses Using X-ray Computed Tomography (XCT)
#### 3.2.2 Impact of Cooling Rate  
Figures 12 and 13 show, respectively, the impact of the cooling rate on the distribution of the \(p\) ratio and the convex hull sphericity (s) as a function of particle sizes. From the plots we can see that the trends observed for the different compositions (Figures 10 and 11) hold true for the different cooling rate samples as well. With a faster cooling rate, the size of the largest particle size decreases, but the trends of the decrease in \(p\) ratio and \(s\) in the 1000-10,000 \(\upmu\)m\({}^{3}\) size range still hold true. As previously discussed, the shape and structure of regions rich in Zn are significantly influenced by the dendritic structures of Mg and the eutectic components that form within the already solidified regions between the Mg dendrites. When there is a small volume of liquid present with a eutectic composition, it tends to take on a more spherical shape, as indicated by the data (see Figure 10). However, as the local volume of the Zn-rich liquid increases, the morphology becomes more complex. This complexity reflects the earlier stages of the solidification process, where the shape of the liquid is molded by the surrounding Mg structures that have already solidified. The similar trends observed in particle morphology at different cooling rates are attributed to the similarity of dendritic structures within the cooling rate range of 0.44 \({}^{\circ}\)C/s to 1.44 \({}^{\circ}\)C/s, as indicated by the minimal variation in dendritic spacing reported by [PERSON] et al. [42]. Figure 12: Correlation of the ratio between particle volume and convex hull volume as a function of particle volume with cooling rate variation: (**a**) the combined plot of \(p\) ratio for the Mg-0.6 Si-3 Zn alloy at three different cooling rates as a function of particle volume shows that as the particle volumes increase, we see a drop in the \(p\) ratio indicating the particles becoming a complex network; (**b**) shows typical particle morphologies at the different size ranges for the 0.44 \({}^{\circ}\)C/s rate, while (**c**) shows the typical particle morphologies at the different size ranges for 0.85 \({}^{\circ}\)C/s, and (**d**) shows the typical particle morphologies at the different size ranges for 1.44 \({}^{\circ}\)C/s. Further, looking closely at the dependency of the value of \(p\) on composition as well as the cooling rate, the data seem to have a tri-linear trend. Based on the results shown in Figures 9a and 11a, after the initial plateau in \(p\) at 0.68 that lasts until a particle volume of 1000 \(\upmu\)m\({}^{3}\), the \(p\) value starts to drop with increases in particle volume. This drop in \(p\) with increases in particle volume is arrested as the particle becomes larger than 10,000 \(\upmu\)m\({}^{3}\) in volume. For particles larger than 10,000 \(\upmu\)m\({}^{3}\) in volume, the \(p\) ratio becomes relatively stable at around 0.05, with a slight tendency to decrease with increases in particle volume. A simple regression of the data between 1000 and 10,000 \(\upmu\)m\({}^{3}\) particle volumes shows that this linear part of the semi log graph can be approximated by the equation  
\[p=-0.63\log_{10}\left(v\right)+2.57 \tag{1}\]  
where \(v\) is the volume of the particle in \(\upmu\)m\({}^{3}\), and \(p\) is the particle-to-hull volume ratio in decimal units. While these values seem to be consistent between the alloys and solidification conditions followed in this research, they may change as the developed morphometric parameters are applied to other compositions and solidification conditions.