**4. Discussion**

Looking at Figure 9, the situations in which particles grow while keeping volume constant are Ostwald ripening. According to classical particle growth theory, a growth of Ostwald ripening [31] is formulated as

$$d\_0^{\dot{s}} - d\_0^{\dot{s}^\*} = kt \tag{1}$$

in a diffusion control situation. Here, *d* and *d*0 are the average particle diameter and initial average particle diameter, *k* is constant and *t* is annealing time. The relationship between the cube of average particle diameter and annealing time in this study is shown in Figure 10. The early stage of particle growth in Al-9.8%Si-3ppmP cast alloy does not correspond to the growth manner expressed in Equation (1). The Si-particles grow proportionally after 3.6 ks of heat treatment. The particle growth in Al-10.1%Si-4ppmP-108ppmSr cast alloy almost obeys Equation (1) though a little difference is seen in the early stage of growth. It is found that Si-particle growth in Al-10.1%Si-4ppmP-108ppmSr cast alloy is faster than that in Al-9.8%Si-3ppmP cast alloy. A very small difference in particle growth rate had been expected because the alloys were simple binary Al-Si system alloys though there was a difference in 108ppm Sr content. However, a six-times difference is recognized in the comparison with the slopes of fitting lines between both of the alloys in the later stage of growth.

By magnifying a three-dimensional volume image, morphology changes of Si-particles in Al-9.8%Si-3ppmP cast alloy are shown in Figure 11. Actually, Si-particles distribute densely as shown in Figure 11a. To understand the morphological changes of particles easily, an image removing surrounding particles is Figure 11b. In the as-cast, the morphology of Si-particles is mildly complex, possessing a fine rod-like shape, which is elongated along the solidification direction, and a partial small plate-like shape, which is broader than the rod part. While heat treatment is progressing, the Si-particles are divided into plural segments (Figure 11c,d). After that, fragmented particles become gradually round and approach into a sphere-like shape that has the smallest surface area (Particle A in Figure 11e). An elongated Particle B shown in Figure 11e gradually shortens in length, and then becomes close to a sphere-like shape. An elongated Particle C seen in the center of the figures

thickens in diameter during heat treatment. However, the tip position does not change so much after 1.8 ks of heat treatment and the small change is observed in particle shortening along longitudinal direction. It is found that the morphology change is slightly different depending on the length of the rod-like shaped particles.

**Figure 10.** The relationship between cube of average particle size and annealing time.

**Figure 11.** Magnification of three-dimensional volume image, illustrating morphology changes of Si-particles in Al-9.8%Si-3ppmP cast alloy. (**a**) as-cast, (**b**) as-cast (one particle), (**c**) heat-treated at 773 K for 450 s, (**d**) heat-treated at 773 K for 900 s. (**e**) heat-treated at 773 K for 1.8 ks. (**f**) heat-treated at 773 K for 3.6 ks, (**g**) heat-treated at 773 K for 7.2 ks, (**h**) heat-treated at 773 K for 14.4 ks.

Figure 12 shows three-dimensional images viewing Figure 11 from the rear side. Particle D fragmented at 900 s heat treatment seen in Figure 12c becomes a sphere-like shape in (d) 1.8 ks and (f) 3.6 ks of heat treatment. The particle is merged with a U-shape particle growing behind its back. The U-shape particle that absorbed Particle D probably will close a gap and be a sphere-shaped particle if heat treatment continues furthermore. A protuberance indicated by the letter E in Figure 12d is not cut off at the neck, shortens in length gradually and finally is absorbed by a plate-shape particle. The morphology change of Si-particles is very complex in Al-9.8%Si-3ppmP cast alloy. It was found that the changes during heat treatment were not only just fragmentation and spheroidizing.

**Figure 12.** Three-dimensional images viewing Figure 11 from the back side. (**a**) as-cast (one particle), (**b**) heat-treated at 773 K for 450 s, (**c**) heat-treated at 773 K for 900 ks, (**d**) heat-treated at 773 K for 1.8 ks, (**e**) heat-treated at 773 K for 3.6 ks, (**f**) heat-treated at 773 K for 7.2 ks, (**g**) heat-treated at 773 K for 14.4 ks.

Figure 13 shows magnified three-dimensional images of Si-particles in Al-10.1%Si-4ppmP-108ppmSr cast alloy. Figure 13a shows the as-cast state displaying peripheral particles. One particle is shown in Figure 13b by removing the peripheral particles. It can be confirmed that Si-particles in Al-10.1%Si-4ppmP-108ppmSr cast alloy are coral-like complex shapes having numerous branches. As well as Al-9.8%Si-3ppmP cast alloy, separation of Si-particles is found at branches and necks surrounded by dashed lines shown in Figure 13c–e. Frequency of separation in Al-10.1%Si-4ppmP-108ppmSr cast alloy is higher than that in Al-9.8%Si-3ppmP cast alloy because the Si-particles in Al-10.1%Si-4ppmP-108ppmSr cast alloy possess many branches in as-cast. Si-particle segmentations formed by separation thicken gradually. Then, the shape approaches a sphere-like form, with a shortening the length of longitudinal direction. Separation also causes a long trunk of Si-particles, as seen in the center of the figure. The trunk becomes segmented and grows to a sphere-like shape. Such behaviors correspond to those that we had expected. However, the morphology change was clearly different with the observed elongated particles in Al-9.8%Si-3ppmP cast alloy. Therefore, the difference of growth rates in the two cast alloys would be brought about by the difference of growth behavior at long elongated Si-particles. That is, the slow growth rate in Al-9.8%Si-3ppmP cast alloy in heat treatment is due to less separation of long elongated rod-shape particles, which are the main morphological features of eutectic Si-particles by the solidification of Al-9.8%Si-3ppmP cast alloy. In addition, the solidification reaction is different in the two cast alloys as seen in the different eutectic Si-particle morphology. Not only is the eutectic Si-particle size in Al-10.1%Si-4ppmP-108ppmSr cast alloy smaller than that in Al-9.8%Si-3ppmP cast alloy, but the eutectic grain formed in the eutectic reaction is also of a fine size in Al-10.1%Si-4ppmP-108ppmSr cast alloy. Therefore, the contribution of grain boundary diffusion in addition to lattice diffusion could also be a factor.

**Figure 13.** Magnified three-dimensional images of Si-particles in Al-10.1%Si-4ppmP-108ppmSr cast alloy. (**a**) as-cast, (**b**) as-cast (one particle), (**c**) heat-treated at 773 K for 450 ks, (**d**) heat-treated at 773 K for 900 ks, (**e**) heat-treated at 773 K for 1.8 ks, (**f**) heat-treated at 773 K for 3.6 ks, (**g**) heat-treated at 773 K for 7.2 ks, (**h**) heat-treated at 773 K for 14.4 ks.

As results of the detail three-dimensional observation of microstructural change through Figures 11 and 13, it was observed that many Si-particles separate and isolate as small segments in the early stage of heat treatment up to 900 s. Considering the relationship between microstructure and mechanical property, it is found that the mechanical properties shown in Figure 5 also start to change after 900 s heat treatment. The non-destructive observation in this study by means of synchrotron radiation nanotomography supports the idea that the connectivity of the strengthening phase affects strength and elongation as reported recently [32,33]. Further investigation and consideration for this will be possible in image-based simulations that are constructed from three-dimensional volume image tomography.
