## 4 Results of Numerical Simulation  
The numerical simulation showing grain growth in the amorphous phase of the Mg\({}_{72}\)Zn\({}_{28}\) alloy required additional assumptions. Appropriate isothermal annealing temperature values were necessary. Based on the research carried out in the work of [PERSON] et al. [30] for the amorphous Mg\({}_{72}\)Zn\({}_{28}\) alloy, the following values were selected: 350 K, 352 K, 353 K, 353.5 K, and 354 K. The total crystallisation time for each of these temperature values was, respectively, 6001 s, 4067 s, 2916 s, 2644 s, and 2447 s. The simulation results are shown in Figure 6. Each graph shows the increase in grain size in the amorphous phase over time for a given isothermal annealing temperature. At the same time, the diffusion of zinc into the grain continued. Therefore, the concentration of this element in the grain increased. It can also be seen how the crystallisation front shifted with the progressing phase transformation. Zinc accumulated at the phase boundary and was displaced by the crystallisation front moving over time. This was mainly due to the inability to diffuse so much zinc into the growing grain. At an isothermal annealing temperature of 354 K, the grain size increased from 20 nm after 50 s to 70 nm after 300 s of isothermal annealing. The ribbon produced from the binary Mg\({}_{72}\)Zn\({}_{28}\) alloy was examined using XRD. The results of these tests are presented in Figure 4. In the plot, only the background is visible,  
Figure 3: Scheme showing how the concentration changes in the grain/amorphous matrix system. represented by the presence of the so-called amorphous \"halo\" with no local peaks. The X-ray diffraction pattern is characterised by a high level of background \"noise\" in relation to the maximum intensity. These observations of the XRD results of the ribbon allow us to identify its amorphous character. The amorphous ribbon of the binary Mg\({}_{72}\)Zn\({}_{28}\) alloy, after heat treatment and isothermal annealing at a temperature of 354 K for 300 s, was subjected to SEM examination. The results of these tests are presented in Figure 5.  
The average grain size was determined using the secant method. The average grain size was approximately 71 nm. Figure 6 shows the results of the numerical simulation showing the impact of the annealing temperature on grain growth. The plots display the zinc concentration in the grain and the amorphous phase on the axis, indicating the distance from the grain centre. Different colours correspond to different annealing times: 50, 100, 150, 200, 250, and 300 s. It  
Figure 4: X-ray diffraction patterns of a Mg\({}_{72}\)Zn\({}_{28}\) cast ribbon. Figure 5: Results of SEM examination of the microstructure of an amorphous ribbon after heat treatment of isothermal annealing at a temperature of 354 K for 300 s.  
can be seen that both the annealing temperature and time affected the grain growth rate and zinc concentration across the whole domain. The key observation is that as the annealing temperature increased, the grain size during the corresponding annealing time also increased. Moreover, the grain size changed more linearly with increasing annealing temperature compared to lower annealing temperatures. The character of the zinc concentration for each annealing time in the grain did not differ significantly at its radius. The zinc concentration in the whole grain was more or less the same. However, as the grain increased or the annealing time increased, the zinc concentration in the grain also increased. This may have been caused by the zinc diffusion through the grain boundary from its reach regions in the amorphous phase. The characteristics of the zinc concentration in the grain did not show any clear functional trend. In the amorphous phase, even for longer annealing times, a higher zinc concentration in the grain was observed; near the grain boundary, a higher zinc concentration was also observed. The zinc concentration in the amorphous phase was highest at the grain boundary in all the cases analysed.