*3.3. Nucleation of Xe Bubbles at the GB*

Early-stage nucleation and growth of Xe bubbles were simulated using the MD method. We simulated the nucleation and growth of Xe bubbles at 1000 and 2000 K. When the system reached a thermodynamic equilibrium after relaxing the system at a given temperature and volume, Xe atoms were regularly added to the bubble. For each Xe atom added, the system needed 10 ps to relax, to ensure there was no change in the properties of the system over time, i.e., the equilibrium state was reached. Thus, the Xe bubble growth was simulated until the bubbles could contain 50 Xe atoms.

Figure 4 lists the differences in the nucleation configurations of Xe bubbles at the Σ5(310) GB and the bulk. At 2000 K, Xe atoms were more regular in the bulk (Figure 4b). This is consistent with a previous report [3], which showed that Xe bubbles evolve into a glassy/amorphous state, a nearly face-centred cubic (FCC) solid structure and subsequently to a high density amorphous or glassy state. At the GB, Xe atoms are absorbed by the GB and multiple Xe atoms form a planar structure, which is similar to the case of rhenium (Re) atoms in metals [55]. The results are consistent with the figure showing the formation energies of an interstitial Xe at the GB. However, comparing the configurations at 1000 K, as shown in Figure 4a, we found that at a smaller bubble size (20 Xe atoms), Xe atoms at the GB remain at the GB plane. However, with the growth of Xe bubbles, most Xe atoms in the larger bubbles are located in the bulk-like region away from the GB plane. This is attributed to the increase in temperature, which causes a change in the lattice constant and the reconstruction of the GB. An increase in temperature decreases the interfacial energy and it is easy to form segregation at the GB, where the energy of Xe atoms in the bulk is higher than that of the atoms at the GB. Thus, Xe atoms spontaneously converge toward the GB.


**Figure 4.** Comparison of the configurations bubbles at the Σ5(310) GB and bulk UO2 at (**a**) 1000 K and (**b**) 2000 K.

Next, to analyze the characteristics of Xe bubbles, the volume (V) and pressure (P) of the bubbles after inserting each Xe atom were calculated. Figure 5 shows the relationship between the volume and pressure of Xe bubbles and the number of Xe atoms in the bubble. The inset of Figure 5a shows that the volume of Xe bubbles increases with an increase in temperature and, during bubble growth, the volume of Xe bubbles increases approximately linearly with an increase in the number of Xe atoms. However, at the same temperature, the volume of Xe bubbles at the GB is higher than that of the bubbles in the bulk. This is because GBs are surface defects in solid materials and there are many defects, such as vacancies, at the GBs. Thus, under the same conditions, the volume of Xe bubbles at the GB is larger than that of the bubbles in the bulk. Also, it reflects the influence of temperature on the volume of Xe bubbles, which is attributed to the increase in temperature, which changes the lattice constant and the reconstruction of GBs [19]. At 2000 K, the Σ5(310) GB consists of a more distorted triangular pattern, the middle gap is larger and the lattice constant increases. In Figure 5b, at the GB, the pressure of the Xe bubble is initially quite high and then drops with increasing Xe concentration until the number of Xe atoms reaches 10. When there are more Xe atoms in the bubbles, the pressure increases with an increase in the number of Xe atoms, which is consistent with that of Xe in the UO2 bulk [4,8]. Then, with an increase in bubble growth, the bubble pressure generally decreases with some fluctuations involving several peaks and drops. This is because the bubbles may be far from the equilibrium, from which they grow because of rapid changes in temperature or insufficient local lattice vacancies [56]; thus, the smallest bubbles have a very high pressure, which results in a density comparable to that of solid Xe [57]. For larger bubbles, the density and pressure are relatively low. Experiments have shown that at the very early stages of bubble development, the bubble density is high with little evidence of deviation from circular bubbles. However, extensive bubble coalescence can greatly reduce the density of bubbles [58]. Under the same conditions, the pressure of Xe bubbles at the GB is less than that of the bubbles in the bulk. The internal pressure of Xe bubbles depends on not only the external stress on UO2 but also the surface tension of the bubble voids [56].

In addition to volume and pressure, we evaluated the configuration of Xe bubbles at 1000 and 2000 K. Snapshots of the radial distribution function of Xe atoms at 1000 and 2000 K are shown in Figure 6. To avoid thermal fluctuations at 1000 and 2000 K, the positions of Xe atoms were obtained by taking an average total time of 50 ps before inserting the next Xe atom into the bubble. As shown in Figure 6, the peak is smaller and fuzzy at high temperatures because the atomic amplitude is large at high temperatures; thus, it is easy to deviate from the equilibrium position and approach the liquid state. With an increase in Xe atoms, Xe in the bubble evolves into a glassy/amorphous state, indicating that the behavior of Xe atoms in UO2 is similar to that of a system with a hard spherical liquid in a solid. This observation is consistent with the results of Geng et al. [32]. For smaller bubbles (1–5 Xe atoms), the distribution of Xe atoms at the GB is more dispersed than that in the bulk, which also shows that Xe atoms at the GB diffuse more easily than those in the bulk, according to the MSD results. In the bulk, there is only one peak of Xe atoms at the beginning, indicating the uniform distribution of Xe atoms in the UO2 matrix. With an increase in Xe atoms, the first peak gradually grows and the second peak appears, indicating the formation of Xe clusters and the emergence of the second layer of atoms in the clusters.

**Figure 5.** (**a**) Volume (Å3) and (**b**) pressure (GPa) of Xe bubbles as a function of the number of Xe atoms in the bubbles. The lines are linear Arrhenius fits.

**Figure 6.** Radial distribution function of Xe atoms in Xe bubbles at (**a**) 1000 K and (**b**) 2000 K. The icons 1, 2, 3, 4 and 5 correspond to the number of Xe atoms 5, 10, 20, 40 and 50, respectively.
