*2.3. MD Simulation Setup*

For all MD simulations, the time step was set as 0.001 ps. For static relaxation, energy minimization was performed and the minimization algorithm was set as the conjugate gradient method (cg), with the stopping criteria for energy tolerance of 10−<sup>25</sup> s−<sup>1</sup> and a force tolerance of 10−<sup>25</sup> eV/Å.

We studied the characteristics of Xe atoms in UO2 GB, including the formation energy of a single Xe atom in the GB, the migration of a single Xe atom in UO2 GB and the nucleation of Xe bubbles, and compared them with those in the bulk. The specific simulation processes for various characteristics are different. Thus we constructed two simulation boxes of different sizes: one contains 38,290 atoms, with a size of 10 *a*<sup>0</sup> × 20 *a*<sup>0</sup> × 10 *a*<sup>0</sup> (*a*<sup>0</sup> is the lattice constant of UO2) and was used to calculate the migration of a single Xe interstitial atom at the GB, and periodic boundary conditions were used in three directions of the box; the second is a 25 *a*<sup>0</sup> × 40 *a*<sup>0</sup> × 25 *a*<sup>0</sup> large system simulation box and was used to simulate the nucleation and growth of Xe bubbles and the energy of Xe atoms at the GB, and periodic boundary conditions were used in three directions of the box. Similarly, in the bulk, a cubic box of 10 *a*<sup>0</sup> × 10 *a*<sup>0</sup> × 10 *a*<sup>0</sup> containing 12,000 atoms was used with periodic boundary conditions to calculate the migration of a single Xe interstitial atom. The other system is 25 *a*<sup>0</sup> × 25 *a*<sup>0</sup> × 25 *a*<sup>0</sup> with 187,500 atoms and was used to simulate the nucleation and growth of Xe bubbles and the energy of Xe atoms. We performed the same simulation process at the GB and in the bulk to better compare the differences in the

behavior of Xe bubbles. A cutoff distance of 1.1 nm was adopted and used for all potentials used in this study.

To determine the stable configuration of Xe atoms in the GB, the stable structures of Xe clusters with different sizes (the number of atoms less than six) were obtained by adding Xe atoms gradually and performing MS simulations for each configuration.

After determining the stable Xe site in the GB, the diffusion mechanism of a single Xe interstitial atom at the GB was studied. The migration of a single Xe atom at the GB was studied using MD and the nudged elastic band (NEB) methods [46–48]. Then, the mean square displacement (MSD) method was used to calculate the diffusion coefficient and diffusion energy barrier of a single Xe atom at the GB.

Finally, to study the evolution mechanism of Xe bubbles, the microcanonical NVT (constant volume and temperature) ensemble was used to simulate Xe bubbles in UO2 GB. To describe the bubble characteristics during bubble evolution, the volume V and pressure P of the bubble after inserting each Xe atom were calculated by MD. The number of Xe atoms in the bubble is too small (only 50 Xe atoms) to obtain a good statistical result. V is the sum of the volumes of all Xe atoms, which were calculated using the Voronoi technology implemented in LAMMPS [49] and the pressure of the Xe bubble was calculated from the sum of the diagonal component of the atomic stress tensor and the bubble volumes as follows [8,50]:

$$P = -\frac{1}{3V} \sum\_{i}^{n} [S\_{11}(i) + S\_{22}(i) + S\_{33}(i)] \tag{1}$$

where *n* is the number of Xe atoms in the bubble, *Sαα*(*i*) is the diagonal component of the stress tensor for atom *i*, *V* is the volume of all Xe atoms.

Before the first Xe atom was randomly introduced into the simulation box, energy minimization was performed and the minimization algorithm was set using the conjugate method (cg), with the stopping criteria for energy tolerance and force tolerance of 10−25. Then, the first Xe atom was randomly added to the system. The initial system with Xe atoms was minimized to avoid long-distance movements of inadvertently overlapping atoms and the temperatures were set to 1000 K and 2000 K, respectively. The velocities of atoms were set and an additional 10 ps simulation was performed using microcanonical NVT (constant volume and temperature). Following this thermal equilibration (i.e., no change in the various properties of the system over time), Xe atoms were sequentially inserted one by one every 10 ps into the Xe bubble center of mass until 50 Xe atoms were contained in the bubble. After inserting each Xe atom, to avoid introducing artificial energy into the system resulting from atoms placed too close together, the system energy was minimized using the conjugate gradient (cg) and then simulated under NVT conditions for 10 ps at 1000 K and 2000 K.

### **3. Results**

### *3.1. Formation Energy and Diffusion Behavior of a Single Xe Atom at the GB*

The interaction between Xe atoms and the microstructure of UO2 fuel is key in fissiongas release. To simulate the redistribution of fission gas atoms in the UO2 microstructure, the interaction range between fission gas atoms and GB must be determined first, to determine the stable structure of migrated Xe atoms at the GB. We calculated the formation energy of a single Xe atom at different positions from the Σ5(310) GB.

There are two distinct regions of a Xe interstitial atom near the GB, as indicated by *E* in Figure 2. The first region is located at a distance of more than 1.5 *a*<sup>0</sup> for interstitial Xe, in which the energy is almost equal to the calculated formation energy of Xe interstitial atoms in the bulk (9.89 eV), indicating that the driving force for the segregation of Xe atoms to GB can be ignored. In the second region, where the distance is within 1.5 *a*<sup>0</sup> from the GB, *E* decreases, having a maximum value of approximately 6.88 eV, which indicates that interstitial Xe can be absorbed by GB. Thus, the interaction range between Xe interstitial atom and GB is approximately 1.5 *a*<sup>0</sup> from the plane of GB. Figure 2 also shows

that the lowest energy point for interstitial Xe is located at the GB. Thus, it is energetically favourable for interstitial Xe to be absorbed by the GB region.

**Figure 2.** Formation energy of interstitial Xe as a function of the distance from the Σ5(310) GB.

The behavior of fission gas in nuclear fuel is the main factor that determines the change in radiation swelling with fuel consumption. Therefore, it is necessary to study the migration of fission gas in UO2 GB.

Herein, we calculated the migration of an interstitial Xe atom in UO2 GB using NEB, i.e., a direct hopping mechanism from one octahedral site to another. NEB showed an energy barrier of 3.87 eV. Thus, whether migration is along or perpendicular to the GB direction, the calculated energy barrier is high. These results suggest that the migration of Xe between the two interstitial sites is not the main mechanism due to the higher energy barrier [51]. The relatively high energy barrier is mainly attributed to the lattice deformation caused by Xe movement between two adjacent interstitial sites, indicating that the octahedral interstitial position in UO2 is highly strained and, therefore, energetically unfavourable [52,53].

In addition to NEB, we employed the MSD method to calculate the diffusion coefficients and diffusion energy barriers of a single Xe atom at the GB. A long simulation time and a short distance between the position were employed to obtain information on the atomic position. The results were obtained by dividing the tracks of the MSD and averaging them several times. As shown in Figure 3, we calculated the diffusion coefficient and diffusion energy barrier separately at the GB and in the bulk. The diffusion energy barrier of the Xe atoms at the GB was 1.40 eV and that in the bulk was 2.46 eV. The magnitude of the preexponential factor was used to describe the ease of diffusion of a Xe atom. We conclude that Xe atom diffuse more easily at the GB than in the bulk.

The diffusion barrier calculated using MSD and the migration barrier calculated using NEB are different. When NEB was used, the single Xe atom showed a high energy barrier in the interstitial site, which is consistent with that of the bulk. However, the calculation results with MSD are different. This is because Xe may not diffuse directionally in UO2. The temperature accelerated dynamics simulations revealed that the dynamics of defect clusters strongly depends on their size and the diffusion direction is not one-dimensional [54].

**Figure 3.** Diffusion coefficients of Xe (**a**) at the Σ5 GB and (**b**) in the bulk UO2. The lines are linear Arrhenius fits.

#### *3.2. Formation Energies of Small Xe Clusters at the GB*

We calculated the formation energies of Xe atoms and atomic clusters at the GB and compared them with those in the bulk. The formation energies of Xe atoms at the GB are shown in Table 2. The formation energy of the Xe bubble is defined as:

$$E^f(n) = E\_{bubble}(n) - E\_{bulk} - nE\_{Xe}^{atom} \tag{2}$$

where *Ebubble*(*n*) is the total energy of the simulation supercell containing *n* Xe atoms in the bubble, *Ebulk* is the total energy of the supercell without bubbles and *Eatom Xe* is the isolated Xe atom energy, which was set to zero. As shown in Table 2, the energy of a single Xe interstitial atom at the GB was 1.3 eV, whereas that in the bulk it was 9.79 eV. The total energy difference (Δ*E*) was 8.49 eV. As the number of Xe atoms increases, Δ*E* increases. Whether at the GB or the bulk, the formation energy increases with an increase in the number of Xe atoms. At all stages, the formation energy of Xe clusters at the GB is smaller than that in the bulk, indicating that Xe bubbles are easier to form at the GB than in the bulk. This is because Xe atoms are at the GB plane, which improves the energy compared to that in the bulk. Similar to previous results [8], at smaller bubble sizes (1–5 Xe atoms), the Xe bubble nucleus at the GB has a much lower formation energy than that of bubbles in the bulk with a similar size.


**Table 2.** Formation energies of Xe atoms at the GB and in the bulk UO2.
