First-Principles Study on the Migration and Release Properties of Xe on the Surface of Uranium Mononitride

Abstract

The fission gas uranium mononitride (UN) causes swelling and affects the properties of fission fuel. Since surface behavior is closely related to the release of gases, it is crucial to study the properties of Xe on the UN surface. Density functional theory was used to study the properties of Xe gas on the UN(001) surface and subsurface layers. Different bulk and surface models of UN were established, and the formation energies of bulk and surface defects, as well as the incorporation energy of surface Xe, were calculated. Differential charge density maps were generated, and the analysis revealed that the migration of Xe atoms on the surface predominantly occurs through a vacancy mechanism. Furthermore, Xe atoms located in the subsurface and interstitial positions are less likely to escape from the surface due to the influence of surrounding atoms. Finally, the Climbing Image Nudged Elastic Band method was employed to calculate migration pathways and the associated migration energies. The modelling results indicated that surface Xe atoms’ migration exhibits a vacancy-assisted mechanism, while surface and subsurface U-vacancies on the UN surface may promote the diffusion of fission gas atoms.

1. Introduction

Uranium mononitride (UN) has been recognized as a promising candidate for Accident-Tolerant Fuel (ATF) in nuclear fission reactors. An ideal replacement for oxide fuels should provide higher safety margins during accidents, featuring a higher melting point, specific heat capacity, thermal conductivity to enhance heat dissipation, and stability at elevated temperatures. Furthermore, it should offer economic advantages through higher uranium density and burnup. Uranium nitride exhibits a higher melting point compared to uranium dioxide, which is currently used in commercial reactors. This means that even under conditions of insufficient cooling, the fuel can remain solid for a longer period, thereby delaying or preventing the release of radioactive materials. Additionally, UN demonstrates superior thermal properties, including higher heat capacity and better thermal conductivity, enabling more efficient heat dissipation and reducing the formation of hot spots that could otherwise affect core stability [1,2,3,4].

The accumulation of fission gases in nuclear fuel compromises stability, thermal efficiency, and safety, demanding advanced design to mitigate swelling and material degradation. During reactor operation, uranium fission generates low-solubility fission gases like Xe. These gases form bubbles, causing fuel expansion and deformation, which compromise stability and safety. Bubbles also increase thermal resistance, reducing heat transfer efficiency and leading to temperature spikes that degrade the fuel and cladding materials [5]. Moreover, the buildup of gaseous fission products causes irradiation swelling in the fuel matrix. Initially, gas atoms exist as interstitials or are trapped by defects. As fission density increases, fission gas atoms agglomerate and grow into bubbles, causing significant fuel swelling which degrades the fuel material’s thermomechanical performance. Thus, it is crucial to predict fission gas behavior and manage fission gas release in UN nuclear fuel to ensure its safe and reliable operation in nuclear power reactors [6,7].

The diffusion behavior of vacancies on the surface of uranium compounds is distinct from that in the bulk, which may affect the release behavior of Xe from the surface of UN. In UO2, extensive research has been focused on U vacancy diffusivity along surfaces. Amato et al. [8], Reynolds [9], Henney and Jones [10], and Maiya [11] utilized mass transfer methods while Marlowe and Kaznoff [12] as well as Zhou and Olander [13] employed tracer techniques. However, it was pointed out that tracers might move from the surface into the interior, which could potentially lead to significant underestimations of surface diffusivities [14,15]. In the end, Zhou and Olander [13] came up with an approach that might possibly circumvent this issue. Nevertheless, the value obtained by Zhou and Olander is much greater than those from mass transfer techniques. At T = 1600 K, the mass transfer techniques yield a value that is six orders of magnitude higher than the bulk U vacancy diffusivity reported by Auskern et al. [16], and the Zhou and Olander value is approximately twelve orders of magnitude higher. Given Xe diffusion is partially correlated with vacancy diffusion [17], surface diffusion of Xe may demonstrate distinct behavior compared with bulk diffusion. Thus, it is crucial to examine surface diffusion behavior of Xe in UN and its implications for fission gas retention and release [18].

The toxicity, radioactivity and the complexity of surface interactions of uranium have presented challenges to the experimental investigation of UN’s surface diffusion behavior [19,20]. However, theoretical computational methods provide a robust and efficient approach to investigate uranium surface properties, as well as the adsorption and dissociation processes of gases. First-principles simulations of Xe behavior in UN have yielded reliable results concerning energy and charge transfer during the study. Klipfel [21] et al. utilized density functional theory (DFT) combined with the generalized gradient approximation (GGA) to investigate the binding of Kr and Xe in UN. Their findings indicate that Schottky defects serve as preferred trapping sites for fission gases in UN, while revealing weak charge transfer between gas atoms and the UN matrix. Several studies have also been conducted on simulating the surface of UN. For instance, Nie [22] et al. investigated the adsorption and diffusion of oxygen on the (001) surface and bulk of UN. Bocharov [23] et al. used density functional theory to study point defects in the (001) surface and subsurface layers of UN. They performed structure optimization using slabs of different thicknesses and, through the analysis of density of states (DOS) and charge density, indicated that the UN surface is more prone to vacancy formation, while the properties of the subsurface and central layers show little difference. Previous studies of UO2 [24] and UN [17] found that Xe atoms predominantly migrate through vacancy-mediated mechanisms. This process involves Xe atoms occupying uranium vacancies and moving through the lattice via vacancies. Xe atoms can also migrate through interstitial sites, but this process is less favorable compared to vacancy-mediated diffusion due to higher migration barriers [24].

This study aimed to reveal the surface physical mechanisms underlying fission gas Xe release in UN materials through first-principles calculations and to explore the role of surface effects in gas capture, accumulation, and release. Benchmark calculations were conducted for the electronic structure of bulk and surface slabs of UN to establish a solid foundation for the study of surface diffusion of Xe. Based on the surface slab models, calculations were performed to simulate the release of Xe at various positions. A vacancy mechanism for surface release was proposed, followed by computations of several common migration pathways and their corresponding activation energies.

2. Method

First-principles calculations of electronic differential density maps are powerful tools in understanding the electronic properties of various systems [25,26,27]. These calculations are typically performed within the framework of density functional theory and can be applied to a wide range of materials and structures. In particular, it provides a good explanation for materials with strong correlation effects [28].

All density functional theory calculations were performed using the Vienna Ab initio Simulation Package (VASP) [29,30]. The generalized gradient approximation in the form of PW91 [31] was employed. The electrons in the ion cores were described by the projected augmented wave (PAW) pseudopotentials included in VASP [32]. A plane-wave kinetic energy cutoff of 500 eV was set. For the sampling of the Brillouin zone [33], the Monkhorst–Pack scheme was used with a 5 × 5 × 5 k-point mesh for bulk calculations and a 6 × 6 × 1 k-point mesh for surface calculations. Both the energy cutoff and the choice of k-points were validated through convergence tests. The convergence criteria for energy and forces were set to 10⁻⁵ eV and 10⁻⁴ eV/Å⁻¹, respectively. The DFT-D3 method incorporates van der Waals interactions and effectively describes dispersion interactions. However, most studies on UN/UN surfaces use the DFT method. In the nuclear fuel field, DFT-D3 is applied in plutonium fuel [34,35,36] and molten salt reactor simulations [37,38,39]. However, uranium has not yet been studied using the DFT-D3 method from our research. Due to its distinct properties compared to plutonium or molten salts, conclusions from related studies cannot be directly applied. For the UN surface, the energy difference between the two methods is approximately one in a thousand. As a preliminary study on this issue, we aim to achieve results comparable to those in the existing literature; hence, we believe the use of DFT is necessary in this study.

To evaluate the properties of the UN bulk phase, an 8-atom UN unit cell and a 64-atom UN supercell were employed. For assessing the surface properties of UN, a slab containing 110 UN atoms was constructed. As shown in the convergence test results (Figure 1), the vacuum layer thickness of the slab was selected as 39.12 Å. In the modeling process, three primary types of Xe sites were considered: Xe on the surface, Xe in the subsurface (the second layer), and Xe at the interstitial sites between the surface and subsurface.

In previous simulations of bulk UN, the inclusion of the Hubbard U parameter has been shown to accurately predict the electronic correlations of U 5f electrons. However, recent research [40] has indicated that simulations of fission gas in UN are more accurately modeled without incorporating the Hubbard U parameter. Our calculations demonstrated that incorporating the U parameter led to discrepancies in the simulated surface properties, deviating from experimental observations. Consequently, we opted to exclude the U parameter in our simulations of the surface. In selecting the exchange–correlation potential, as shown in

Table 1. Calculated parameter settings.

Functions	Lattice Constants (Å)	Magnetic Moment (μB)
PW91	4.87	1.17
PBE	4.868	1.25

, PW91 demonstrated superior performance over PBE in calculating magnetic moments and lattice constants; hence, PW91 was chosen for the subsequent investigation of surface properties. Given that PW91 predicted the ground state of UN to be ferromagnetic (FM), we conducted our calculations using the FM structure.

Surface atoms are influenced by the bulk atoms and also interact with external gases. In modeling, a vacuum layer is constructed to simulate the interaction between surface atoms and external gases. Therefore, it is necessary to select an appropriate thickness of the vacuum layer and number of atomic layers to ensure convergence of the results while maintaining computational efficiency [41]. In this study, models were established with atomic layer counts ranging from 5 to 9 and vacuum layer thicknesses from 24.45 Å to 48.9 Å (corresponding to 10–20 atomic spacings). Subsequent computations determined that the final settings for the calculations were 7 atomic layers with a vacuum layer thickness of 39.1 Å (16 atomic spacings).

The formation energy of defects was calculated using the following formula:

$$E_f=E_{defect}-E_{perfect}\pm \sum n_A{\mu }_A+E_{el}$$

where $E_{defect}$ and $E_{perfect}$ represent the total energies of the system with and without the defect, respectively; $n_A$ and ${\mu }_A$ describe the number of atoms inserted or removed, respectively, in the formation of the defect and their corresponding chemical potentials. $E_{el}$ is the elastic correction term, which was neglected in this study due to its negligible value.

Similarly, the incorporation energy after introducing a Xe atom was calculated using the following formula:

$$E_{i\left(Xe\right)}=E_{Xe\in defect}-E_{perfect}\pm \sum n_A{\mu }_A+E_{el}$$

where $E_{Xe\in defect}$ denotes the total energy of the supercell containing a Xe atom located at the defect site.

The Climbing-Image Nudged Elastic Band (CINEB) method was employed in this work. It is a computational technique used to find the minimum energy path (MEP) and transition states between initial and final states in a system. This method is particularly useful in studying the activation energies and migration paths of atoms and molecules on various surfaces and within different materials [42,43,44]. It was employed to determine the migration energy of neutral indium atoms in silicon [45,46], providing detailed energy configurations and paths, which indicates it is suitable for this work.

3. Results

Structural relaxation was performed on the constructed models and generated differential charge density plots to analyze the electronic properties. Following relaxation, as illustrated in Figure 2e, the interstitial Xe atom migrated toward the surface layer, causing substantial displacements of neighboring atoms and significant alterations in bonding configurations. These observations suggest that the UN surface exerted a constraining influence on the interstitial Xe atom, impeding its immediate release. Conversely, in the model where Xe occupied a surface vacancy, depicted in Figure 2a,b, the Xe atom escaped from the surface layer into the external environment after relaxation. Regarding the model structure featuring a subsurface U vacancy, particularly post-structural optimization as shown in Figure 2d, minimal displacements of N atoms surrounding the subsurface U vacancy were observed. This implies that the N atoms had attained a lower-energy equilibrium position within the local environment shaped by the U vacancy, thereby indicating the stable persistence of the subsurface U vacancy. The differential charge density plot of the subsurface N vacancy, presented in Figure 2c, reveals evident charge transfer near the N vacancy, particularly within the surface layer. This signifies a slight enhancement in bonding and an increase in interaction forces. Moreover, the relaxation outcomes exhibited negligible atomic displacements, suggesting that the subsurface N vacancy, akin to the U vacancy, constituted a relatively stable configuration. The differential charge density plots, as shown in Figure 3a,b, demonstrate a relatively pronounced charge transfer between the surface and subsurface layers, indicative of strengthened bonding. As a result, the subsequent departure of Xe atoms from the surface necessitated surmounting a higher energy barrier. The differential charge density around the Xe atoms, illustrated in Figure 3a,d,e, shows negligible charge transfer between the Xe atoms and their surrounding atoms. This observation implies that the Xe atoms do not engage in chemical bonding, aligning with the inert nature of Xe as a noble gas.

The defect formation energies and incorporation energies were calculated based on the previously mentioned formulas. According to

Table 2. Defect formation energies of different slab models (eV).

Model	This Work	Reference [23]
Surface U vacancy	1.273	1.44
Surface N vacancy	3.753	3.70
Subsurface U vacancy	2.756	3.09
Subsurface N vacancy	4.413	4.33

, the formation energies of defects in the surface layer and subsurface layer are closely aligned with values calculated in the referenced literature, indicating that the parameters used in our calculations effectively reflect the properties of the surface. The formation energy of U vacancies is lower than that of N vacancies, implying that U vacancies form more readily than N vacancies. Moreover, the formation energies of U and N vacancies in the surface layer are consistently lower across different reference energies compared to those in the subsurface layer, making them easier to form. According to

Table 3. Incorporation energies of different slab models (eV).

Models	Incorporation Energies (eV)
Xe occupies a surface U atom vacancy	1.19
Xe occupies a surface N atom vacancy	4.10
Xe occupies an interstitial site between the surface and subsurface layers	8.63
Xe occupies a subsurface U atom vacancy	7.78
Xe occupies a subsurface N atom vacancy	12.75

, the incorporation energy of Xe atoms in U and N vacancies at the surface layer is notably low, whereas the incorporation energy increases significantly in the subsurface layer. Additionally, it is observed that the energy required for Xe to occupy an N atom vacancy is significantly higher than that required to occupy a U atom vacancy.

Four surface migration pathways were constructed for the calculations. Given that the previous discussion has established that Xe migrates via a vacancy mechanism, we only considered surface migration paths assisted by vacancies. These pathways are designated as V_UU, V_UN, V_NU, and V_NN, as illustrated in Figure 4. In the V_UN configuration, the U vacancy is located on the surface, while the N vacancy is situated in the subsurface; conversely, in the V_NU configuration, the N vacancy is on the surface and the U vacancy is in the subsurface. In the calculations performed using the CINEB method, we opted to fix the z-axis degree of freedom of the Xe atom to simulate the shielding effect of the casing on Xe gas in the actual operating environment. We computed the energy barriers for different models, as shown in Figure 4. The V_UN model does not exhibit a saddle point, whereas the migration barriers for the V_UU, V_NU, and V_NN models are 0.18 eV, 2.55 eV, and 0.70 eV, respectively.

4. Discussion

Based on the energy relaxation results of the surface structures shown in Figure 2, only the Xe atoms occupying surface U and N vacancies tend to escape to the surface, while atoms in other positions are restrained within the surface by the influence of other atoms. Due to the comparable defect formation energies between the subsurface and the bulk phase, it can be inferred that the environments of the subsurface and the bulk are relatively similar. The incorporation energy of Xe calculated for U and N vacancies in the UN surface layer is much lower compared to the subsurface layer. This is primarily due to the fact that during the relaxation process, Xe leaves the vacancy and moves above the UN surface, causing a displacement of surface atoms due to the incorporation of Xe. As the interaction force between surface layer atoms and Xe decreases, the incorporation energy becomes very small, indicating that if Xe atoms can move to surface defects, they tend to move outward from the surface, i.e., undergo a release process. During the relaxation process, it was observed that Xe atoms did not escape the surface but remained at 2 to 3 Å from the surface layer, suggesting that the surface has a certain adsorption effect on Xe atoms. Furthermore, the incorporation energy of Xe in surface N vacancies is greater than that in U vacancies, a conclusion similarly drawn for the subsurface layer. This is attributed to the smaller size of N atoms, requiring more work when Xe enters the N vacancy, thus resulting in higher incorporation energy.

Interstitial atoms exert a strong influence on their surroundings, moving closer to the surface layer but not further outward due to interaction with neighboring atoms. Since the volume of interstitial sites is smaller than that of U or N vacancies, Xe must exert more influence on surrounding atoms when occupying these positions, leading to higher incorporation energy than when Xe occupies N vacancies. The incorporation energy of Xe occupying U or N vacancies in the subsurface layer is notably higher than the three aforementioned positions: surface U vacancies, surface N vacancies, and interstitial sites between the surface and subsurface layers. This is mainly because all three of these positions move to the surface or outward during the relaxation process, and since there are no atoms beyond the surface, there is no corresponding interaction force, resulting in a significant decrease in incorporation energy. At this point, the incorporation energy of Xe in N vacancies remains noticeably lower than that in U vacancies, again due to the smaller volume of N vacancies requiring more interactions with surrounding atoms.

In this study, another focus of our work was on the role of vacancies in assisting Xe migration on the surface. Previous studies in the bulk phase showed that the presence of various defects such as Schottky defects and Frenkel pairs can significantly impact the migration of Xe atoms. For instance, the Schottky defect, which involves two vacancies as nearest neighbors, is a favorable site for Xe incorporation, facilitating its migration [47]. By optimizing different migration paths and calculating the energy barriers, we proposed possible migration pathways for Xe. Four intermediate images between the initial and final configurations were constructed by linear interpolation. The reaction coordinate ranging from 0 to 6 corresponds to the first state, the intermediate images, and the last state, respectively. From Figure 5, it can be observed that the energy barriers for Xe migration towards the surface are relatively low across different vacancy-assisted migration mechanisms, indicating that Xe indeed tends to move via vacancy, which aligns with other studies conducted in bulk materials and the conclusions proposed earlier in this work. Additionally, as hypothesized, the shorter distance required for migration due to different vacancies should result in lower migration energies. Notably, the V_UN model exhibits no energy barrier, suggesting that if there is a U vacancy on the surface layer, Xe will directly release when moving to the subsurface N vacancy. Conversely, the V_NU model presents a higher energy barrier. For dissimilar vacancies, the energy barrier is expected to be higher; however, the V_UU model shows a very low barrier of only 0.18 eV, while the V_NN model has a somewhat higher barrier. After a comprehensive comparison and analysis, we conclude that U vacancies are more favorable for Xe migration compared to N vacancies. This finding provides guidance for manipulating fuel properties to enhance gas release, such as the potential for higher gas release rates in U-deficient UN.

5. Conclusions

Based on the comprehensive analysis of the above data, we can gain some understanding of the behavior of fission gases on the UN surface: (1) In subsurface positions, there is a significant incorporation energy, leading to a high energy barrier that hinders further outward movement. (2) Xe occupying surface vacancies tends to be released onto the surface but still experiences a certain degree of adsorption. Additionally, the strengthening of bonding among surface atoms acts to inhibit the escape of subsequent Xe atoms. Furthermore, other studies have suggested that it is appropriate not to include the U parameter in calculations for the bulk phase of UN. This paper demonstrates that simulations of the UN surface without the U parameter can also yield relatively accurate results. In addition, the CINEB calculations revealed that the optimal migration path is via the subsurface N atom and the surface U atom vacancies, and the migration energies for different pathways have been calculated.

While the current work is limited to the atomic scale, bridging the gap to practical engineering applications requires calculations at multiple scales. Future work could build upon the results of this model to develop new potential functions and incorporate them into multiscale simulation approaches for further calculations. Our team plans to conduct relevant work to address this need.