First-Principles Study on Thermodynamic Stability of UO₂ with He Gas Incorporation via Alpha-Decay

Abstract

Using first principles calculations we investigated the thermomechanical stability of spent nuclear fuels (SNF), especially how mechanical properties of $UO_2$, such as, bulk, shear and Young's moduli and Poisson's ratio vary through alpha-decay of U into Th with generation of He gas. Our results indicate that substitution of U by Th through alpha decay ($U_{1-x}Th_xO_2$) does not significantly affect the stability of the grain in a fuel matrix. In addition, we studied the transport properties of He in and boundaries of the $U_{1-x}Th_xO_2$ grain. Helium preferentially resides at the grain boundaries through diffusion. Our study can contribute to substantial reduction of environmentally risk and enhancement of our sustainability by safe control of radioactive materials.

1. Introduction

Uranium dioxide (UO₂) has been a key nuclear fuel, yet in spite of many benefits of nuclear energy, the safe and economic, disposal of radioactive waste is global concern [1,2]. Especially, highly radioactive spent nuclear fuels (SNFs) are the central materials for the handling [3]. Recently, reducing the radioactivity and volume through pyroprocessing technology was studied, but still long-term storage or permanent disposal of SNFs in deep earth is the major way.

In the disposed SNFs, a large quantity of helium (He) gas can be generated and coalesced into bubbles in the fuel matrix, which may lead to swelling of the materials [4]. To improve the safety level, there have been many experimental or theoretical studies on the behavior of He in the fuel matrix [5-7]. For example, the stability of He was evaluated by calculating its incorporation energy in the grain [8,9]. It was reported that He thermodynamically does not stay in the interstitial sites of bulk UO₂ since the incorporation energy is very high (0.72~ 0.89 eV per He atom). In addition, activation energies of He for diffusion in bulk UO₂ were calculated as 2.2~4.15 eV [5,8] and measured as 1.87~2.23 eV [9,10] meaning the process is, in fact, difficult.

Since an alpha decay of U into Th is of importance for the production of He [11], a pure UO₂ may not be a relevant model system for studying SNF stability. In this study, we calculated moduli and Poisson’s ratio of U_1-xTh_x O₂ to identify variation of the mechanical strength as a function of Th composition. Using first-principles calculations the incorporation energy and diffusion energy barrier of He in the U_1-x Th_xO₂ at x = 0.0, 0.13 and 0.25 were predicted. Since UO₂ pellet is a poly-crystalline [12], we set up a Σ3(111) tilt grain boundary model, which is one of common grain boundaries [13,14] to investigate stability of SNF.

2. Computational Details

All calculations in this study were carried out as density functional theory calculations as implemented in Vienna ab initio simulation package (VASP) [15]. The generalized gradient approximation (GGA) with Perdew-Burke-Ernzerhof (PBE) [16,17] was used for exchange correlation functional, and the projected augmented wave (PAW) potentials [18] were used to substitute core electrons of U, Th, O and He. Convergence criteria for ionic relaxation for all calculations were energy difference less than 10^-5 eV compared to the right before step. For bulk properties, such as incorporation energies and mechanical properties, the k-points sampling was conducted on a 4 × 4 × 4, while for grain boundaries 5×5×3 mesh was used. We performed calculations including the effective Hubbard correction in the GGA+U approximation to correct the localized f electrons of uranium, using U_eff = 3.96 eV [14,19] with U-ramping scheme [20,21] to optimize ground state of UO₂.

3. Results and Discussion

3-1. Stability of fuels in a grain of U_1-xTh_xO₂

As both UO₂ and ThO₂ have cubic fluorite structures (a = 5.47 for UO₂, a = 5.60 for ThO₂) [22], 1 × 1 × 1 unit cells of U_1-xTh_xO₂ were employed to model U_1-xTh_xO₂ matrix. An octahedral interstitial site (OIS) [23,24] was considered for He atom in the matrix as shown in Fig. 1. Supercells of 2 × 1 × 1 and 2 × 2 × 1 of UO₂ were used, and in each supercell a U atom was replaced by Th atom to describe U_0.87Th_0.13O₂ and U_0.94Th_0.06O₂, respectively, via alpha decay.

Fig. 1. Model systems of (a) UO₂ (b) U_0.94Th_0.06O₂ (c) U_0.87Th_0.13O₂ and (d) U_0.75Th_0.25O₂ to calculate incorporation energies. Yellow, green and red atoms indicate U, Th and O, respectively. The blue circle in (a) indicates OIS.

The incorporation energy was defined as Eq. (1) ,

\(E_{\text {inc}}=E_{\text {bolk}+\text {He}}-\left(E_{\text {bulk}}+E_{\text {he}}\right)\)              (1)

E_inc represents incorporation energy, E_bulk+He is total energy of the model U_1-xTh_xO₂ with He at the OIS, E_bulk is the total energy of the U_1-xTh_xO₂ without He and E_He is the energy of non-interacting He atom.

Incorporation energies of U_xTh_1-xO₂ were 0.89, 0.85, 0.91 and 0.90 eV at x = 0.0, 0.06, 0.13 and 0.25 as shown in Fig. 2(a), and activation barriers of He in UO₂ and U_0.03Th_0.97O₂ obtained by nudged elastic band (NEB) method [9,10] are illustrated in Fig. 2(b). Since E_inc in U_1-xTh_xO₂ are positive He is thermodynamically unstable in U_1-xTh_xO₂. It implies that He prefers diffusing out of the grain and into grain boundary or more open space. Diffusion activation barriers of He between OIS in UO₂ and U_0.97Th_0.03O₂ were estimated as 2.39 eV and 2.40 eV, respectively, as shown in Fig. 2(b). These values are similar because bond strength Th-O are stronger than U-O. In the diffusion process in UO₂ the diffusing He only elongates U-O bond length from 2.39 to 2.52 Å, while Th-O bond length remains the same. This feature is the same in U_0.97Th_0.03O₂ even when He moves beside Th.

Fig. 2. (a) Incorporation energy of U_1-xTh_xO₂ at x = 0.89, 0.85, 0.91 and 0.90. (b) Energy variations of UO₂ and U_0.97Th_0.03O₂ when He atom diffuses from OIS to another nearest OIS.

3-2. Mechanical strength of U_xTh_1-xO₂

For a cubic crystal, three independent elastic constants (C₁₁, C₁₂, and C₄₄) [25] are defined. C₄₄ can be obtained from applying a monoclinic volume-conserving distortion matrix, Δ_mono to a relaxed unit cell [25].

\(\Delta_{\text {mono }}=\left(\begin{array}{ccc} 1 & 8 / 2 & 0 \\ 8 / 2 & 1 & 0 \\ 0 & 0 & 4 /\left(4-\delta^{2}\right) \end{array}\right)\)

where δ = -0.1, -0.05, 0.05 and 0.1. The energy increase per unit cell volume E(δ) is given by Eq. (2),

\(E(\delta)=C_{44} \delta^{2}+O\left(\delta^{3}\right)\)              (2)

Tetragonal shear constant C' [C' = ((C₁₁+C₁₂)/2)] can be obtained from applying a tetragonal volume-conserving distortion matrix, Δ_tetra to a relaxed unit cell [25].

\(\Delta_{\text {tern}}=\left(\begin{array}{ccc} 1+\delta & 0 & 0 \\ 0 & 1-\delta & 0 \\ 0 & 0 & 1 /\left(1-\delta^{2}\right) \end{array}\right)\)

where δ = -0.1, -0.05, 0.05 and 0.1. The energy increase per unit cell volume E(δ) is also related to the tetragonal shear constants as Eq. (3),

\(E(\delta)=C^{\prime} \delta^{2}+O\left(\delta^{3}\right) \)              (3)

Bulk modulus, B, can be similarly obtained from a volumechanging matrix to a relaxed unit cell. Here, we used the volume changes of -10 %, -5 %, 5 % and 10 % to the relaxed unit cell volume, respectively. From the tetragonal shear constant and bulk modulus, C₁₁ and C₁₂ were obtained [B = (C₁₁ + 2C₁₂)/3].

From these elastic constants, we calculated shear moduli (G) and Poisson’s ratio (ν). The G can be calculated by the Hershey-Kröner averaging method relation [25,26] by which poly-crystalline effect is considered as shown in Eq. (4).

\(8 G^{3}+\left(9 B+4 C^{\prime}\right) G^{2}-3 C_{44}\left(B+4 C^{\prime}\right) G-6 B C_{44} C^{\prime}=0\)              (4)

Young’s modulus (E) is obtained from the relation between bulk and shear moduli [25,27] as Eq. (5).

\(E=\frac{9 B G}{3 B+G}\)              (5)

Poisson’s ratio ν is obtained from bulk and Young’s moduli [25] as Eq.(6).

\(v=\frac{3 B-E}{6 B}\)              (6)

We calculated elastic constants to obtain moduli and Poisson’s ratio with respect to x in U_1-xTh_xO₂, where x is 0, 0.25, 0.5, 0.75 and 1.0, which are represented in Table 1.

Table 1. Shear modulus, Young’s modulus

Our calculations for Poisson ratios for UO₂ (0.251) and ThO₂ (0.220) are slightly smaller than experimentally measured ones (0.291~0.302 [28] and 0.279 [29], respectively). It can be ascribed to the anharmonicity in the moduli of the materials. From the results, the mechanical strength of UO₂ does not significantly change when the substitutional defects of U ions to Th ions are generated.

3-3. Stability of He in the grain boundary

Σ3(111) tilt grain boundary is one of the most abundant coincident site lattice (CSL) boundaries [14]. This boundary consists of two (111) planes as shown in the Fig. 3. Grain boundary energy and surface energy are defined as Eq. (6-1) and (6-2).

Fig. 3. Structure of Σ3 (111)/[110] tilt grain boundary and Position of a helium atom along the grains.

\(E_{x b}=\frac{E_{t \alpha}(g b)-N_{s b} E_{b u l k}}{2 A_{g b}}\)              (6-1)

\(E_{\text {arf }}=\frac{E_{\text {tot }}(\text { surf })-N_{\text {surf }} E_{b u l k}}{2 A_{\text {surf }}}\)              (6-2)

where E_tot (gb) and E_tot (surf) indicate total energy of grain boundary and surface of UO₂, respectively, and E_bulk does the energy of a pristine bulk UO₂ per formula unit. N_x and A_x represent number of UO₂ units in the system x and a cross-sectional area of the system x. If E_gb − 2E_surf < 0 two grains with grain boundaries are more stable than two non-interacting grains with terminated by (111) surfaces. As we calculated grain boundary energy E_gb = 0.057 eV/Ų and surface energy E_surf = 0.053 eV/Ų the Σ3(111) tilt grain boundary is more stable than two grains with terminated with (111) surfaces supporting experimental observation.

We calculated incorporation energies of a He atom for three different locations near the grain boundary shown in Fig. 3. The grain boundary incorporation energy of He was calculated as 0.85 eV, which is, in comparison with a pristine bulk, lower by 0.05 eV. It means He more easily resides at the more open grain boundary than inside bulk grain (the distance between two U atoms at the grain boundary increases from 3.17 Å to 3.30 Å. Therefore, we found that helium becomes more stable as helium migrates to the Σ3(111) tilt grain boundary.

4. Conclusion

We performed first-principles modeling to investigate the influence of Th and grain boundary on mechanical properties and stability of UO₂ nuclear fuels. By calculating incorporation energy of He and mechanical properties we concluded that Th does not affect the stability of SNFs significantly. Among the various grain boundaries, we chose Σ3 (111)/[110] tilt grain boundary to see stability of He. We found that this grain boundary is a more thermodynamically stable position for He than bulk grain.