First-principles investigations on the anisotropic elasticity and thermodynamic properties of U₃Si₂–Al

Abstract

U₃Si₂ has been tested as a new type of nuclear fuel, and Al has been proven to improve its oxidation resistance. However, there is no research on its anisotropic mechanical and thermal properties. The mechanical and thermal properties of Al-alloyed U₃Si₂ nuclear fuel are calculated on the basis of first principles. Through the phonon dispersion curves, two kinetic stable structures sub-U₃Si_1.5Al_0.5 and sub-U_2.5Si₂Al_0.5(I) are screened out. It is found that the toughness of these two compounds after alloying are significantly improved compared to U₃Si₂. The three-dimensional Young's modulus shows that, the sub-U₃Si_1.5Al_0.5 formed by Al alloying in U₃Si₂ maintains a higher mechanical isotropy, while sub-U_2.5Si₂Al_0.5(I) shows higher mechanical anisotropy, which is consistent with the value of A^u. The calculation result shows that the lattice thermal conductivity of sub-U₃Si_1.5Al_0.5 and sub-U_2.5Si₂Al_0.5(I) after alloying exhibits high isotropy as the temperature increases.

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1 Introduction

Uranium dioxide (UO₂) is currently the most widely used fuel for commercial light water reactors.^1–4 Traditional UO₂ fuel has the advantages of high melting point, outstanding radiation performance and exceptional compatibility with cladding materials, but it has the disadvantage of low thermal conductivity.^2,5 In the context of the Fukushima nuclear accident, the research on the accident-tolerant fuel system (ATF) has attracted the interests of many researchers.^6,7 Replacing UO₂ with new nuclear material is one of the focuses for the ATF development and ensures that the new fuel has high thermal conductivity and other advantageous properties.⁸

One type of ideal candidate material for ATFs is uranium-silicon fuel,^9,10 and there have been studies on the microstructure changes of uranium-silicon alloys under irradiation conditions.¹¹ The currently known types of uranium silicon compounds are U₃Si₂, U₃Si, USi₂, USi and so on. U₃Si₂ has ultra-high thermal conductivity and high uranium-containing density among uranium silicon compounds, which has attracted many researchers' attentions.^10,12,13 Due to poor mechanical performance and antioxidant capacity, alloying the U₃Si₂ is an effective solution for further application within light water reactors.¹⁴ Wood et al.¹⁵ studied the effect of adding Al elements into U₃Si₂ on their performance through thermal reanalysis, and the investigation showed that U₃Si₂ oxidation behavior could be postponed, which confirmed that U₃Si₂ antioxidant resistance was improved by adding Al elements.

Experimental and theoretical studies on the U–Si–Al ternary system have been carried out around the world over past years. Marín et al.¹⁶ mixed the U₃Si₂ powder with Al to produce intermetallic compound U(Al,Si)₃ at melting temperature of aluminum, indicating that this strategy could be adopted to manufacture U₃Si₂–Al dispersed fuel. Wang et al.¹⁷ observed the occurrence and spread of interlayer cracks within U₃Si₂–Al dispersed fuel plates by scanning electron microscopy and analyzed the mechanisms of the crack formation; and proved that their mechanical properties, especially fatigue properties, were greatly influenced by the processing method of the fuel plates. Through transmission electron microscope, Gan et al.¹⁸ observed the average diameter, number density and volume fraction of the fuel matrix interaction layer (Si–Al)/U of U₃Si–Al fuel plate after irradiation, as well as the average diameter, number density and volume fraction of small bubbles within the fuel particles; and discussed their effects on fuel performance. Rabin et al.¹⁹ established a thermodynamic model of the U–Si–Al ternary system using computational phase diagram method, and conducted novel experiments in the aluminum-silicon-depleted uranium area; which showed that under equilibrium conditions, the silicon content of U(Al,Si)₃ is significantly higher than previously reported ones. Zenou et al.²⁰ used electron crystallography and powder X-ray diffraction techniques to study the structure and bonding of a stable ordered U(Al,Si)₃ phase, and results showed that Si was locating closer to U than Al; and when the energy was stable, the U–Si bond energy was stronger than the U–Al bond energy. Mirandou et al.²¹ studied two U₃Si₂–Al fuel plates by scanning electron microscopy and transmission electron microscopy, and the results showed that the final structure was composed of U(Al,Si)₃ and some modified U₃Si₂ phase which had the same crystal structure with U₃Si₂. Suparlina et al.²² examined the properties of U₃Si₂–Al fuel plate after irradiation, and the results showed that the properties were satisfied within the limits by safety standards.

Nuclear fuel with enhanced safety requires excellent mechanical property and high thermal conductivity under harsh conditions. The research on the mechanical anisotropy and thermodynamic properties of Al-alloyed U₃Si₂ ternary compounds have not been studied so far. In this work, including one Si-vacancy position, two U-vacancy positions and three interstitial sites with all different symmetries are selected as Al atom alloying sites. The atomic coordinates for three interstitial sites are (0, 0, 0.5), (0.319, 0.181, 0), (0.611, 0.111, 0.5). Through first-principles calculation of the phonon spectrum, the dynamically stable structures are screened out. The mechanical anisotropy as well as thermodynamic properties of U₃Si₂–Al ternary compounds are studied. This work will serve as a reference for the design of a new generation of ATF and the establishment of the U–Si–Al ternary phase diagram.

2 Method

The theoretical calculations in this paper are based on density functional theory,^23,24 using Vienna ab initio simulation package (VASP)²⁵ to perform first-principles calculations. During the calculation process, the electron wave function uses the plane wave super-soft pseudopotential method.²⁶ Generalized Gradient Approximation (GGA) combined with Perdew–Burke–Ernzerhof (PBE) functional²⁷ is employed to describe the exchange–correlation interactions between electrons. Since the 5f electron of U atom has a strong local Coulomb repulsion, the spin polarized PBE+U²⁸ method is used to eliminate this correlation.

In this work, the optimized total energy cutoff energy of the original unit cell was set to 500 eV, and the k-point meshes were constructed using the Monkhorst–Pack scheme. The 9 × 9 × 9 k-point mesh was used for the primitive cell for Brillouin zone sampling.²⁹ The structural optimization were under the conditions that the total energy difference between two iterations was less than 10–5 eV and the force on the atom was less than 0.01 eV Å⁻¹. The elastic properties in this article are achieved by an effective stress–strain method.³⁰ Periodic boundary conditions were adopted in this work.

The theoretical phonon spectrum was obtained using the Phonopy package.³¹ The lattice thermal conductivity is obtained by using phono3py,³² where the phonon lattice thermal conductivity and phonon relaxation time were simulated by using the finite displacement method.³³ The formula for lattice thermal conductivity is as follow eqn (1):

Here V is the volume of the unit cell, V_λ is the group velocity, τ is the SMRT of the phonon mode λ, and C_λ is regarded as the mode-related phonon heat capacity. The thermal conductivity of electrons is achieved by solving the Boltzmann transport equation in BoltzTraP2.³⁴

3 Results and discussion

3.1 Structural parameters

U₃Si₂ belonging to the p4/mm space group as a typical tetragonal crystal system at 0 K contains ten atoms in each primitive unit cell, as shown in Fig. 1. The PBE and PBE+U methods were used when optimizing the original structure.

Fig. 1 The crystal structure of U₃Si₂ (blue atom: U, green atom: Si).

The results shown in previous literatures^35,36 are also listed in Table 1 for reference. As shown in Table 1, the lattice constant a (a = b) is 7.24 Å, the value of c is 3.90 Å, and the volume is 204.43 ų for primitive U₃Si₂ by using the method of PBE, which have obvious deviations from the previous experimental and calculated results. When using the PBE+U method to optimize the original U₃Si₂, the calculation results show that the lattice constant a and c reach 7.39 Å and 3.95 Å, respectively; and the volume is 215.72 ų. The structural parameters calculated by the PBE+U method are closer to the experimental and calculated values, which proves that the calculation method of PBE+U is effective. Further, the non-spin polarized (non-SP) DFT + U calculations have also been performed to make comparisons with spin polarized (SP) calculations; and the results are listed in Table 1. It can be seen that the results obtained from the spin polarized DFT + U are consistent with the reported experimental results. Therefore, in the following calculations, the PBE+U method is adopted and U = 2.0 with spin polarized (where the magnetic moments for U, Si and Al atoms are 2.46 μ_B per atom, −0.025 μ_B per atom, and −0.20 μ_B per atom, respectively) is chosen as the calculation parameter.

Table 1 Optimized structural parameters of U₃Si₂ with PBE and PBE+U pseudopotential

Structural parameters	This work			Literature
	PBE	PBE+U (U = 2.0)		Expt^36	PBE+U^35
	SP	SP	non-SP
a = b (Å)	7.24	7.35	7.19	7.32	7.48
c (Å)	3.90	3.95	3.84	3.89	3.98
v (Å^3)	204.43	213.39	198.51	208.43	222.68

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3.2 Structural design

One Al atom substitutes U or Si positions of U₃Si₂ and the three resulting structures named sub-U_2.5Si₂Al_0.5(I), sub-U_2.5Si₂Al_0.5(II) and sub-U₃Si_1.5Al_0.5 are shown in Fig. 2(a)–(c). The three configurations that one Al atom staying in three different interstitial sites of U₃Si₂ are shown in Fig. 2(d)–(f), which are named int-U₃Si₂Al_0.5(I), int-U₃Si₂Al_0.5(II) and int-U₃Si₂Al_0.5(III). These six defect positions shown in Fig. 2 are based on the optimized structure of the initial unit cell of U₃Si₂. For a crystal structure with dynamic stability determined by phonons, the value of its phonon dispersion curve must be a real number.³¹

Fig. 2 The six structures of U₃Si₂ after Al alloying: (a) sub-U_2.5Si₂Al_0.5(I), (b) sub-U_2.5Si₂Al_0.5(II), (c) sub-U₃Si_1.5Al_0.5, (d) int-U₃Si₂Al_0.5(I), (e) int-U₃Si₂Al_0.5(II) and (f) int-U₃Si₂Al_0.5(III) (blue atom: U, green atom: Si, red atom: Al).

By calculating the phonon dispersion curves, two stable structures are obtained. Fig. 3(a) and Fig. 3(b) are the phonon dispersion curves of sub-U₃Si_1.5Al_0.5 and sub-U_2.5Si₂Al_0.5(I), respectively. The phonon dispersion curves of the remaining four are unstable structures due to the appearance of obvious imaginary frequency. As shown in Fig. 3, for sub-U₃Si_1.5Al_0.5 and sub-U_2.5Si₂Al_0.5(I), these phonon dispersion curves without imaginary frequency. In order to further determine the dynamic stability of the two structures, the corresponding phonon density of states is also calculated in Fig. 3. The phonon density of states is shown in Fig. 3 without imaginary frequency, implying that the optimized structure of sub-U₃Si_1.5Al_0.5 and sub-U_2.5Si₂Al_0.5(I) are dynamically stable. In the following calculations, these two stable structures are used as research target structures.

Fig. 3 Phonon dispersion curve of (a) sub-U₃Si_1.5Al_0.5 and (b) sub-U_2.5Si₂Al_0.5(I).

3.3 Mechanical properties

For nuclear fuels, the mechanical anisotropy might cause cracks under the actions of applied stress. In this section, the toughness of the material is firstly estimated through bulk modulus (B) and shear modulus (G), where B can reflect the incompressibility of the material, and the G can estimate the degree of deformation caused by shear force. The mechanical anisotropy of the material is then evaluated through the three-position Young's modulus diagram and A^u. The B and G of U₃Si₂, U₃Si_1.5Al_0.5 and sub-U_2.5Si₂Al_0.5(I) are calculated in this work, as shown in Table 2, where U₃Si₂ bulk modulus and shear modulus are 87.69 GPa and 65.37 GPa, which are basically consistent with the calculation results of Wang et al.¹² and Chattaraj et al.³⁷ In comparison, both of the B and G values for sub-U₃Si_1.5Al_0.5 and sub-U_2.5Si₂Al_0.5(I) are lower than the values of U₃Si₂ to a certain extent, indicating that their incompressibility and deformation resistance are weaker than U₃Si₂.

Table 2 Bulk modulus (B), shear modulus (G), B/G, Poisson's ratio (σ) and A^u for the configurations of U₃Si₂, sub-U₃Si_1.5Al_0.5 and sub-U_2.5Si₂Al_0.5(I)

	Bulk modulus (GPa)	Shear modulus (GPa)	B/G	Poisson's ratio (σ)	A^u
U3Si2	87.69	65.37	1.34	0.20	0.13
U3Si2 (ref. 12)	92.01	67.63	1.36	0.20	—
U3Si2 (ref. 37)	83	57	1.46	0.22	—
Sub-U3Si1.5Al0.5	84.58	54.10	1.56	0.24	0.32
Sub-U2.5Si2Al0.5(I)	68.94	40.45	1.70	0.26	1.27

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As proposed by Pugh et al.³⁸ the ductility of a material can be defined through the value of B/G; and 1.75 is specified as a critical value. The materials with their values below 1.75 are treated as brittle, while the ones with values above 1.75 are treated as ductile. The data in Table 2 show that U₃Si₂, sub-U₃Si_1.5Al_0.5 and sub-U_2.5Si₂Al_0.5(I) are all brittle materials, but the B/G values of sub-U₃Si_1.5Al_0.5 and sub-U_2.5Si₂Al_0.5(I) after alloying are significant increase and very close to 1.75. Therefore, we can understand that alloying could improve the toughness of the material. In addition, Poisson's ratio is also calculated using the following eqn (2):

According to Frantsevich's theory, the value of σ could also be used as a criterion for judging whether the material has ductility, and 0.26 is defined as the critical value.³⁹ The calculation results of U₃Si₂, sub-U₃Si_1.5Al_0.5 and sub-U_2.5Si₂Al_0.5(I) σ are shown in Table 2. The σ values of the three compounds are all less than or equal to 0.26, indicating that they are all brittle materials. However, sub-U₃Si_1.5Al_0.5 and sub-U_2.5Si₂Al_0.5(I) have a significant increase compared with U₃Si₂, which is consistence to the conclusion of B/G.

The anisotropy of elasticity could be characterized by anisotropy index A^u, which was proposed by Ranganathan and Ostoja-Starzewski⁴⁰ and is derived by eqn (3):

where A^u = 0 indicates isotropic for a structure, and the degree of anisotropy is determined based on the deviation from 0. Next, the three optimized compound crystal structures are taken into consideration, among which U₃Si₂ belongs to the tetragonal structure, while both of sub-U₃Si_1.5Al_0.5 and sub-U_2.5Si₂Al_0.5(I) belong to the orthogonal structure. For orthorhombic crystals, the corresponding B^V and B^R are the bulk modulus; G^V and G^R are the shear modulus can be calculated by eqn (4)–(7):⁴¹

B^V = [C₁₁ + C₂₂ + C₃₃ + 2(C₁₂ + C₁₃ + C₂₃)]/9 (4)

G^V = [(C₁₁ + C₂₂ + C₃₃) − (C₁₂ + C₁₃ + C₂₃) + 3(C₄₄ + C₅₅ + C₆₆)]/15 (5)

B^R = 1/[S₁₁ + S₂₂ + S₃₃ + 2(S₁₂ + S₁₃ + S₂₃)] (6)

G^R = 15/[4(S₁₁ + S₂₂ + S₃₃) − 4(S₁₂ + S₁₃ + S₂₃) + 3(S₄₄ + S₅₅ + S₆₆)] (7)

For the tetragonal structure, the same variables from shear modulus, as well as the variables from bulk modulus can be calculated by eqn (8)–(11):^42,43

B^V = [2(C₁₁ + C₁₂) + 4C₁₃ + C₃₃]/9 (8)

G^V = (M + 3C₁₁ − 3C₁₂ + 12C₄₄ + 6C₆₆)/30 (9)

B^R = C₁₁ + C₁₂ + 2C₃₃ − 4C₁₃/C₁₁ + C₁₂ + 2C₃₃ − 4C₁₃ (10)

G^R = 15/[(18B_V)/C² + 6/(C₁₁ − C₁₂) + 6/C₄₄ + 3/C₆₆] (11)

S_ij is the elastic compliance constant, and C_ij is the elastic constant. From the A^u value in Table 2, it can be concluded that U₃Si₂ has the lowest anisotropy is, followed by sub-U₃Si_1.5Al_0.5, and finally sub-U_2.5Si₂Al_0.5(I).

Next, the three-dimensional Young's modulus diagram is used to visually explore the anisotropy of elasticity. For tetragonal and orthogonal structures, the three-dimensional Young's modulus is expressed using eqn (12) and (13):^44,45

The isotropic materials would display a three-dimensional spherical structure; otherwise, the solid material is anisotropic one. The larger the deviation to spherical structure, the higher the anisotropy. Fig. 4 shows the three-dimensional Young's modulus calculated from the above formula. The curved surface for U₃Si₂ is close to a circle, and its A^u value is 0.13; while the A^u values of sub-U₃Si_1.5Al_0.5 and sub-U_2.5Si₂Al_0.5(I) are 0.32 and 1.27 respectively, indicating that U₃Si₂ has the strongest isotropy, followed by sub-U₃Si_1.5Al_0.5, and the most anisotropy belongs to sub-U_2.5Si₂Al_0.5(I).

Fig. 4 3D Young's modulus (E) and projections of the 3D Young's modulus of (a) U₃Si₂ (b) sub-U₃Si_1.5Al_0.5 (c) sub-U_2.5Si₂Al_0.55(I).

In order to further study the anisotropy of elasticity, 2D plane projections along the directions of [100], [010] and [001] have also been added into Fig. 4. For U₃Si₂, the maximum Young's modulus is 150 GPa, while sub-U₃Si_1.5Al_0.5 is 130Gpa, and sub-U_2.5Si₂Al_0.5(I) is 110 GPa. Therefore, it can be concluded that the sub-U₃Si_1.5Al_0.5 still maintains high isotropy, and the Young's modulus of the structure does not change much, while the sub-U_2.5Si₂Al_0.5(I) shows out of the strong anisotropy, the Young's modulus decreases significantly. The conclusions are consistent with the result determined by the A^u value.

3.4 Thermal conductivity

The thermal conductivity is an important property of nuclear fuel, since it determines the surface and centerline temperature of nuclear fuel during operation, as well as its maximum allowable direct heating rate. Phonons play an important role in dynamic behavior and thermal properties. The thermal conductivity of U₃Si₂ mainly comes from the contribution of phonons and electrons due to its metallic nature. In this section, we employ first-principle calculations to derive the phonon–phonon interaction strength. By solving the linearized Boltzmann transport equation, the lattice thermal conductivity in different directions can be accurately calculated; where the electronic transport properties are approximated with the relaxation time τ = 0.68 × 10⁻¹⁴ s evaluated using ρ at 300 K obtained from experiments.⁴⁶

The anisotropic lattice thermal conductivity of U₃Si₂, sub-U₃Si_1.5Al_0.5 and sub-U_2.5Si₂Al_0.5(I) is calculated, as shown in Fig. 5(a–c), the inset is the curve of K_zz/K_xx with temperature. It is worth noting that smaller values of K_l in the x- and z-directions have recently been reported in ref. 47 with the consideration of the occupation matrix control (OMC). In comparison, the results obtained within this work using VASP with the consideration of the U ramping method; and this is known as a source of discrepancies in DFT + U calculations. As shown in the Fig. 5(a–c), as the temperature increases, the lattice thermal conductivity of the three compounds decreases drastically until it becomes flat. This is because the scattering between phonons and phonons strengthens with the increase of temperature. It can be seen from the illustration that the K_zz/K_xx of U₃Si₂ is 1.3 at 50 K. As the temperature increases, it rises to a saturation value of 1.65 at about 300 K, and then gradually stabilizes. The general trends of sub-U₃Si_1.5Al_0.5 and sub-U_2.5Si₂Al_0.5(I) are similar to those of U₃Si₂. The same, but the saturation value is different, the former is 2.95 and the latter is 10.5. Therefore, we understand that through alloying, the lattice thermal conductivity still maintains a high isotropy as the temperature increases. This isotope of the thermal conductivity of the phonon is caused by the dispersion phase space and the similar group velocities in different directions.⁴⁸ Fig. 5(d) shows the calculated electronic thermal conductivity (K_e/τ), U₃Si₂, sub-U₃Si_1.5Al_0.5and sub-U_2.5Si₂Al_0.5(I); and it can be seen from the figure that the electronic thermal conductivity of the three compounds all increase linearly with the increase of temperature. In addition, for the compound of sub-U₃Si_1.5Al_0.5 configuration, the increase rate of the electronic thermal conductivity with the increase of temperature is lower than that of U₃Si₂, while the increase rate of the compound of sub-U_2.5Si₂Al_0.5(I) configuration is higher than that of U₃Si₂.

Fig. 5 The anisotropy in lattice thermal conductivity (K_l) of (a) U₃Si₂, (b) sub-U₃Si_1.5Al_0.5 and (c) sub-U_2.5Si₂Al_0.5(I) obtained by using relaxation time approximation (RTA). (d) The electronic thermal conductivity (K_e) with dependent relaxation time.

4 Conclusion

This paper studies the mechanical and thermal properties of Al-alloyed U₃Si₂ nuclear fuels on the basis of first principles. Through the phonon dispersion curve, two kinetic stable structures sub-U₃Si_1.5Al_0.5 and sub-U_2.5Si₂Al_0.5(I) are screened out. We found that the toughness of the two compounds after alloying were significantly improved compared to U₃Si₂. The three-dimensional Young's modulus show that, the sub-U₃Si_1.5Al_0.5 formed by Al alloying in U₃Si₂ maintains a higher mechanical isotropy, while sub-U_2.5Si₂Al_0.5(I) shows higher mechanical anisotropy, Which are consistent with the value of A^u. The calculation result shows that the lattice thermal conductivity of sub-U₃Si_1.5Al_0.5 and sub-U_2.5Si₂Al_0.5(I) after Al alloying exhibits high isotropy as the temperature increases.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The authors acknowledge the financial support of National Natural Science Foundation of China (Grant no. 51872302, 21875271), National Key Research and Development Program of China (No. 2016YFB0700100), Key Research and Development Program of Zhejiang Province (Grant no. 2019C01060).

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