Density Functional Theory-Based Study of UC₂ and Cr-Doped UO₂

Abstract

A density functional theory-based study of UC₂ and Cr-doped UO₂ using the phono3py and VASP computational simulation packages is presented. Furthermore, the temperature-dependent thermal conductivities are compared to the traditional urania fuel. Doping of urania with Cr allows for improved fission gas retention, reducing the fission gas release and lowering the oxidation rate of UO₂. The thermal conductivity calculated using the random alloy method with one U atom replaced by Cr in a supercell (CrU₃₁O₆₄) shows a slight decrease; however, this may be compensated for by larger grain sizes in the presence of Cr. The reduction of thermal conductivity for the 0.61 wt.% Cr substation in urania is presented. Investigated here, the UC₂ metallic high-temperature fcc phase looks promising due to additional electronic contribution to conductivity. Furthermore, we found that the temperature-dependent phonon-assisted thermal conductivities for UC₂ and UO₂ are very similar. The elastic properties of UC₂ were also evaluated and compared with UO₂. The presented analysis provides information for further improvement of the design of nuclear fuels.

1. Introduction

Urania (UO₂) has been the dominant nuclear fuel in commercial light water reactors (LWR) over the past 50 years. Despite its success, one limiting factor for improving burn-up is the fission product release during the irradiation [1,2,3]. Especially the gaseous fission products, such as Kr and Xe, with very low solubility, accumulate along the grain boundaries causing fuel swelling. This ultimately stimulates fuel cracking and premature fuel degradation [2,4,5,6]. One of the proposed solutions is doped fuels (e.g., ADOPT^TM fuel [7]), which incorporate small amounts of dopants, such as Cr and Cr₂O₃, during the fabrication [8,9,10,11,12]. It is known that these modified fuels have coarser grains than pure UO₂, and this allows for enhancing the fission product retention and suppressing the fuel swelling [9,12,13,14,15].

Most often, Cr-doped UO₂ is produced by sintering a mixture of raw Cr₂O₃ and UO₂ powders under a controlled process at elevated temperatures [4,13]. As reported in the literature, depending on the sintering temperature and other fabrication conditions, a couple of options exist. In some cases, the sintered fuel may contain dissolved Cr metallic particles, in another cases it forms a solid solution with UO₂. In some other cases, Cr is distributed along the grain boundaries of the fuel [16].

In addition, precipitates of chromium oxides such as Cr₂O₃ and CrO [17] within the fuel matrix have been observed [12,13,18]. Various studies analyzed properties in Cr-doped UO₂ fuels in relation to the preparation methodologies [9,13], microstructural behaviour [13,14], dissolution mechanism [10,11], and oxidation of chromium [18]. There are numerous experimental and computational records on the structural, chemical, electronic, mechanical, and thermal properties of Cr₂O₃-doped UO₂ fuels in the literature [10,11,15]. However, there are inconclusive, conflicting results regarding the thermal conductivity of such Cr-modified fuels, and such information is crucial for the safe operation of the reactor; therefore, any enhancement of such critical property will be required for future reactors.

Alternatively, metallic ceramic fuels are of special interest as their thermal conductivity remains high even at very high temperatures due to significant electronic heat transport. They are important because urania fuel, which is used in conventional nuclear reactors, is not the optimum choice for some designs of new-generation reactors due to its low thermal conductivity [19].

ADOPT^TM fuel [7] is considered as an evolutionary solution, while metallic, high-density fuel like UN is a revolutionary future solution due to its much higher thermal conductivity [20]. UC₂ metallic fuel is of interest since it has a similar crystal structure as UO₂ and a high melting temperature (2720 K), which is attributed to high carbon-metal bonds [21]. Although UC₂ has a lower uranium density than UC, it can potentially be considered for advanced fuel fabrication once the stoichiometric instability issue is resolved. Density functional theory (DFT)-based calculations become a more and more common methodology to evaluate the mechanical and thermal conductivity of novel fuel materials. The DFT calculations for bcc UC₂ confirmed its metallic character; however, they predicted a slightly lower bulk modulus (180.5 GPa) [22] than the experiment (216 GPa) [23]. It is expected that the high-temperature phase of UC₂ is fcc, but it has not been investigated theoretically.

Here, we unveil the impact of Cr on the thermal conductivity of Cr-doped UO₂ (i.e., CrU₃₁O₆₄). It is of interest to compare fcc UC₂ with UO₂ since they have the same (cubic, Fm$\overline{3}$m symmetry) structure. The thermal conductivities and mechanical properties will be also investigated.

2. Methodology

The Vienna DFT simulation package VASP was employed in the simulation [24]. We assumed electronic configurations of 5f³, 6d¹, 7s² for U, 2s², and 2p⁴ for O and 2s² and 2p² for C atoms. The exchange–correlation functionals were selected within the generalized gradient approximations as parameterized by Perdew et al. [25]. It is established that, with the exception of the band gap, DFT describes very well the physical properties of UO₂. Optical properties are not investigated here, but for completeness, we used the simplified (rotationally invariant) approach to the DFT + U, introduced by Dudarev et al. [26] and implemented in VASP as option 2. The elastic properties and equilibrium lattice constants are compared with and without HU correction equal to 3.5 eV, since it reproduced the experimental band gap in UO₂ [27]. All the presented results for the VASP calculations are obtained using a plane-wave cut-off energy of 500 eV and a supercell of 2 × 2 × 2 with 96 atoms. In the supercells of UO₂, there are 32 uranium and 64 oxygen atoms; in the case of CrO₂, the doping there is substituted for U atoms.

The norm-conserved pseudopotentials were used for evaluation of electronic thermal conductivity of UC₂ a with the functional for solids developed for GGA of the Perdew, Burke, and Ernzerhof functional developed for solids (PBEsol) [28], as implemented in Quantum Espresso code [29] with a kinetic energy cutoff of 120 Ry for wave functions.

2.1. Elastic Constants

The elastic constants were evaluated using the Elastic code [30] (Release v5.1.0-14-g6596fd4) with a command option and adapted for a high-performance batch submission script. This code is generalized to non-cubic systems.

2.2. Lattice Assisted Thermal Conductivity

We used the random alloy method [31,32] to model CrU₃₁O₆₄ by randomly replacing a uranium atom in the UO₂ supercell with chromium and then solving the corresponding BTE for that particular system using the phono3py [33] simulation package to evaluate lattice assisted thermal conductivity. In the random alloy method, one of the uranium atoms in the (2 × 2 × 2) supercell was randomly displaced with a Cr atom, and the thermal conductivity was evaluated for the new crystal structure U₃₁CrO₆₄ utilizing the phono3py code [33]. The respective fine meshes with the sizes of $[20, 20, 20]$ and $[5, 5, 5]$ were used in the calculation of pure UO₂ and U₃₁CrO₆₄. We need to note here that the calculation of pure UO₂ with phono3py considers the symmetry along the primitive axis (0 1/2 1/2, 1/2 0 1/2, 1/2 1/2 0) to reduce the computational cost. In Figure 1 the results are presented for the grid of meshes from 4 to 20 (red triangles) for pa = half. There is also the option “auto” with smaller primitive cells, but as indicated in Figure 1 by black circles, the results were not very reliable, although they converge to a similar value for $[20, 20, 20] \mathrm{m}\mathrm{e}\mathrm{s}\mathrm{h}.$ For the calculations for 96 atoms without symmetry (U₃₁CrO₆₄), we had to use a reduced mesh. To eliminate a systematic error from the usage of reduced mesh to investigate the effect of Cr we performed calculation for a full cell of 96 atoms of UO₂ with the mesh [$5, 5, 5$] and compared it with the results for $[4, 4, 4]$ mesh in Figure 1 as indicated by the solid dark blue line.

2.3. Electronic Thermal Conductivity

The electronic contribution to the thermal conductivities (κ_e) of metallic UC₂ was calculated using the Wiedemann–Franz law [34] and the electrical conductivities (σ) are evaluated here using the BolzTrap2 code [35]:

$${\kappa }_e={\displaystyle \frac{\pi }{3}}{\left({\displaystyle \frac{k_B}{e}}\right)}^2\sigma T$$

(1)

Using the EPW code [36,37] we determined that the scattering rate is increasing linearly with temperature (ds_R/dT = 1.23373 × 10¹¹ s⁻¹K⁻¹) for the selected 6,7 bands within the k_BT energy range for predominantly 5f electrons around Fermi Energy. The absolute value of the electrons’ relaxation time at 850 K temperature (τ₈₅₀ = 3.734 × 10⁻¹⁵ s) was evaluated using non-magnetic thermal electronic conductivity (κ_e) calculated at a temperature of 850 K from the BoltzTrap2 code (in relaxation time units: 5.72918 × 10⁺¹⁵ Wm⁻¹K⁻¹ s⁻¹) and obtained (κ_e) from solving the Boltzmann transport equation (BTE) using the EPW code (21.39 Wm⁻¹K⁻¹) at the same temperature. The derived correlation for the electrons’ relaxation time (τ_T) as a function of temperature is in the form:

$${\mathsf{\tau }}_T={\left({\mathsf{\tau }}_{800}^{-1}+(T-850)\hspace{0.17em}d{s}_R/dT\right)}^{-1}={\left(2.67814\times {10}^{14}+(T-850)\hspace{0.17em}1.23373\times {10}^{11}\right)}^{-1}$$

(2)

We selected a temperature of 850 K since it was not dependant on the selected fsthick parameter in the present calculations. In contrast to previous EPW calculations [36], we were not able to select a stable range for the fsthick value for all temperatures, therefore we present here only the electronic thermal conductivity calculated using Ziman’s formula for metals (Equation (54) in [37]), which is implemented in the EPW code with a number of carriers evaluated to coincide with the electrical resistivity at 850 K, as evaluated via the EPW BTE solver.

The EPW code was developed as a module using the Quantum Espresso (QE) code [29]. Similarly to previous calculations for UN [36], we used norm-conserving pseudopotentials and generalized gradient approximation (GGA) of the Perdew, Burke, and Ernzerhof functional developed for solids (PBEsol)18 [28], as implemented in QE.

2.4. Total Thermal Conductivity

The total thermal conductivity is calculated as a sum of the electronic thermal conductivity obtained using the BoltzTrap2 code with the relaxation time defined by Equation (2) and extrapolated to higher temperatures lattice conductivity calculated using phono3py [33]. Similarly to the calculation of resistivity using Zimann’s formula, as implemented in the EPW code, phono3py outputs results only up to 1000 K, therefore we used the trend lines to extrapolate values to higher temperatures.

3. Results

3.1. Initial Ground State Structures

Stoichiometric UO₂ crystallizes in a cubic structure (space group $Fm\overline{3}m$) containing one uranium and two oxygen atoms in the unit cell. We study here fcc UC₂ high-temperature phase with the same cubic structure and primitive unit cell (Figure 2a). The optimized lattice parameter using the VASP code of UO₂ is 0.537 nm, which is only slightly lower than the reported experimental (see

Table 1. The equilibrium lattice constant and elastic constants of UO₂ and UC₂ are listed and compared with experimental data, where available. The shown polycrystalline aggregate of the bulk modulus (B), Young modulus (Y), and shear modulus (G) in the Voigt approach for U0₂ and UC₂ are very similar. In the second column the respective quantities are also shown in the brackets, with an HU = 3.5 eV correction for UO₂.

Compound	UO2 (HU 2.5 eV)	Exp. UO2	UC2	Exp.: UC2
acalc [nm]	0.537 (0.5457)	0.54582 [38,39] (0.543 at 0 K)	0.538	0.541 [41]
B [GPa]	208.8 (192.6)	208.9	198.7
C11 [GPa]	400.9 (369.0)	389.3	267.5
C12 [GPa]	112.7 (104.5)	118.7	164.3
C44 [GPa]	82.9 (82.2)	59.7	145.8
G [GPa]	107.4 (102.2)		108.12
Y [GPa]	275.0 (260.5)		274.6
G/B	0.51 (0.53)		0.54

) and theoretical values [38,39,40]. The larger equilibrium lattice constants were found in the present QE calculations (0.5496 nm) are in good agreement with the experiment. The lattice constants of UC₂ (0.538 nm) evaluated here are very close to those reported previously [41], as shown in Table 1.

In Figure 2b, the supercell of Cr-doped UO₂ (i.e., CrU₃₁O₆₄) and the optimized cubic lattice constants are slightly smaller, as expected (0.536 nm).

In Table 1, the equilibrium lattice constants, elastic constants, and mechanical moduli of UO₂ and UC₂ are listed and compared with experimental data, where available. The calculations were performed using a GGA functional, with the exception of those listed in the second column, the respective quantities (shown in the bracket) with a GGA + HU correction of 3.5 eV are included for UO₂.

While the C₁₁ and C₁₂ elastic constants are in good agreement with experiment values, C₄₄ is over predicted (82.9 GPa and 82.2 GPa with HU correction versus 59.7 GPa) for UO₂. The calculated lattice constants are comparable for UO₂ and UC₂, but C₄₄ is predicted to be much higher (145.8 GPa versus 82.9 (82.2) GPa) for UC₂ than for UO₂. This is probably due to the high affinity of 2p² electrons of the carbon atoms with strong anisotropy, since the polycrystalline aggregate of bulk modulus (B), Young modulus (Y), and shear modulus (G) in the Voigt approach are comparable. As shown in Table 1, UC₂ is predicted to be more brittle according to Cottrell’s prediction for cubic metal and alloy crystals that indicates intrinsic brittleness occurs when G/B > 0.5 and ductility when G/B < 0.4. While the lattice constant of UO₂ agrees slightly better with the experiment with the HU (3.5 eV) correction included, 0.537 nm (0.5457 nm) versus 0.54582 nm, the bulk modulus is slightly underpredicted for the latest (208.8 GPa (192.6 GPa) versus 208.9 GPa. The effect of HU correction is insignificant; therefore, we omit it in further calculations as it also requires higher computational resources.

The phonon density of states (DOS) is one of the fundamental concepts used to investigate the dynamic stability of a crystalline structure. Hence first, we evaluated the phDOS of UO₂ (solid black line) along the high symmetry path of the first Brillouin zone using phono3py code, which is shown in Figure 3. As shown in the figure, all the phonon vibrations have positive frequencies, indicating the dynamical stability of the considered structure. Additionally, the phonon DOS of UO₂ is experimentally and theoretically well-established in the literature, and our predicted phonon DOS compares well with the available data [42,43]. Figure 3 also provides the phDOS for the CrU₃₁O₆₄ structure (short-dashed grey line) which was created by replacing one of the uranium atoms with chromium via the random alloy method. The dot-dashed red line in the figure represents the phDOS for pure Cr with the equilibrium lattice constant of 0.283 nm. The addition of Cr is predominantly visible in the PhDOS of CrU₃₁O₆₄ in additional states in the gap with frequencies higher than 200–250 $\mathrm{c}{\mathrm{m}}^{-1}$.

In Figure 4 the total phDOS is marked for UC₂ by the red line. The partial phDOS of U is indicated by a dark blue long-dashed line, while for the C atom it is marked by a short-dashed black line. Since C atoms have a lower mass than U atoms, they mainly contribute to high-frequency optical phonons, while the acoustic phonons’ contribution is predominantly from the U atoms. We confirm the stability of fcc UC₂ as there are no negative frequencies observed.

3.2. Thermal Conductivity of Pure UO₂ and U_1-xCr_xO₂

First, the temperature dependence of the lattice thermal conductivities of UO₂ over temperatures from 300 to 1000 K is evaluated with the highest accuracy [20, 20, 20] mesh; pa = half, as discussed in Section 2.2. The solid red line in Figure 5 represents the temperature-dependent lattice thermal conductivity of UO₂ estimated here, employing the phono3py simulation packages versus experiments shown by the red stars and recommended values by Fink et al. [44]. As discussed in Section 2.2 to reduce the systematic error in the evaluation of Cr doping, we performed additional calculations using less dense mesh [5, 5, 5] for the UO₂ supercell with 96 atoms. It can be seen from Figure 5 that the thermal conductivity of pure UO₂ (indicated by a long-dashed dark blue line) decreases when the Cr atom is added, as shown by the dot-dashed green line for the conductivity of CrU₃₁O₆₄. However, all lines fit the form ${(a+bT)}^{-1}$, where $a$ and $b$ are temperature-independent constants.

As shown in Figure 5, the thermal conductivity of Cr-doped UO₂ corresponds to the 0.61 wt.%Cr random alloy method calculations using the phono3py simulation package. The weight percentage of chromium ($wt.\%\mathrm{C}\mathrm{r}$) is calculated as the percentage of the ratio of the total weight of Cr atoms to the sum of the total atomic weight of the unit cell, as given by Equation (3).

$$wt.\%\mathrm{C}\mathrm{r}={\displaystyle \frac{M_{Cr}\times x}{M_U\times (1-x)+M_{Cr}\times x+2M_O}}\%$$

(3)

where $M_{Cr}=51.991$, $M_U=238.029$, and $M_O=31.998$ represent the atomic masses of chromium, uranium, and oxygen in atomic mass units, respectively. The drop in thermal conductivity with Cr doping does not support the experimentally observed increase of thermal conductivity by Camarano et al. [45]. The thermal conductivities at 300 K of Cr₂O₃ doped UO₂ from Camarano et al.’s experiments increased from 7.16–7.60 W.m⁻¹K⁻¹ for 1% wt% Cr₂O₃ doping to from 7.86–8.27 W.m⁻¹K⁻¹ at 2%. This discrepancy may be attributed to the microstructural conditions of the samples, which critically affect the experimentally measured thermal conductivity. We believe the increase may result from an increase in the grain size, as was observed for ADOPT^TM fuel, while here our calculations are for the single crystal. Additionally, there is a strong dependence of the κ_ph of UO₂ on stoichiometry deviation, which may have been reduced during doping [45]. Our conclusions, however, agree with recent, extensive investigations [46] that we found during the final publication of our work. Additionally, lower thermal conductivity was confirmed for 5000 ppm Cr–UO₂ [47] in earlier publications.

The calculated lattice thermal conductivity of UC₂ shown by the dotted blue line in Figure 5 resembles that of UO₂.

3.3. Conductivity of Metallic UC2

In contrast to UO₂, UC₂ is metallic as shown in Figure 6. The total density of states per formula unit is indicated by a black solid line. The projected density of states of the 5f electrons (indicated by a dashed blue line) was too low around the Fermi energy to lead to a stable magnetic ordering. There is a significant contribution from the 2p electrons of the C atoms, with high affinity (two 2p electrons in shell) around the Fermi energy, as indicated by the short-dashed line.

Using the EPW code with BTE solver and the Wiedemann–Franz law represented by Equation (1), we evaluated the electronic thermal conductivity of UC₂ at 850 K, as indicated by the grey circle in Figure 7. Using this value, we were able to evaluate a number of carriers (nc = 0.811) that can be an input parameter to Ziman’s formula for metals in the EPW code and obtained electronic thermal conductivity, as shown by black dot-dashed line.

Furthermore, as described in Section 2.3 we used the electron’s relaxation time derived here (Equation (2)) to calculate electrical conductivity using the BoltzTrap2 code and xml main output file from VASP. The electronic thermal conductivity of UC₂ was obtained next using Equation (1) and is shown in Figure 7 by a long-dashed red line. We do not show the EPW results for the other temperatures, since there was some noise observed that was dependent on the fsthick parameter value. The fsthick parameter used here is equal to 3 eV and also leads to coinciding values calculated by the EPW code and denoted by a dashed red line: κ_e for 300 K and 500 K. In Figure 7, we reproduced from Figure 5 the lattice-assisted thermal conductivity (short-dashed green line) as calculated using VASP and phono3py and extrapolated from 1000 K to 1600 K. The total thermal conductivity of UC₂ is calculated as a sum of κ_e, marked by a long-dashed red line, and κ_ph (short-dashed green line) and is shown as a solid dark blue line. When comparing the obtained total thermal conductivity of UC₂ with the thermal conductivity of UO₂ (Figure 5) we note significant enhancement due to the electronic (κ_e) contribution.

4. Discussion

The thermal conductivity of Cr-doped UO₂ was investigated using the random alloy method to check the effect of doping, as described in the methodology section. However, the computational cost for the random alloy method is very demanding because it requires several hundreds of displaced supercells to assess the harmonic and anharmonic scatterings. Therefore, the random alloy method is employed only for 0.61 wt.%Cr (i.e., for CrU₃₁O₆₄) and the results are compared in Figure 5 with pure UO₂ for the same mesh [5, 5, 5] to reduce the systematic error due to computational limitations.

The agreement of the phono3py predictions for stoichiometric urania with the values from the literature [44] (red star) is fair. The observed deviation at the high-temperature region could be due to the consideration of the polaron term in Fink’s recommendation.

Interestingly the calculated κ_ph for UC₂ is very similar to that of UO₂, as shown in Figure 5.

Our calculations predict that doping urania with Cr does not enhance the total thermal conductivity in a single crystal, but the total thermal conductivity in UC₂ is significantly enhanced by electronic contribution (κ_e), evaluated here using various methods in the EPW and BoltzTrap2 codes. However, our calculations may over-predict κ_e due to the limitation of DFT usage for strongly correlated systems with 5f electrons on U atoms, as was observed for UN previously.

5. Conclusions

The effect of Cr additives on the thermal conductivity of urania was investigated. The results show that, at any temperature, the thermal conductivity decreases with the Cr dopant in the fuel, as modelled for a single crystal. However, this decrease is not that significant at higher temperatures, where κ_ph is already very low.

Interestingly, although we found the κ_ph of UO₂ and UC₂ to be very similar, the total thermal conductivity of UC₂ is much higher due to the electronic contribution, which additionally increases with temperature. Therefore, we conclude that metallic fuel will be much safer and more economical, as higher thermal conductivity also reduces thermal strain and increases the longevity of the fuel.