Systematic study of the structural, energetic and elastic properties of U_1−yAm_yO_2−x compounds using empirical interatomic potentials

Abstract

Uranium–americium mixed oxides are envisaged as fuels or blankets for the transmutation of americium in fast neutron reactors, which requires an accurate knowledge of their properties. Here, the Cooper–Rushton–Grimes (CRG) potential terms related to Am⁴⁺ are reoptimized to lower the very high melting temperature of AmO₂ predicted by the potential (≈3350 K) while keeping other properties unchanged. The resulting terms are used alongside U⁴⁺ parameters as well as previously optimized terms for the Am³⁺ and U⁵⁺ oxidation states in combination with molecular dynamics to perform a systematic study of the structural, thermodynamic and elastic properties of hypostoichiometric uranium–americium mixed oxides U_1−yAm_yO_2−x with 0 ≤ y ≤ 1 and 0 ≤ x ≤ 0.1 over a wide temperature range (250–4500 K). The properties calculated include the lattice parameter and related properties (density, thermal expansion coefficient), the enthalpy increment and related properties (specific heat capacity, enthalpy of fusion), the melting temperature and the elastic constants. Our study shows a peculiar evolution of properties with the americium content with a change in the evolution for 50% americium due to the various oxidation states of the cations present in these compounds. Results presented in this paper provide significant insight into the properties of uranium–americium mixed oxides for all compositions, in particular for high americium contents, non-stoichiometric oxides and at high temperature and complement the few experimental data available for these materials.

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I. Introduction

Uranium–americium mixed oxides are envisaged as fuels or blankets in fast neutron reactors to allow for the transmutation of americium in view of the reduction of its quantity in nuclear waste.¹ In this context, the knowledge of thermodynamic, structural and elastic properties of (U,Am)O₂ fuels is of crucial importance for fuel manufacturing, as well as for safe reactor operation. Beyond the use of low americium content fuel in reactor, highly Am-enriched uranium–americium oxides are being considered for space applications in radioisotope thermoelectric generators.^2–10 The knowledge of thermodynamic properties for all Am contents, including high ones, is therefore crucial. In addition, the knowledge of elastic properties, in particular the elastic constants C₁₁, C₁₂ and C₄₄, is necessary since these are direct parameters of micromechanical laws for single crystals, which are then homogenised for the simulation of the behaviour of the polycrystalline material.

Experimental investigations on americium-bearing compounds are difficult due to the high radioactivity of this element and the data reported in the literature on uranium–americium oxides are relatively scarce. The U–Am–O ternary phase diagram data were obtained by Horlait et al.,¹¹ Vălu et al.,¹² Prieur et al.,^13–15 Lebreton et al.¹⁶ and more recently Epifano et al.^17–20 A CALPHAD model of the ternary system was then developed by Epifano et al. using these experimental data.²¹ This work pointed out that further investigations of this system are required, especially for high Am contents and oxygen to metal ratio (O/M) below 2 to improve the CALPHAD modelling of americium-bearing mixed actinide oxides.

Atomic scale modelling has proved itself to be an important complement to experimental characterizations to obtain data and bring insight into the behaviour of fuel materials over a wide range of compositions and conditions.^22,23 We use it in this study to complement the few experimental data available on uranium–americium oxides.

The Cooper–Rushton–Grimes (CRG) potential²⁴ has yielded excellent results on several pure and mixed actinide oxides,^25–37 and was adapted for the study of uranium–americium mixed oxides in a previous paper.³⁸ It yields, however, a very high melting temperature for AmO₂ (≈3350 K), which could impact the results on uranium–americium mixed oxides with high Am content. We propose here an estimate of the probable melting temperature range for a hypothetical americium dioxide based on thermodynamic considerations and compare it to the very scarce data available.^39,40

Based on this range estimate, we reoptimized the interaction terms related to Am⁴⁺ to lower the AmO₂ melting temperature while keeping structural, thermodynamic and elastic properties virtually unchanged, as they are in line with experimental data. We then used the updated parameters in combination with U⁴⁺, Am³⁺ and U⁵⁺ terms to calculate the structural, thermodynamic and elastic properties of U_1−yAm_yO_2−x with 0 ≤ y ≤ 1 and 0 ≤ x ≤ 0.1.

The paper is organized as follows. We briefly recall the potential formalism and calculation details, which are presented in detail in the paper dedicated to the optimization of the Am³⁺ and U⁵⁺ terms.³⁸ We then describe precisely the method used to include the melting temperature in the fitting procedure and the generation scheme used to incorporate vacancies in our supercells. We then present and discuss the results of the systematic study of U_1−yAm_yO_2−x compounds: the structural and energetic properties extracted directly from simulations, the derivative properties, i.e., thermal expansion coefficient and specific heat capacity, the melting temperatures, and the elastic properties. Finally, we conclude and present the outlooks of this work.

II. Methods

II.A. Potential formalism

The Cooper–Rushton–Grimes (CRG) potential²⁴ is an embedded atom method (EAM) potential^41,42 with the following formal definition for the potential energy of an atom i as follows:The first term is the pair potential, where φ_αβ(r_ij) is the sum of the long-range Coulomb and of the short-range Morse⁴³ and Buckingham⁴⁴ potentials. The second term is the N-body contribution representing the embedding energy of the atom i in the electronic density owing to its neighbours, enabling one to take into account the impact of the atomic environment on the potential energy of the atom i.

This potential formalism is remarkably good at reproducing known properties of actinide oxides, in particular the lattice parameter, the Cauchy violation of the elastic constants C₁₂ ≠ C₄₄ and the bulk modulus temperature dependence, probably due to the strongly ionic iono-covalent bonding in these systems. The terms originally published were optimized on experimental lattice parameters and elastic properties, while ensuring that fluorite was the most stable structure. The terms subsequently added for Am³⁺ and U⁵⁺ were optimized using lattice parameters and enthalpy increments^12,17 for several stoichiometric uranium–americium mixed oxides at room temperature^11,16 and at 1000 K.^16,19

II.B. Calculation details

II.B.1. Building the supercells. The compositions studied in this paper are U_1−yAm_yO_2−x with 0 ≤ y ≤ 1 and 0 ≤ x ≤ 0.1. As in our previous study,³⁸ the uranium–americium mixed oxides supercells were generated using a custom Python script computing pair distances by positioning cations as homogeneously as possible and Am³⁺ and U⁵⁺ as close as possible to each other. One hundred configurations were generated for each composition. The five most homogeneous ones were chosen from the computation of the distances between the various cations and used in production runs. The size of the supercells depends on the properties calculated (see next section). The concentrations of the cations with various oxidation states determined by electronic structure calculations⁴⁵ are represented in Fig. 1 for compounds with oxygen to metal ratio O/M = 2.00. It can be seen that Am⁴⁺ cations are present only for compositions with Am/(Am + U) > 0.5. This specific behaviour of (U,Am)O_2±x was observed experimentally by Epifano et al., who measured the multi-valence states in the mixed oxides using XAS.¹⁸ The CALPHAD model, derived from experimental oxygen potential and phase diagram data, reproduces this trend as well.²¹

Fig. 1 Concentrations of the various cationic oxidation states in (U,Am)O₂ compounds as a function of americium content, reproduced from ref. 45.

For the study of hypostoichiometric oxides, oxygen vacancies were introduced randomly considering their relatively low concentration and the possibility for oxygen to diffuse. Moreover, DFT+U calculation results on AmO_2−x showed that configurations with oxygen vacancies nearest neighbours to Am³⁺ were not necessarily the most stable.⁴⁶ The charge neutrality was ensured by the reduction of one or several species. The order in which species are reduced is known from DFT+U calculations:⁴⁵

• Below 50% Am, only U⁵⁺ can be reduced to U⁴⁺ since all americium cations are already in the (+III) oxidation state and U³⁺ terms are not included in the potential. This implies that there is a limit on the oxygen to metal ratio imposed by the americium content.

• Above 50% Am, Am⁴⁺ ions are reduced first to Am³⁺, followed by U⁵⁺ ions to U⁴⁺ if necessary. The latter occurs only for large deviations from stoichiometry for 0.50 < Am/(Am + U) ≤ 0.60.

II.B.2. Determination of structural, thermodynamic and elastic properties. All calculations are performed using the large-scale atomic/molecular massively parallel simulator (LAMMPS) code.⁴⁷
II.B.2.a. Heat until it melts and moving interface methods. The Heat Until it Melts (HUM)^48,49 and moving interface (MI)^50,51 methods are presented in detail in our previous paper, only the main points are recalled here. The HUM method allows for the determination of the structural and energetic properties of interest as a function of temperature. We equilibrated the supercells for 100 ps in the NPT ensemble at 0 bar (as is standard in the community given the negligible pressure dependence of studied properties and to ensure U = H), from 250 K to 4500 K every 25 K, and averaged the properties over the last 50 ps for each temperature. These simulations were performed on 6 × 6 × 6 replications of the conventional fluorite cell, i.e., 2592-atom supercells for stoichiometric systems.

Due to overheating effects caused by the finite-size of the system, the melting temperature detected in a HUM run is overestimated by approximately 20%.^52,53 The MI method is thus used to estimate a pseudo-melting temperature by simulating a solid–liquid interface, imposing the final temperature and pressure and monitoring the position of the interface. This monitoring was performed by computing the local entropy and the centro-symmetry parameter, as well as performing common neighbour analyses using LAMMPS to obtain the profile of these properties along the longest dimension of the supercell. The position of the interface at the end of the simulation indicates if the applied temperature is above or below the melting temperature yielded by the potential. The melting temperature is found when at the end of the simulation, both phases account for approximately half the supercell each. These simulations were performed in the NPT ensemble with a 25 K temperature step and 1 ns final relaxation time on 8 × 8 × 16 replications of the conventional fluorite cell, i.e., 12 288-atom supercells for stoichiometric compounds.

For both types of calculations, we used a 1 fs timestep, a 0.1 ps temperature damping parameter and a 0.5 ps pressure damping parameter for the Nosé–Hoover thermostat and barostat, respectively. Simulations were performed on five supercells with varying random configurations of U and Am positions in the cationic sublattice for each composition and on three seeds for the initialization of the velocities per supercell, totalling fifteen repetitions for a given set of conditions. Properties were then averaged over these simulations, taking twice the standard deviation as the uncertainty.

II.B.2.b. Calculation of elastic properties. The supercells equilibrated at a finite temperature for a given pressure using the HUM method can be used to compute the elastic constants. To do so, small deformations are applied to the simulation cell. These must be small enough so that the stress versus strain relationship respects Hooke's law, i.e., remains linear. This allows for the determination of the elastic tensor which, in the case of cubic materials, is entirely described by the C₁₁, C₁₂ and C₄₄ constants using the Voigt notation. These can be used to compute various elastic moduli, as well as the Zener ratio, which measures the isotropy of cubic materials, in the Voigt–Reuss–Hill (VRH) scheme,^54–56 therefore yielding a complete representation of the linear elastic properties of the material.

In practice, we used the lattice parameters yielded by the HUM method every 100 K to rescale the perfect supercell at each temperature before performing a 2 × 2 × 2 replication due to the sensibility of elastic properties on the supercell size. We then equilibrated the resulting 12 × 12 × 12 replication supercell at the corresponding temperature and zero pressure in the NPT ensemble for 30 ps to dislodge atoms from their equilibrium position. The LAMMPS “ELASTIC_T” script was used on the resulting structure to apply a ±4% deformation successively along all crystallographic axes and tilt factors in order to compute C₁₁, C₁₂ and C₄₄. The Zener ratio A is then calculated as:

As for the previous types of calculations, we used a 1 fs timestep, a 0.1 ps temperature damping parameter and a 0.5 ps pressure damping parameter for the Nosé–Hoover thermostat and barostat, respectively. Simulations were performed on five supercells with varying uranium and americium positions for each composition. Properties were then averaged, taking twice the standard deviation across the five simulations as the uncertainty.

III. Reoptimization and validation of Am⁴⁺ related potential parameters

III.A. Estimated melting temperature range

A single experimental value of AmO₂ melting temperature was published by McHenry in 1965.⁵⁷ The measurement was performed without control of the oxygen content and yielded 2386 K at standard pressure. CALPHAD-based calculations^58,59 and experimental characterizations,^17,58 however, show that the reduction process of AmO₂ begins at temperatures as low as 1100 K. The calculated phase diagram also indicates that the melting should occur for an O/M ratio of approximately 1.76. The melting temperature of stoichiometric americium dioxide can therefore only be estimated. Casalta estimated the melting temperature of AmO₂ at 2520 K without providing the method.³⁹ More recently, Kato et al. proposed a value of 2773 K⁴⁰ based on the effect of Am on the melting temperature of mixed actinide oxides.

The melting temperature of stoichiometric americium dioxide predicted by the CRG potential using the MI method is about 3350 K. Since the melting temperature of AmO₂ seems to have a strong influence on that of the mixed uranium–americium oxides with high Am contents⁶⁰ and as a starting point for the precise modelling of AmO_2−x compounds, a reoptimization of the Am⁴⁺ related parameters of the potential is needed to lower this value. We propose here a method to determine a probable melting temperature interval for a hypothetical stoichiometric AmO₂. We use the enthalpy of formation and Gibbs free energy of formation of the actinide dioxides to infer the stability of these compounds. Using values from,^61,62 we compute the ratios T_m/Δ_fH° and T_m/Δ_fG° for all actinide dioxides in the series ThO₂ to AmO₂. Results are reported in Table 1 and shown in Fig. 2. It is observed that the values are similar for all actinide dioxides except for AmO₂, giving an additional hint that the experimental value by McHenry was not obtained on the stoichiometric compound.

Table 1 Computed ratios T_m/Δ_fH° and T_m/Δ_fG° for actinide dioxides

	ThO2	PaO2	UO2	NpO2	PuO2	AmO2
T m/ΔfH° (K mol kJ^−1)	−2.977	−2.891	−2.876	−2.848	−2.858	−2.560
T m/ΔfG° (K mol kJ^−1)	−3.123	−3.036	−3.024	−3.007	−3.023	−2.721

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Fig. 2 Ratios T_m/Δ_fH° (top) and T_m/Δ_fG° (bottom) for the actinide dioxides series.

Considering the various degrees of uncertainty associated to these ratios, a simple linear fit on the values of other actinide dioxides is used to determine a probable upper limit for these two ratios for americium dioxide. Values obtained are −2.81 K mol kJ⁻¹ and −2.97 K mol kJ⁻¹ for T_m/Δ_fH° and T_m/Δ_fG°, respectively, corresponding to a melting temperature of 2620 K and 2605 K, respectively. The latter is chosen as the lower bound for our melting temperature interval estimate.

The lower limit for these two ratios, which corresponds to the upper limit of the temperature interval, is more difficult to assess due to the transition in the chemical nature of cation-oxygen bonds taking place between plutonium and neptunium dioxides.^63,64 It can be seen that the value of the two ratios decreases slightly from NpO₂ to PuO₂, but we have no indication on their behaviour for heavier actinides. We thus take the extreme values, i.e., those of ThO₂, as being the lower limit for AmO₂. The corresponding melting temperatures are 2780 K and 2735 K for T_m/Δ_fH° and T_m/Δ_fG°, respectively.

We thus consider that the melting temperature of a hypothetical americium dioxide is in the 2605–2780 K range. This rule of thumb gives results in line with the previous evaluations while diminishing the width of the interval spanned by the previous two estimates by Casalta and Kato et al. cited above. Despite the large uncertainties in this evaluation, it is clear that the current CRG potential largely overestimates the melting temperature of a hypothetical stoichiometric americium dioxide.

III.B. Fitting procedure

We used the DAKOTA software⁶⁵ to reoptimize the potential parameters in order to lower the melting temperature of americium dioxide while keeping other properties such as the room-temperature lattice parameter and bulk modulus, the thermal expansion and the enthalpy increment unchanged, as they are in excellent agreement with the available experimental data. Based on previous work by Cooper et al. to lower the melting temperature of PuO₂ predicted by the original potential parameters,²⁵ we largely increased the N-body parameters of Am⁴⁺ while adjusting the Am⁴⁺–O terms to ensure that other properties were changed as little as possible. We employed the single-objective genetic algorithm (SOGA) using the JEGA package⁶⁶ implemented in DAKOTA to optimize all parameters simultaneously, running the optimisation with five different initialisation seeds. Properties were calculated as follows: (1) room-temperature lattice parameter, (2) room-temperature bulk modulus, (3) melting temperature, (4) lattice parameter and enthalpy increment at 1000 K. After each step, the result was examined and the set of parameters was discarded if the computed properties were unacceptably far from their reference value. The tolerance for each property was chosen large enough to leave space for the subsequent optimization of the parameters, while rejecting sets having no chance of converging towards a valid solution. Values used for the optimization as well as tolerance criteria are reported in Table 2.

Table 2 Fitting database for the reoptimization of Am⁴⁺ related terms

Property	Reference value	Tolerance
300 K lattice parameter (Å)	5.375^67	0.05
300 K bulk modulus (GPa)	205^68	50
Melting temperature (K)	Range: 3100–3400	Must be in the interval
1000 K lattice parameter (Å)	5.414^67	No check
1000 K enthalpy increment (kJ mol^−1)	55.8^69	No check

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We used several loss functions to guide the optimization: first the absolute relative error on numerical properties and second a 0 or 1 value corresponding to the identification of the solid to liquid phase transition or not, respectively. We applied various weights to the linear combination of these loss functions to find the best set of parameters. A weight value of 5 was chosen for the loss function regarding each room-temperature lattice parameter, bulk modulus and 1000 K lattice parameter. A weight value of 1 was used for the loss function of the enthalpy increment due to the rather high rigidity of the potential formalism with regard to energetic properties. A weight value of 10 was applied to the detection of the melting.

To decide on the “best” interatomic potential parametrization after the optimization, we selected the subset of all the sets of parameters predicting the melting in the estimated temperature range and defined a global objective function (GOF) combining the various loss functions:

with N the number of loss functions, δ^r_i the absolute relative error of the loss function i, and σ(δ^r_i) the standard deviation of δ^r_i over all sets of parameters evaluated during the optimization. This form allows one to add loss functions with various dimensions and to take into account their variability during the optimization. We finally selected the potential parametrization with the lowest GOF.

As for the optimization of the Am³⁺ and U⁵⁺ terms,³⁸ simulations for the optimization of the interatomic potential coefficients were carried out on 2592-atom supercells with a time step of 2 fs. The T_damp and P_damp parameters of the Nosé–Hoover thermostat and barostat were fixed at 0.1 ps and 0.5 ps, respectively. For simulations at room temperature, the cells are equilibrated for 8 ps and the properties are averaged over the last 4 ps. For higher temperature properties, the equilibration time was set at 10 ps, with properties averaged over the last 4 ps. The room-temperature bulk modulus was determined by applying ±4% deformations on the supercell equilibrated at 300 K. The relatively high-cost of a moving interface calculation is incompatible with the iterative scheme of the parameters optimization. The alternative is to equilibrate the supercell at two temperatures corresponding to the lower and upper bounds of our acceptable interval multiplied by an overestimation coefficient calibrated on previous HUM and MI calculations. We found this coefficient to be extremely consistent over the melting temperatures calculated on low-concentration uranium–americium mixed oxides,³⁸ with a value of 1.205 ± 0.005. We thus performed the two 30 ps equilibrations at 3100 K and 3400 K corresponding to corrected values of 2575 K and 2825 K, respectively. The interval was slightly broadened compared to the values determined in Section III.A to account for the uncertainties intrinsic to this method. We used the existence of the farthest peaks of the radial distribution functions or lack thereof to deduce the phase of the supercell. By ensuring that AmO₂ was solid at the lower bound of the range and liquid at its upper bound, we successfully incorporated the melting temperature in the optimization procedure.

III.C. Final set of parameters

The final Am⁴⁺–O pair parameters as well as terms used for the other species are given in Table 3, while Table 4 regroups all the N-body terms with the updated values for Am⁴⁺.

Table 3 Pair interaction terms of the potential, in bold the terms reoptimized in this study

Species α	Species β	A αβ (eV)	ρ αβ (Å)	C αβ (eV Å^6)	D αβ (eV)	γ αβ (Å^−1)	r ^0 αβ (Å)
U^4+	U^4+	18600	0.2747	0.0
U^4+	O^2−	448.779	0.387758	0.0	0.6608	2.05815	2.38051
U^4+	Am^4+	18600	0.2677	0.0
Am^4+	Am^4+	18600	0.2609	0.0
Am ^4+	O ^2−	361.3445	0.41441	0.0	0.85916	2.01777	2.31482
O^2−	O^2−	830.283	0.352856	3.884372
Am^3+	U^4+	18600	0.248879	0.0
Am^3+	U^5+	18600	0.234726	0.0
Am^3+	Am^3+	18600	0.286433	0.0
Am^3+	Am^4+	18600	0.272028	0.0
Am^3+	O^2−	367.4422	0.381697	0.0	0.80836	2.16288	2.325338
U^5+	U^4+	18600	0.297631	0.0
U^5+	U^5+	18600	0.256063	0.0
U^5+	Am^4+	18600	0.286782	0.0
U^5+	O^2	1157.0704	0.357723	0.0	1.85097	1.917152	1.916728

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Table 4 N-Body interaction terms of the potential, in bold the terms reoptimized in this study

Species	G (eV Å^1.5)	η (Å^5)
U^4+	1.806	3450.995
U^5+	1.7921	3448.0359
Am^3+	0.3553	1630.7277
Am ^4+	1.3726	6034.3348
O^2−	0.690	106.856

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III.D. Validation of the updated interaction terms

III.D.1. Validation on AmO_2−x. We first applied the new terms on a HUM simulation, as well as elastic constants calculations, on stoichiometric AmO₂ to ensure that the potential is stable across the whole range of temperatures studied. Fig. 3 shows the comparison between the initial CRG potential, the new terms and the available data for the lattice parameter,^58,67 enthalpy increment⁶⁹ and bulk modulus^64,68,70,71 as a function of temperature. The available 0 K DFT+U calculated values are plotted at T = 300 K in the bulk modulus figure. The 141.5 GPa DFT+U value by Lu et al.⁷² was disregarded here due to its large deviation with other studies, probably due to convergence issues in the DFT+U calculations.⁷⁰ Experimental data regarding the lattice parameter is truncated above 1073 K as the calculated O/M ratio deviates rapidly from 2.00 above this temperature.^17,58,59 We observe that the evolution with temperature of the structural, energetic and elastic properties indicates a physical behaviour of both sets of parameters.

Fig. 3 Comparison of properties yielded by both sets of parameters and available experimental data for the lattice parameter^58,67 (top), the enthalpy increment⁶⁹ (middle) and the bulk modulus^64,68,70,71 (bottom) as a function of temperature.

We see that both the initial and new parametrizations are in excellent agreement with the available experimental data. They begin to behave differently above 2000 K and exhibit different melting temperatures, which can be identified by the discontinuity in the enthalpy increment. As expected, the new terms yield a much lower melting point. To confirm this, we used the moving interface method with fifteen different seeds for the initialization of the velocities and found the new melting point to be (2792 ± 5.7) K. This is slightly higher than the upper bound of the interval determined in Section III.A and in excellent agreement with the estimate by Kato et al.⁴⁰

III.D.2. Validation on (U,Am)O₂. Fig. 4 shows the evolution of the lattice parameter and the enthalpy increment as a function of temperature for U_0.3Am_0.7O₂ predicted by the original CRG and the updated parametrization, as well as the experimental data measured by Epifano et al.^17,19 The small discrepancy observed between the experimental and calculated lattice parameter (0.015 Å at 300 K) is due to the slight mismatch between the value of Epifano et al.⁵⁹ and that of Lebreton et al.,¹⁶ which was included in the fitting database for the Am³⁺ and U⁵⁺ terms, as well as to the small difference in the Am content between the molecular dynamics simulations and the experiments (70% against 67%). Regarding the enthalpy increment, the values measured in ref. 17 showed a rather large deviation from the Neumann–Kopp law, which the authors explained by a probable reduction of the sample. This excess enthalpy at high temperature in the experimental data accounts for part of the difference with the calculated values for a perfectly stoichiometric compound. As expected given the small differences previously observed on AmO₂ for T < 1500 K, there is little to no change between both parametrizations. The evolution of energetic and structural properties with temperature is reproduced well by both sets of parameters.

Fig. 4 Comparison of properties yielded by both sets of parameters and available experimental data for the lattice parameter¹⁹ (top) and the enthalpy increment¹⁷ (bottom) as a function of temperature for U_0.3Am_0.7O₂.

As a final validation, we investigate the difference between both parametrizations on the 0 K enthalpy of mixing, shown in Fig. 5. The enthalpy of mixing of uranium–americium mixed oxides is defined as:

ΔH_mix = H^{U_1−yAm_yO₂} − (1 − y)H^UO₂ − yH^AmO₂ (4)

Fig. 5 Comparison of the 0 K enthalpy of mixing computed using the original CRG potential (red diamonds), the Arima potential⁷³ (yellow squares), the CRG potential augmented with Am³⁺ and U⁵⁺ oxidation states (green hexagons),³⁸ the CRG potential with Am³⁺; U⁵⁺ and reoptimized Am⁴⁺ parameters (blue circles, this work), DFT+U⁴⁵ (purple crosses) and CALPHAD⁷⁴ (black line).

It can be observed that only the interatomic potentials including the Am³⁺ and U⁵⁺ oxidation states yield negative values for the mixing enthalpy – indicative of a stable solid solution – also calculated in DFT+U,⁴⁵ as values calculated with two potentials taking into account only the (+IV) oxidation states yield unphysical positive values.⁷³ Moreover, the values obtained with the new parametrization are in much better agreement with the DFT+U-calculated ones due to the change in the absolute value of AmO₂ enthalpy content. The results are also in good agreement with the calculated values at 300 K with the CALPHAD model.⁷⁴ Since the enthalpy of mixing was not included in the fitting procedure, this is an important result for the validation of the reoptimized parameters.

IV. Results and discussion

We apply our parametrization to the calculation of properties of interest of (U,Am)O_2−x oxides. We present here the results, separating the properties directly obtained as a function of temperature during HUM calculations, their derivative properties, the melting temperature estimates yielded by the MI method and the elastic properties.

IV.A. Direct properties

IV.A.1. Structural properties. The calculated room-temperature lattice parameter is shown in Fig. 6 for U_1−yAm_yO_2−x with 0 ≤ y ≤ 1 and 0 ≤ x ≤ 0.1 and compared to experimental data measured by Lebreton et al.¹⁶ and DFT+U calculations performed by Talla-Noutack et al.,⁴⁵ whose results were rescaled to match the experimental lattice parameter of pure oxides. The results for Am contents up to 50% and with O/M = 2.00 are taken from our previous study.³⁸ The lattice parameter increases when the O/M ratio decreases, which is common in oxides.⁷⁵ The magnitude of this expansion depends on the americium content. Medium concentrations are less impacted, leading to a quasi-linear room-temperature lattice parameter for (U,Am)O_1.90 compounds. U_0.5Am_0.5O_2−x compositions have a peculiar behaviour, the lattice-parameter remaining quasi-constant and even decreasing slightly for some O/M ratios. This is probably due to the larger uncertainty (not shown here for clarity) in the calculation of structural properties observed at lower temperature for this americium content, as discussed in ref. 38.

Fig. 6 Room-temperature lattice parameter of (U,Am)O_2−x compounds, including data of ref. 38, compared to experimental data measured by Lebreton¹⁶ and DFT+U calculations performed by Talla-Noutack.⁴⁵

Fig. 7 shows the evolution of the lattice parameter and density as a function of temperature for stoichiometric uranium–americium mixed oxides. We can clearly see a change in the trend of the evolution of structural properties as a function of americium content occurring at an Am content of 50%: the lattice parameter (resp. the density) remains relatively similar from UO₂ to U_0.5Am_0.5O₂ while decreasing (resp. increasing) rapidly between 50% Am and AmO₂. This is due to the change in the cationic oxidation states taking place at 50% Am, shown in Fig. 1, which impact the interatomic distances.³⁸ At higher temperature, the two properties show a subtle inflection point with a temperature dependence on the americium content. This is associated with the Bredig transition predicted by the interatomic potential in these compounds. This transition is discussed in more detail in Section IV.B relative to the derivative properties.

Fig. 7 Evolution of the lattice parameter (top) and density (bottom) as a function of temperature for stoichiometric uranium–americium mixed oxides, including data of ref. 38.

Then, around 3300–3600 K, the discontinuity in the density increment curves indicates the melting. Although the melting temperature T_m is overestimated by this simulation technique, it can already be seen that the melting temperature is very similar for all Am contents between 50% and 100% and that it is between the melting temperature of U_0.5Am_0.5O₂ and that of UO₂. Melting temperatures are discussed extensively in Section IV.C, which describes the results of the moving interface method.

The effect of the oxygen to metal ratio on the lattice parameter as a function of temperature is shown in Fig. 8 for three Am contents (20%, 50% and 80% Am). The same trend is observed in the variation of the cell parameter with the O/M ratio for the three Am contents: an increase up to approximately 1900 K and a decrease above this temperature. Quantitatively, however, these changes depend significantly on the americium content. Low americium contents are impacted both at low and high temperature; medium americium contents are virtually unaffected by the O/M ratio and high Am-contents are affected only at low temperature. We can also see O/M-dependent changes in the inflexion point associated with the Bredig transition (see Section IV.B).

Fig. 8 Lattice parameter as a function of temperature for U_0.8Am_0.2O_2−x (top), U_0.5Am_0.5O_2−x (middle) and U_0.2Am_0.8O_2−x (bottom) for oxygen to metal ratios from 1.90 to 2.00.

IV.A.2. Energetic properties. The enthalpy increment of stoichiometric (U,Am)O₂ is shown in Fig. 9 for several americium contents. No impact of the Am content is seen up to 2000 K. As with the lattice parameter and density, this property shows a subtle inflection point for which the temperature depends on the americium content, associated with the Bredig transition. The enthalpy of fusion Δ_fusH° can be estimated approximately by fitting two linear functions to the high-temperature solid part and to the liquid part of the enthalpy increment. It is difficult, however, to assess precisely the temperatures at which the melting process begins and ends, in part due to the averaging of properties over different supercells and velocity seeds in which the melting temperature varies slightly. A conservative approach enables us to estimate an enthalpy of fusion in the 38 kJ mol⁻¹ to 52 kJ mol⁻¹ range for all americium contents, without a clear evolution trend with the Am content. There is no experimental value for AmO₂: the only two measurements of enthalpy of fusion of actinide dioxides are for UO₂, with values of 74 kJ mol⁻¹ by Hein and Flagella⁷⁶ and 76 kJ mol⁻¹ by Leibowitz et al.⁷⁷ In addition, the enthalpy of fusion of PuO₂ was estimated to be (64 ± 6) kJ mol⁻¹ by Konings et al.⁶² using enthalpy measurements performed by Ogard⁷⁸ in combination with the entropy of fusion of UO₂.

Fig. 9 Enthalpy increment as a function of temperature for stoichiometric uranium–americium mixed oxides (top, magnification on bottom), including data of ref. 38.

The impact of the oxygen to metal ratio on the enthalpy per atom is similar for all temperatures and compositions: the larger the deviation from O/M = 2.00 in the hypostoichiometric domain, the higher the enthalpy, as shown in Appendix A (Section A.1) for three Am/(Am + U). This is consistent with the enthalpy values calculated using the CALPHAD method in (U,Pu)O_2−x by Fouquet-Métivier et al.⁷⁹

IV.B. Derivative properties

The general thermal expansion coefficient α and specific heat capacity c_p are defined as:From the values of the lattice parameter and enthalpy increment calculated as a function of temperature, we computed these two derivatives using the finite difference method:We included the three neighbouring values (±25 K, 50 K and 75 K) in the derivation to slightly refine the derivatives.

The evolution of α and c_p with respect to temperature is shown in Fig. 10 for stoichiometric (U,Am)O₂ compounds.

Fig. 10 General thermal expansion coefficient (top) and specific heat capacity (bottom) as a function of temperature for (U,Am)O₂ compounds. Values up to 50% Am/(Am + U) are taken from ref. 38.

We consider both properties concurrently due to their nearly identical evolution as a function of temperature, focussing on americium contents above 50%. Their evolution with temperature is qualitatively identical to that of low americium contents mixed oxides:³⁸ first a regular rise up to 1700 K, followed by a peak. The influence of the Am content up to 1700 K is more pronounced for the general thermal expansion coefficient than for the specific heat capacity. Medium americium contents have a generally higher thermal expansion coefficient than the pure oxides in this temperature region. The comparison of the specific heat capacity calculated with the values derived from enthalpy measurements performed by Epifano et al.¹⁷ is not conclusive due to the slight but systematic underestimation of this property by the CRG formalism, as well as to the possible changes in the experimental O/M ratio affecting the measurements.

The peaks observed in the derivative properties are related to the Bredig transition,⁸⁰ observed in many compounds with fluorite structure including UO₂.^81–85 It is also predicted by atomic scale calculations using various interatomic potentials for UO₂,^24,86–88 including machine-learning potentials trained on DFT+U data,⁸⁹ using the CRG potential for PuO₂ and using ab initio molecular dynamics in UO₂ and PuO₂.⁹⁰ The CRG potential also predicts this transition in (U,Pu)O₂,^25,32,35 (U,Th)O₂,²⁶ (U,Am)O₂³⁴ and (U,Gd)O₂.²⁹ The Bredig transition is commonly admitted to be a superionic transition with the temperature T_B of the maximum of the peak empirically considered to be approximately T_B ≈ 0.8 × T_m, but the origin of the oxygen diffusion acceleration is still a matter of debate. Several modelling^91–93 and experimental works^81,83,84 were dedicated to the clarification of the underlying mechanisms with no consensus reached to this day.

In the case of (U,Am)O₂ compounds, the amplitude and temperature of the peak are strongly dependent on the Am content. From UO₂ to U_0.5Am_0.5O₂, the temperature and amplitude of the peak decrease strongly and it becomes difficult to identify the peak for medium americium contents. For Am-rich compounds, the amplitude of the peak increases with the americium content while its temperature is relatively stable due to the low impact of the Am content on the melting temperature (see next section). The change in the trend of the evolution of the Bredig peak amplitude and temperature with americium content is a direct consequence of the change in the cationic oxidation states occurring at 50% Am, as seen in Fig. 1. The Bredig transition in pure americium dioxide is not a feature of the reoptimized Am⁴⁺ potential terms, it was yielded by the original CRG terms with the same amplitude at a higher temperature (2800 K).³⁴ It is noteworthy that both parametrizations yield this behaviour, as it confirms that the only difference between the two sets of parameters is the melting temperature.

We also studied the effect of the O/M ratio on the derivative properties. Fig. 11 presents the general thermal expansion coefficient for all O/M ratios studied for two mixed oxides, as well as for AmO_2−x. The effect of an increasing deviation from O/M = 2.00 on α is twofold. First, for temperatures below 1700 K, it results in a decreasing general thermal expansion coefficient. The intensity of this decrease depends on the americium content. Second, the Bredig peak is shifted at higher temperature, while its amplitude decreases. While all compositions seem to be affected qualitatively in the same manner, the temperature and amplitude of the peak also depends on the americium content, the peak disappearing entirely for AmO_1.90. The fact that the Bredig peak disappears for low O/M ratios, which is also the case in the empirical potential calculations of (U,Pu)O_2−x,⁹⁴ could explain why this transition has not been observed yet in (U,Pu)O₂ and (U,Am)O₂ mixed oxides.

Fig. 11 General thermal expansion coefficient as a function of temperature for U_0.8Am_0.2O_2−x (top), U_0.2Am_0.8O_2−x (middle) and AmO_2−x (bottom) for oxygen to metal ratios from 1.90 to 2.00.

The evolution of the specific heat capacity as a function of temperature is shown in Fig. 12 for the same compositions. The effect of the O/M ratio on c_p is different from what is observed for the thermal expansion α. The simulations show no impact of the O/M ratio on the specific heat capacity values up to the onset of the peak. When O/M decreases, the peak is shifted towards higher temperatures for all Am contents. This is in line with the experiments performed on UO₂ by J. P. Hiernaut et al.,⁸³ which showed that successive shots – associated to a reduction of the sample – yielded higher temperatures for the Bredig transition. The amplitude of the peak, however, evolves differently for the three Am contents. For 20% Am, the amplitude is virtually unchanged, as the difference between the c_p values at the maxima corresponds to the difference between the c_p values at the onset of the transition. For 80% Am, the peak decreases while flattening rapidly into a plateau. Finally, for AmO_2−x, the decrease is more pronounced and the peak transforms into a plateau only at lower O/M values around 1.93.

Fig. 12 Specific heat capacity as a function of temperature for U_0.8Am_0.2O_2−x (top), U_0.2Am_0.8O_2−x (middle) and AmO_2−x (bottom) for oxygen to metal ratios from 1.90 to 2.00.

Based on our calculations as well as on the results of empirical potential calculations conducted on (U,Pu)O_2−x⁹⁴ and (U,Pu,Gd)O_2−x²⁹ using the same potential formalism, the conclusion as follows can be drawn concerning the evolution of the specific heat capacity with composition:

• As seen in AmO_2−x, U_1−yAm_yO_2−x with y > 0.5, (U,Pu)O_2−x and (U,Pu,Gd)O_2−x, the inclusion of oxygen vacancies associated with cations in the (+III) oxidation state results in a significant decrease of the Bredig peak amplitude, while increasing slightly its temperature.

• From the results on (U,Am)O₂ and (U,Pu,Gd)O₂ compounds, the inclusion of (+III) cations associated to (+V) ones while keeping O/M = 2.00 (without O vacancies) results in a decrease of both the amplitude and temperature of the Bredig transition.

• Compounds with a cationic sublattice composed of (+V) and (+III) cations in equal proportions, such as U_0.5Am_0.5O₂, do not seem to exhibit a Bredig transition.

• Finally, as seen in U_1−yAm_yO_2−x with y < 0.5, the inclusion of oxygen vacancies accommodated by the reduction of a (+V) cation to (+IV) results in a shift of the Bredig peak towards higher temperatures while leaving its amplitude unchanged.

From these we infer that, using the CRG formalism:

• The presence of (+III) cations tends to lower the amplitude of the Bredig peak.

• The presence of (+III) and/or (+V) cations tend to lower the Bredig peak temperature.

• Oxygen vacancies tend to increase the Bredig peak temperature.

Since we cannot perform simulations separating the effect of oxygen vacancies from that of changes in the oxidation states due to the charge neutrality of the supercells in our simulations, some of these effects could be coupled and/or could cancel each other out.

IV.C. Melting temperature estimates

Fig. 13 compares the estimates of the melting temperature for each composition yielded by the original Am⁴⁺ parameters and the reoptimized terms. The results for compounds with less than 50% Am are unaffected by Am⁴⁺ parameters since this oxidation state is not present for these compositions. The slope discontinuity exhibited by both parametrizations is once again due to the change in oxidation state at Am/(Am + U) = 0.5 (see Fig. 1). We observe a very large difference between the two sets of parameters on T_m for Am-rich compositions, our parametrization yielding much lower melting temperatures. For AmO₂, the new value is (2792 ± 5.7) K, as opposed to (3346 ± 5.7) K with the original CRG potential. We can also see that the melting temperature of Am-rich compositions yielded by our parameters varies only slightly with Am content, with a possible local maximum reached for compositions around 70% Am for all studied O/M ratio. An important observation is that the melting temperature is close to the linear interpolation between UO₂ and U_0.5,Am_0.5O₂ for Am contents lower than 50% and between U_0.5,Am_0.5O₂ U_0.5 and AmO₂ for higher contents. The confirmation of the melting temperatures for (U,Am)O₂ with high Am contents could thus enable one to obtain a better estimate for the melting temperature of AmO₂.

Fig. 13 Estimates of the melting temperatures of U_1−yAm_yO_2−x using the original CRG Am⁴⁺ parameters (top) and the set of terms obtained in this work (bottom). Available experimental data^20,62 is shown as empty squares for comparison.

The difference in the effect of the O/M ratio between the two parametrizations should also be noted. Using the original Am⁴⁺ parameters, the deviation from O/M = 2.00 had opposing effects on the melting temperature for low and high Am contents, increasing T_m for Am/(Am + U) ≤ 50% while decreasing it for Am-rich compositions. In contrast, the reoptimized potential predicts that a decrease in the O/M ratio increases the melting temperature for all Am contents, which is consistent with the shift of the Bredig transition temperature (see Section IV.B). This is in line with the results of the CALPHAD calculations using the TAF-ID database, which predict the congruent melting to be in the hypostoichiometric region for all Am contents in uranium–americium mixed oxides.⁷⁴ This is also similar to the solid/liquid phase transition in MOX calculated using the CALPHAD method by Fouquet-Métivier et al.⁷⁹ They proposed a stabilising mechanism due to an increase in the entropy that lowers the Gibbs free energy of the system and thus increases T_m for hypostoichiometric domains for all plutonium contents. They show that this entropy increase is due to the presence of plutonium in a mixed Pu⁴⁺/Pu³⁺ oxidation state to accommodate the oxygen vacancies in the hypostoichiometric compositions. Given the similarity in the oxidation states of plutonium and americium in actinide oxides, this explanation could also be valid for the U–Am–O system.

IV.D. Elastic properties

There is no measurement or first-principle calculation of the elastic properties of uranium–americium mixed oxides, these are therefore first estimates. The evolution with temperature of C₁₁, C₁₂, C₄₄ and the Zener ratio for stoichiometric (U,Am)O₂ compounds is shown in Fig. 14. The plots representing several isotherms of the elastic constants for (U,Am)O₂ compounds can be found in Appendix B (Section B.1.1).

Fig. 14 Evolution with temperature of (a) C₁₁ (b) C₁₂ (c) C₄₄ (d) the Zener ratio for (U,Am)O₂ compounds.

Elastic constants tend to decrease with temperature, indicating a softening of the materials. C₁₂ exhibits a non-monotonous evolution with temperature: its value increases between 2300 K and 2800 K for UO₂ and AmO₂ as well as for low and high Am contents, in line with measurements by Hutchings on UO₂,⁸¹ while it decreases steadily for U_0.5Am_0.5O₂ from 300 K to 3000 K. We therefore infer that this increase is a direct consequence of the Bredig transition.

For all Am contents, the Zener ratio is low at room temperature, showing a significant anisotropy of the (U,Am)O₂ compounds. It is higher for medium Am contents. For low and high Am contents, the Zener ratio remains relatively constant up to approximately to 2000 K, i.e., until the onset of the Bredig transition, due to the concomitant decrease of C₁₁, C₁₂ and C₄₄. Above 2000 K, the Bredig transition affects the elastic constants in different ways, leading to a strong increase of the Zener ratio with temperature, which indicates a significant increase in the material isotropy. Medium compositions, in particular U_0.5Am_0.5O₂, exhibit a rather steady evolution of all three independent elastic constants with temperature but with different temperature dependences, leading to a more progressive increase of the Zener ratio.

The variation of the elastic constants and Zener ratio as a function of the O/M ratio is similar for all Am contents. Fig. 15 shows the variation of the elastic constants and of the Zener ratio of U_0.8Am_0.2O_2−x with temperature for all O/M ratio studied as an example. Additional results for two other americium contents (50% and 80% Am) can be found in Appendix B (Section B.1.2).

Fig. 15 Evolution with temperature of (a) C₁₁ (b) C₁₂ (c) C₄₄ (d) the Zener ratio for U_0.8Am_0.2O_2−x compounds with 0 ≤ x ≤ 0.1.

The modifications induced by the O/M ratio on the lattice parameter cause an opposite variation in C₁₁: the increase of the lattice parameter with decreasing oxygen content at low temperature induces a decrease in C₁₁ and the opposite behaviour is observed at high temperature. The amplitude of the increase in C₁₂ at high temperature can be correlated to the intensity of the Bredig peak in the general thermal expansion coefficient (see Fig. 11). The effect of the oxygen content on C₄₄, indicative of the resistance to shear stress, is generally less significant but non negligible in particular for U_0.5Am_0.5O_2−x. Finally, the Zener ratio increases when O/M decreases for temperatures lower than 2000 K, but the effect of the hypostoichiometry becomes negligible at higher temperatures for which stoichiometric materials are already nearly isotropic.

It can be noted that the behaviour of the elastic properties with the O/M ratio observed for Am contents below 50% is similar to that obtained by Porto et al. in (U,Pu)O_2−x using the CRG potential.⁹⁴ They took into account the same oxidation states as in (U,Am)O_2−x for the cationic sublattice: U⁴⁺, U⁵⁺, and Pu³⁺ and Pu⁴⁺ instead of Am³⁺ and Am⁴⁺.

V. Conclusion

We reoptimized the interactions terms involving Am⁴⁺ cations of the CRG potential to lower the calculated melting temperature of AmO₂ and used this parametrization in combination with U⁴⁺ parameters as well as previously optimized Am³⁺ and U⁵⁺ terms to perform a systematic study of the structural, thermodynamic and elastic properties of U_1−yAm_yO_2−x with 0 ≤ y ≤ 1 and 0 ≤ x ≤ 0.1.

Significant differences are observed between Am contents less than and greater than 50%, which can be correlated to the changes in the cationic oxidation states present in these compounds. Of particular interest is the prediction by the potential of a Bredig transition for the majority of the studied compositions, including AmO₂, as well as for most hypostoichiometric compositions. The existence of a Bredig peak, however, is unclear for medium americium contents. In addition, the melting temperature is now more in line with basic thermodynamic data and the effect of the O/M ratio on this phase transition is consistent with CALPHAD predictions using the TAF-ID. Finally, elastic properties of (U,Am)O_2−x compounds were evaluated for the first time.

These results will be useful for the CALPHAD modelling of Am-bearing oxides, as well as for the simulation of (U,Am)O₂ fuel pellet at the mesoscopic and macroscopic scale. Additional experimental characterizations and/or electronic structure calculations on selected compositions, however, would be needed to confirm these results, especially for high americium contents and at high temperature.

Data availability

Data are available in the Zenodo repository of the PATRICIA H2020 European project: https://zenodo.org/communities/patricia/records?q=&l=list&p=1&s=10&sort=newest.

Conflicts of interest

There are no conflicts to declare.

Appendices

Appendix A

A.1. Enthalpy variation with O/M ratio. Fig. 16 shows the variation of the enthalpy in (U,Am)O_2−x as a function of the O/M ratio for three Am contents.

Fig. 16 Enthalpy as a function of temperature for uranium–americium mixed oxides including 20% Am (top), 50% Am (middle) and 100% Am (bottom).

Appendix B

B.1. Elastic properties.
B.1.1. Isotherms. At low temperature, the elastic properties exhibit three types of evolution with the Am content, as shown in Fig. 17: a “V” (or “U”) shape, a “W” shape, or a “Δ” shape. This is consistent with the fact that the cationic concentrations in these oxides are linearly dependent on the americium content for the two sections UO₂ to U_0.5Am_0.5O₂ and U_0.5Am_0.5O₂ to AmO₂ (see Fig. 1).

Fig. 17 Isotherms of (a) C₁₁ (b) C₁₂ (c) C₄₄ (d) the Zener ratio for (U,Am)O₂ compounds.

B.1.2. Effect of the O/M ratio on the elastic properties. Fig. 18 shows the effect of the O/M ratio on the evolution with temperature of C₁₁, C₁₂, C₄₄ and the Zener ratio in (U,Am)O_2−x for 2 Am contents.

Fig. 18 Evolution with temperature of C₁₁, C₁₂, C₄₄ and the Zener ratio for U_0.5Am_0.5O_2−x (left column) and U_0.2Am_0.8O_2−x (right column) for 0 ≤ x ≤ 0.1.

Acknowledgements

The authors would like to thank Bruno Michel and Anna Smith for fruitful discussions. This work was supported by the basic research Program (RTA) of the CEA Energy Division and is part of the PATRICIA project, which has received funding from the Euratom research and training programme 2019–2020 under grant agreement no 945077. It was performed using HPC resources from GENCI at TGCC and CINES under the allocations 2023 – A0150906922 and 2023 – R0151010339, as well as resources from CCRT funded by the CEA/DES simulation programme.

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