Thermodynamic model of the oxidation of Ln-doped UO2

Ln-doped UO2 is often considered as a model system of spent nuclear fuel (SNF) helping to reveal effects of fission and activation products on its chemical stability. Comparing thermodynamics of UO2-UO3 and LnO1.5-UO2-UO3 systems provides a means to understand the phenomenon of an increased resistivity of Ln-doped UO2 to oxidation in air relative to pure UO2. Here a thermodynamic model is developed and is applied to investigate detailed phase changes occurring along the oxidation of Ln-doped fluorite to U3O8. The study proposes that an enhanced resistivity to oxidation of Ln-doped UO2 is likely caused by a thermodynamically driven partitioning of Ln between a fluorite-type phase and a U3O8 polymorph, which at ambient temperatures becomes hindered by slow diffusion.


Non-stoichiometry in the fluorite solid solution
The fluorite phase, U 1-z Ln z O 2+δ , is modelled here as a mixture of the endmembers UO 2 , LnO 1.5 , UO 2.5 , U 1/2 Ln 1/2 O 2 , and U 1/3 Ln 2/3 O 2 (see "Methods").The model appears to be sufficiently robust to describe the dependence of the non-stoichiometry parameter, δ, on T, z, and P O 2 .Figure 1 shows the model fit to the data for the pure UO 2 -UO 3 system.Figure 2 shows the fit to the data for the GdO 1.5 -UO 2 -UO 3 system, where the composition, z, is varied within the range of 0 < z < 0.7.The fit to G O 2 vs. δ, where �G O 2 = RTln(P O 2 /P 0 ) ; P 0 = 101325 Pa , is improved significantly relative to 32 , particularly, at z > 0.5.This is the result of extending the interval of δ to (−1/2 < δ < 1/3) and of including the new endmember U 1/3 Ln 2/3 O 2 .The model parameters are given in Table 1.Importantly, some of the fitted parameters appear to be close to the values estimated from the data on UO 2 , LnO 1.5 , γ-UO 3 and U 3 O 8 with the additivity rule (see Table S1 in Supplementary materials).This is observed, for example, in the case of UO 2.5 , but not in the cases of U 1/2 Ln 1/2 O 2 and U 1/3 Ln 2/3 O 2 .The fitted standard free energies of the latter endmembers (−76.0 and −72.0 kJ/mol, respectively) appear to be significantly lower than the corresponding estimates based on additivity.This implies that these endmembers are significantly stabilized due to certain interactions occurring between UO 2.5 and LnO 1.5 and between UO 3 and LnO 1.5 .A good fit was obtained by defining Margules parameters only for UO 2 -UO 2.5 and UO 2 -GdO 1.5 interactions.All other interactions were set athermal.The same model provided a reasonably good fit to G O 2 vs. δ data for systems with NdO 1.5 33 and LaO 1.5 34,35 .This is illustrated in Figs.S1 and S2 (Supplementary materials).Thus, practically, the model is applicable to systems with Ln = {La, Nd, Gd} with no modification.
www.nature.com/scientificreports/Oxidation of pure UO 2   The equilibrium oxidation of pure UO 2 at 973-1273 K 38,39 occurs through the following sequence of transformations.First, the O/M ratio increases within the mono-phase fluorite, F, reaching a certain limiting value, then, a two-phase F + U 4 O 9 mixture develops.After the fluorite phase in the mixture becomes extinct, a mono-phase U 4 O 9 remains stable within a certain interval of log(P O 2 /P 0 ) , and, finally, a two-phase assemblage with varying   36 .Note, that δ can be written as δ = (x − y)/2, as it is a function of the mole fractions, x and y, of the UO 2.5 and LnO 1.5 endmembers.δ can be also evaluated as δ = O/M − 2. A negative/positive deviation from δ = 0 implies the presence of either oxygen vacancies or oxygen interstitials.Note also, that δ characterizes non-stoichiometry only of a monophase fluorite, while O/M − 2 is also applicable to a poly-phase system.G O 2 is a convenient function to visualize effects of the temperature and/or the partial pressure of O 2 on the chemical potential of O 2 , while the dimensionless quantity log(P O 2 /P 0 ) is more convenient when the temperature and the pressure effects need to be distinguished.Both quantities are used throughout the text.
The symbols F1 and F2 distinguish different states within the same fluorite phase.The transformation U 4 O 9 → F2 is driven by an increase in the chemical potential of oxygen caused by the decrease in temperature.The stabilization of F2 fluorite with δ ~ 0.33 and the disappearance of U 4 O 9 (δ = 0.25) is the consequence of a larger δ value achievable in fluorite.The model allows to qualitatively explain the experimentally observed appearance of fluorite-like tetragonal phases with δ ~ 0.33 at T < 823 31 .Although the tetragonal β-U 3 O 7 phase differs structurally from fluorite 40 , its thermodynamic properties are likely similar to F2 fluorite with δ ~ 0.33.Indeed, the data of Grenthe et al. 41 indicate that U 3 O 7 is stable just by 0.7 ± 2.0 kJ/mol relative to the mixture of UO 2.25 and UO 2.67 .Similarly, in our simulations, F2 fluorite (δ ~ 0.33) is marginally stable relative to the same phases at temperatures below 873 K.At ~ 1373 K U 4 O 9 becomes unstable relative to the F1 phase (with δ ~ 0.25).Thus, above 1373 K the phase sequence simplifies to F1 → F1 + U 3 O 8 .

Oxidation of Ln-doped fluorite
Figure 4 shows the fit to the data of Stadlbauer et al. 29 for UO 2 doped with 4.8 mol % of LaO 1.5 .The fit required defining the standard Gibbs free energies of the endmembers {Ln 3 1/4 U 5 1/4 }U 5 1/2 O 9/4 and {Ln 3 2/9 U 6 7/9 }O 8/3 for the γ-and α'-phases, respectively.Stadlbauer et al. 29 concluded that the phase relations in the doped system remain essentially the same as in the pure system.Thus, one would expect to see the phase sequence F1 However, the predicted phase relations (Fig. 4) are more complicated at high O/M ratios.At O/M ~ 2.60 fluorite reappears, and the sequence is modified as: Table 1.Adopted thermodynamic parameters for the solid solution phases.Cations taken in curly brackets contribute to the configurational entropy of an endmember.This contribution is not included in the standard entropy values of the endmembers given in the Table but is counted within the general equation describing the entropy of mixing of a relevant phase.Upper symbols in structural formulas denote charges of cations.The ratio of O/M ~ 2.0 within the Ln-rich F3 phase -the consequence of an exceptionally strong stability of the U 1/2 Ln 1/2 O 2 endmember -has an important impact on phase relations at high P O 2 .When log(P O 2 /P 0 ) approaches zero, the partitioning of Ln into the M 3 O 8 phase becomes favourable (Fig. 5).This partitioning is driven by the tendency of the system to achieve the largest total O/M ratio of ~ 2.67, which would be impossible

Lattice parameter variation as a function of oxygen partial pressure, temperature, and Ln-concentration
The lattice parameter, a, of fluorite depends on z and on δ.This dependence is modelled here via an ion-packing model which is linked to the thermodynamic model.Thus, within this inter val the phase sequence with the increase in z is as follows: An interesting feature is the change in the slope in the a vs. z relationship occurring in oxidized samples at z ~ 0.67.In strongly oxidized samples (i.e., at log P O 2 /P 0 ∼ 0, T = 1123 K) , the change of slope correlates with a rapid ingrowth in the fraction of the LnO 1.5 component, i.e., with the fraction of vacancies.Thus, below and above the composition of z = 0.67 the fluorite phase is markedly different; within the interval of 0.5 < z < 0.67 fluorite closely maintains the stoichiometric relationship (O/M = 2) -a feature, which reflects the stability of the endmembers U 1/2 Ln 1/2 O 2 and U 1/3 Ln 2/3 O 2 , while at z > 0.67 it is hypo-stoichiometric.A similar change in the slope of the a vs. z relationship is also seen in the systems with La, Gd, Eu, and Y 25,27,42,43 .The dependence of the lattice parameter in the system of LaO 1.5 -UO 2 -UO 3 is shown in Figure S5.Indeed, graphs analogous to Fig. 7 or Figure S5 can be easily constructed with the presently developed model for any LnO 1.5 -UO 2 -UO 3 system for which the ionic radii of Ln in 6-, 7-, and 8-fold coordination are known.Table S2 lists the relevant radii for systems with La, Nd, Gd and Y.The transformation at z ~ 0.67 suggests that in any oxidized LnO 1.5 -UO 2 -UO 3 system, where Ln is in +3 state, the solid composition at z ~ 0.67 is given by the formula U 6,8 1/3 Ln 3,8 2/3 O 2 .This observation allows determining the ionic radius of an eightfold coordinated U +6 from a = 4 √ 3 (2/3)R 3,8 Ln + (1/3)R 6,8 U + R O (see "Methods").The value of R 6,8 U = 0.755 Å fitted to this relationship appears to be considerably smaller than the Shannon's value of 0.86 44 ..Dashed lines (red online) are the 1373 and 1673 K isotherms computed at log P O 2 /P 0 = 0 .These isotherms correspond to the synthesis conditions in the study of Keller & Boroujerdi 26 .The other experimental data are from Lee et al. 46 , Fukushima et al. 47 , Ohmichi et al. 48and Une & Oguma 33 .

Discussion
The remarkable thermodynamic stability of compounds with the composition of U 0.5 Ln 0.5 O 2 has been previously reported based on solution calorimetry data 23,24 .The formation energies of U 1-z Ln z O 2 (Ln = Y, Nd, La) relative to stable oxides (UO 2 , UO 3 and LnO 1.5 ) fall onto a linear trend ΔH f,ox = -(103.8± 4.3)z kJ/mol 24 .Using the values of �G 0 U 1/2 Ln 1/2 O 2 = -76.0kJ/mol, G 0 LnO 1.5 = 3.3 kJ/mol (Table 1) and G 0 UO 3 = -113.9kJ/mol (Table S1 in Sup- plementary materials) we obtain for Ln = {La, Nd}: ΔG ~ ΔH = (-76-(0/4-113.9/4+ 3.3/2)) = -49.2kJ/mol, which agrees well with the solution calorimetry trend at z = 0.5, as well as with ab initio calculations 49 .Considering the large endothermic formation energy of U 1/2 Ln 1/2 O 2 and U 1/3 Ln 2/3 O 2 endmembers, the tendency of samples with 0.5 < z < 0.67 to preserve stoichiometry in oxidizing conditions is well understood.The O/M = 2 appears to represent the maximum oxidation state of fluorite at z = 0.5.A further oxidation of the U 0.5 Ln 0.5 O 2 compound to hyper-stoichiometry would require the oxidation of U +5 to U +6 , which would be associated with a significant lattice contraction.Such an oxidation/contraction is not observed experimentally, suggesting that U +6 might be too small to be easily accommodated into the fluorite lattice.On the other hand, when an excess of LnO 1.5 over the U 0.5 Ln 0.5 O 2 composition is added to fluorite, the advantage of a further annihilation of vacancies makes the U +5 → U +6 oxidation favourable; the system gains an equivalent amount of oxygen that is immediately consumed in the endothermic vacancy annihilation process.Consequently, the stoichiometric states extend to z = 0.67.This suggests that U +6 can be stable in fluorite only at an excess of LnO 1.5 , i.e., when z > 0.5.A rapid increase in the lattice parameter at z > 0.67 can be attributed to the fact that at z > 0.67 oxygen vacancies cannot be avoided.The significantly larger effective size of the oxygen vacancy (see "Methods") relative to the ionic radius of oxygen explains the positive slope.
The tendency of Ln-doped UO 2 to keep O/M ~ 2 is also seen in Fig. 7 as the clustering in a vs. z data along the linear trend extending from stoichiometric UO 2 towards fully oxidized samples with z ~ 0.5.In Fig. 7 this stoichiometric trend is outlined as an attractor for 1123 K isochores computed at different partial pressures of oxygen.The convergence of isochores implies the stability of stoichiometric states within a wide interval of log(P O 2 /P 0 ) . Figure S6 illustrates the evolution in the endmember fractions at 1273 K along the transition from reduced to oxidized samples.This evolution clearly shows the growing importance of the U 1/2 Ln 1/2 O 2 and U 1/3 Ln 2/3 O 2 components both with an increase in log P O 2 /P 0 and with an increase in the mole fraction of LnO 1.5 .
Consistently with the synthesis studies performed in oxygen or in air at 1173 < T < 1823 K 25,26 , the present model predicts a co-existence of U 3 O 8 with a fluorite phase, where the latter is significantly enriched in LnO 1.5 (Fig. 7).This fluorite phase is predicted to be closely stoichiometric.The M 4 O 9 phase is stable in association with M 3 O 8 only within a narrow interval of log P O 2 /P 0 .At highly oxidizing conditions M 4 O 9 is destabilized relative to the fluorite (F3) phase.The reason for the destabilization is the partitioning of Ln into M 4 O 9 coupled with the limited ability this phase to incorporate Ln +3 .Due to the same reason the M 4 O 9 phase becomes unstable relative to fluorite even at a very small total LnO 1.5 fraction when the total O/M ratio is large (Figs. 5 and 6).Our study predicts, however, that at an excessively high chemical potential of O 2 (e.g., in air at temperatures below 973 K) the association of F + M 3 O 8 becomes unstable relative to a mono-phase M 3 O 8 system (Figs. 5 and  6).The reason for this instability is the tendency of the Ln-rich fluorite to be stoichiometric.Thus, to achieve a total O/M ~ 2.67, the system must get rid of fluorite completely.This prediction is consistent with the results of the recent study of Potts et al. 50.
The comparison of Figs. 3 and 4 shows that in the doped case a higher oxygen pressure is required to oxidize the system to the same O/M ratio, i.e., the doped system is more stable against oxidation.However, within the range of 2.25 < O/M < 2.5, where the main retardation effect is thought to take place, the difference in log(P O 2 /P 0 ) is less than one unit.Thus, it is unlikely that the oxidation resistance is a pure thermodynamic effect.Kinetic factors should be given a closer attention.The model predicts that at ~ 873 K in the doped system the equilibrium transformation goes along the sequence F1 which includes several biphasic states.Figure 5 shows that these states involve partitioning of Ln between the phases, which is particularly strong in the cases of M 4 O 9 + M 3 O 8 and F3 + M 3 O 8 co-existence.This partitioning implies that nucleation and growth of a nearly pure U 3 O 8 from an Ln-bearing M 4 O 9 requires redistributing LnO 1.5 back into the M 4 O 9 phase.At T < 873 K the sequence of phases is modified to F1 The predicted stabilization of F2 in the doped system is important because this phase can possibly emulate the appearance of γ-M 4 O 9 described by Thomas et al. 14 .It is conceivable that at a low temperature the redistribution of Ln between the phases would be controlled by the speed of solidstate Ln-diffusion.Likely, a crystallization of a small amount of U 3 O 8 would be concurrent with the formation of a thin Ln-enriched layer in M 4 O 9 or in γ-M 4 O 9 , which would have protective properties; a continuing growth of U 3 O 8 would require a further enrichment of this layer making it less compositionally suitable for re-crystallizing into a Ln-poor U 3 O 8 .Even at a low doping level such a mechanism would eventually cause the formation of a layer sufficiently enriched in Ln, from which the growth of Ln-poor U 3 O 8 would be practically impossible.In this respect a pure system differs remarkably from any doped system.The oxidation of pure UO 2 requires oxygen diffusion only, which is known to be several orders of a magnitude faster than U self-diffusion in UO 2 51 .On the other hand, a recent computational study 52 showed that diffusion coefficients of U +4 , La +3 and Y +3 differ only within an order of a magnitude, meaning that the Ln diffusion in UO 2 is many orders of magnitude slower than O-diffusion.This proposition could explain a surprisingly large effect of small loads of trivalent dopants on rates of UO 2 oxidation in air.The idea is that the resistance occurs not because the doping hinders the ability of fluorite and/or M 4 O 9 to sorb oxygen, but because a dopant that forms a stable solid solution within fluorite or M 4 O 9 , cannot be easily transferred into a relatively Ln-poor more oxidised phase, i.e., U 3 O 8 , and must diffuse back into the parent cubic phase.
The proposed retardation mechanism is linked to the thermodynamic tendency of the cubic phases to be enriched in Ln relatively to orthorhombic or hexagonal U 3 O 8 .As discussed above, at a very high oxygen potential www.nature.com/scientificreports/this partitioning scheme becomes less polarized, i.e., the concentration of Ln increases in the M 3 O 8 phase, while the fraction of Ln-rich fluorite decreases in the system (Figs. 5 and 6).As oxidation experiments are typically performed at low temperatures in air where the Ln-rich M 3 O 8 becomes stable, the proposed retardation mechanism involving Ln diffusion in M 4 O 9 could be questioned.However, considering that the experiments are carried out with an excess of a preliminary reduced material, the effective partial pressure of oxygen at the reaction front should be significantly lower than the one in air.Likely, the oxygen pressure is buffered by the M 4 O 9 /M 3 O 8 co-existence.The retardation mechanism is discussed here mostly in the context of MO 2 oxidation in air.An adequate description of oxidative dissolution would require the modelling of an aqueous phase containing dissolved forms of U +6 .We could speculate, however, that an oxidation in an aqueous system would similarly include a dopant partitioning between an aqueous phase and an M 4 O 9 -like phase.A recent oxidative dissolution study of an unirradiated homogeneous mixed oxide fuel (MOX) sample with the composition of (Pu 0.27 U 0.73 )O 2 53 showed that the dissolution is incongruent, i.e., the ratio of the total dissolved Pu to the total dissolved U is about two orders of a magnitude lower than the Pu/U ratio in the solid, while the surface of the sample is enriched in Pu up to the composition of (Pu 0.39 U 0.61 )O 2+δ .The formation of a Th-enriched protective layer has been also inferred in an electrochemical study of oxidation in the (U 1-z Th z )O 2 system 54 .Thus, the dopant partitioning (and the diffusion of a dopant within an M 4 O 9 -like phase), as a factor, cannot be excluded in dissolution experiments too.Clearly, a further understanding of the retardation effect would benefit from detailed spectroscopic and microanalytical studies of dopant partitioning in phases that crystallize at the oxidation front.

Thermodynamic approach
The modelling of LnO 1.5 -UO 2 -UO 3 systems is typically done with the CALPHAD methodology where solid solutions are described with the Compound Energy Formalism (CEF) [55][56][57] .In CEF endmembers are generated considering all possible occupations of available sublattices by admissible species 58 .In the case of the fluorite model with four cations in the M sublattice and two anionic sublattices this procedure gives 16 endmembers, many of which are not of neutral charge.In CEF the Gibbs free energies of charged endmembers are carefully constrained such that in all equations they appear only in charge-neutral combinations 59 .Thermodynamic functions of some individual endmembers may be chosen arbitrarily and their importance in determining real solid solution properties is difficult to visualise.The present model is more transparent.We employ only neutral endmembers and formulate site occupancies as linear functions of the endmember fractions.This means, for example, that the fractions of vacancies and O −2 interstitials in MO 2+δ fluorite are linked to the fractions of the neutral endmembers LnO 1.5 and UO 2.5 , respectively.We also allow for the presence of mixed endmembers, such as Ln 1/2 U 1/2 O 2 and Ln 2/3 U 1/3 O 2 .Such endmembers cannot be easily included in CEF.Further, we introduce short- range order (SRO) constraints; we reduce the randomness of cation and anion distribution within sublattices by limiting the space available for mixing of species to a subset of available sites.For example, when a vacancy is introduced into a fluorite MO 2 solid solution, it is not allowed to substitute for any lattice oxygen.Rather, the oxygen-sublattice splits into two imaginary sublattices.One of them accommodates vacancies, while the other remains fully occupied by O −2 anions.The fraction of sites over which the O/V mixing is allowed is considered as a model parameter.Decreasing this fraction emulates entropy reduction due to the vacancy/vacancy avoidance.Finally, we allow for a complete exclusion of certain species from mixing with other species.Particularly, in the fluorite model the fraction of U +5 cations that is needed to balance the fraction of O −2 interstitials is excluded from mixing with other cations.Such an exclusion emulates local cation-anion association, causing an additional entropy reduction.The main advantage is that the number of parameters needed to be defined is decreased significantly compared to CEF.
These non-standard model features prevent us from using available Gibbs free energy minimization software.Thus, the calculations are performed with our own code.Several simplifications are adopted.The thermodynamic properties of all endmembers in all phases are defined relative to a mechanical mixture of a stoichiometric UO 2 fluorite and a hypothetical ordered LnO 1.5 with the pyrochlore, Ln 2 Ln 2 O 6 VV, structure.The Gibbs free energies of these two endmembers are set to zero at all temperatures.The Gibbs free energy of any other endmember i is expressed as a function of the temperature and the partial pressure of oxygen via the equation G i is set equal to the Gibbs free energy change in a reaction by which an endmember is obtained from an equiva- lent mixture of UO 2 , LnO 1.5 (pyrochlore) and O 2 gas.Here G 0 i , S 0 i and Cp 0 i are fitting parameters and n i is the number of moles of O 2 gas consumed or added when the endmember i is built from a mixture of UO 2 and LnO 1.5 .For example, the endmember Ln 1/2 U 1/2 O 2 of the fluorite solid solution is formed via the reaction Thus, for i = Ln 1/2 U 1/2 O 2 n i = 1 8 .Similarly, the endmember Ln 1/4 U 3/4 O 9/4 of the M 4 O 9 -type solid solution is obtained via the reaction (1) (2) .The chemical potential of oxygen at a given temperature and a given partial pressure of O 2 is computed via the equation where S 0 O 2 = 205.1373J/K/mol and Cp 0 O 2 = 29.355J/K/mol and where P 0 is the standard pressure of 101325 Pa, T 0 = 298.15K 60.Practically, the Gibbs free energy of an endmember and of a phase is made dependent not only on the temperature, but also on the partial pressure of oxygen.With this simplification modelling of the gas phase is not needed as its thermodynamic effect is included in the definition of the free energies of the solids.
The Gibbs free energy of a solid solution phase is described with a model that combines features of molecular mixing and sublattice models.The reference Gibbs free energy of a phase is modelled with the equation where X i is the endmember fraction.The endmembers as chemical components are thought to be split into molecular cation and anion entities, which are allowed to mix separately within sublattices.In this respect {U 4 }, {Ln 3 1/2 U 5 1/2 }, {Ln 3 2/3 U 6 1/3 }, {U 5 } and {Ln 3 } are legitimate molecular cation entities of the fluorite phase, which can substitute each other within the cationic sublattice.The enthalpy of mixing within a sublattice is described with the regular mixing model where x i is the fraction of molecular entity i within a sublattice.In a case of a binary solid solution, such as M 4 O 9 or M 3 O 8 , the mixing is assumed to occur only within the cationic sublattice, and the fractions of molecular species are set equal to the endmember fractions.The x i = X i relation is also obtained in the case when enthalpic interactions are set to zero in the anionic sublattice.Such a model is adopted for fluorite.The configurational entropy is built from contributions from different sublattices as in CEF.But, when computing the entropy, the molecular entities, such as {Ln 3 2/3 U 6 1/3 }, are split into their elemental constituents.The free energy of a phase is described as follows where the last term combines entropic contributions from all sublattices.The free energy minimization is performed via a simple grid approach, where ich parameter is varied with a small increment over the whole parameter space.Special cases are discussed below.

Fluorite solid solution
The model of Ln-doped fluorite is improved relative to the previous study 32 in two important aspects.First, an additional stoichiometric endmember, U 1/3 Ln 2/3 O 2 is introduced.This endmember, together with U 1/2 Ln 1/2 O 2 , accounts for the tendency of vacancies and interstitials to annihilate, favouring the stoichiometric composition.While the U 1/2 Ln 1/2 O 2 endmember allows maintaining the stoichiometric relationship up to the limit of z = 0.5, the U 1/3 Ln 2/3 O 2 endmember, due to the presence of U +6 , allows extending the stoichiometric relation (O/M = 2) to z = 2/3.Second, the hypo-and hyper-stoichiometric limits are extended to δ = − 0.5 and δ = 0.33, respectively.The upper limit is smaller than the theoretically possible value of δ = 0.5 (X UO2.5 = 1) because of an additional SRO constraint, which is discussed below.UO 3 is excluded from the list of independent endmembers.This is justified by the observation that the O/M ratio in fluorite never exceeds 2.5.
The parameter space is built via a stepwise admixing of new components/endmembers to UO 2 fluorite as shown in Fig S7 (Supplementary materials).First, the LnO 1.5 component is added.A completely reduced solid solution is thus built of z moles of LnO 1.5 and 1 -z moles of UO 2 giving the general formula of U 1-z Ln z O 2−0.5z .Then, one mole of LnO 1.5 and one mole of UO 2 are allowed to react with 1/8 mol of O 2 producing two moles of the endmember U 1/2 Ln 1/2 O 2 .If the reaction progress is denoted r, then the fraction of this endmember per mole of M cations is 2r.Then, two moles of LnO 1.5 are allowed to react with one mole of UO 2 and with 1/6 mol of O 2 producing three moles of the endmember U 1/3 Ln 2/3 O 2 .If the reaction progress is denoted d, then the fraction of this endmember is 3d.The remaining fraction, y, of LnO 1.5 is then y = z -r -2d, while the remaining fraction, q, of UO 2 is q = 1 -z -r -d.Further, this remaining fraction q can oxidize to UO 2.5 along with the reaction UO 2 + 1/4 O 2 = UO 2.5 .If the reaction progress is denoted x, the fraction of UO 2.5 is x, and the rest fraction of UO 2 is 1 -z -r -d -x.The non-stoichiometry parameter, δ, is a simple function of the fractions of the endmembers UO 2.5 and LnO 1.5 , i.e., δ = (x − y)/2.
The structural formula becomes x ]O 2+0.5(x−y) , where the square brackets embrace species contributing to endmember fractions.With the notation "UO 2 " = 1, "LnO 1.5 " = 2, "UO 2.5 " = 3, " U 1/2 Ln 1/2 O 2 " = 4, " U 1/3 Ln 2/3 O 2 " = 5, the endmember fractions are given: 2r , and X 5 = 3d .These fractions appear in Eqns. 5 and 6.The variables r, d and (3) In this compound the cation coordination number is 6, and ¼ of oxygen sites is vacant.These vacant sites form an (empty) BCC sublattice within the oxygen lattice of a hypothetical M 4 (O,V) 8 fluorite.Filling in all these sites gives the stoichiometry of MO 2 , where the M cation is eightfold coordinated.A partial filling produces 6-, 7-and eightfold coordinated cations.Thus, the UO 2 fluorite and the LnO 1.5 pyrochlore are logical endmember choices for a model in which the M cations adopt exclusively the coordination numbers 6, 7 and 8.A model with the latter constraint is more reasonable from energy grounds than, for example, a model of perfect randomness, where the M cation can adopt all coordination numbers between 0 and 8. Indeed, atomistic simulation studies have shown that vacancies in fluorite-type compounds due to the Coulombic repulsion tend avoiding each other at short near-neighbour distances 61,62 , making cation coordination numbers smaller than 6 much less probable compared to the random case.To emulate this repulsion and to exclude cation coordination numbers smaller than 6, the vacancy/oxygen mixing is restricted to a BCC sublattice consisting of a quarter of available oxygen lattice sites.As the fraction of vacancies over the whole oxygen lattice is 0.25y, within the sublattice it is y (i.e., four times larger).The entropy (per one mole of M cations) is given by the equation Similarly, the distribution of oxygen interstitials within vacant interstitial sites is assumed to be non-random.The O i /V i mixing is restricted to a sublattice consisting of 1/3 of the available interstitial sites.This is motivated by the observation that UO 2.33 (U 3 O 7 ) represents a fluorite-related compound with the largest O/M ratio.As the concentration of interstitials over the whole interstitial lattice is 0.5x, it is three times larger within the sublattice.Thus, the entropy contribution due to the interstitials (per one mole of M cations) becomes Equation 9 limits the admissible fraction of the UO 2.5 component to x = 2/3.
The cations are assumed to be randomly mixed within the M site, however, the fraction, x, of U +5 cations is excluded from mixing with other cations.These are the U +5 cations that balance the excess negative charge of O −2 interstitials.We assume that these U +5 cations are tightly associated to the interstitials and thus do not contribute to the configurational entropy.The other cations mix randomly with each other over the fraction 1 − x of M sites.The fractions of Ln +3 , U +4 , U +5 , and U +6 cations that are allowed to be mixed with each other are, thus, given by t

The entropy equation is
The entropy model could be compared to CEF.In CEF the structural formula of fluorite would be written (M) 1 (O) 3/2 (O, V) 1/2 (O/V) 1/3 .The difference is that in the present model a certain fraction of U +5 is excluded from mixing with other cations affecting the configurational entropy.The enthalpy of mixing is modelled differently from CEF.We assume that the excess effects within sublattices are due to interactions between molecular species which could be represented by combinations of ions with different charges.In the M sublattice these species are {U 4 }, {Ln 3 }, {Ln 3 1/2 U 5 1/2 }, {Ln 3 2/3 U 6 1/3 }, and {U 5 }.In the anionic sublattices the species are O −2 and V.The mixing within the anionic sublattices is assumed athermal.Because of the latter assumption the model of mixing in fluorite is simply mapped onto the regular model of mixing of the five endmembers.

M 4 O 9 solid solution
High-temperature galvanic-cell experiments on UO 2 -LaO 1.5 29 and UO 2 -LuO 1.5 63 systems showed that up to ~ 15 mol % of LnO 1.5 could be incorporated into the ordered U 4 O 9 phase at 1273 K.At a higher content of LnO 1.5 superlattice reflections disappeared 29 .On the other hand, according to the data of Stadlbauer et al. 29 , the lattice parameter of U 1-z La z O 2.23 samples increased linearly within the range of 0 < z < 0.3 showing no break at the order/disorder transition.Based on this observation, we postulate a solubility mechanism of the type 2U +4 = Ln +3 + U +5 in M 4 O 9 assuming the endmembers U 4 1/2 U 5 1/2 O 9/4 and {Ln 3 1/4 U 5 1/4 }U 5 1/2 O 9/4 and a theoretical solubility limit of 25 mol % of LnO 1.5 .In this model we assume the O/V arrangement within the interstitial site to be fully ordered.The configurational entropy is thus due to M cations only.As in the fluorite model, the U +5 cations that are needed for balancing O −2 interstitials are excluded from mixing with Ln +3 .This exclusion implies that the mixing of Ln +3 with U cations occurs over half of the M sites.If the Ln/M fraction in the solid solution is z, the fraction of {Ln 3 1/4 U 5 1/4 }U 5 1/2 O 9/4 endmember is 4z.The structural formula is written {U 4 (1−4z)/2 Ln 3 z U 5 z }U 5 1/2 O 9/4 , where the curly brackets unite cations that are allowed to mix with each other.The configurational entropy (per one mole of M cations) is The enthalpy of mixing is modelled using a single Margules parameter for the interaction between the endmembers U 4 1/2 U 5 1/2 O 9/4 and {Ln 3 1/4 U 5 1/4 }U 5 1/2 O 9/4 .The standard properties of the endmembers are defined according to the Eq. 1, relative to UO 2 fluorite and a hypothetical LnO 1.5 pyrochlore.The U 4 O 9 phase exists in α-, β-and γ-forms.The structural transitions occur at ∼323 K (α ↔ β) and at ∼873 K (β ↔ γ) 64 .The present

Ion-packing model of fluorite-type phases
The lattice parameter of doped UO 2 fluorite varies as a function of composition, non-stoichiometry, and the type of Ln cation [46][47][48] .These variations could be adequately modelled with the aid of an ion-packing model 32,48,69 , which uses the geometrical relationship between the lattice parameter, a, and the sum of the average radii of cations, R C , and anions, R A

Figure 2 .
Figure 2. Model fit to the experimental data on �G O 2 = RTln(P O 2 /P 0 ) vs. δ = (x − y)/2 for the system GdO 1.5 -UO 2 -UO 3 .The experimental data are from Lindemer & Sutton36 .Note, that δ can be written as δ = (x − y)/2, as it is a function of the mole fractions, x and y, of the UO 2.5 and LnO 1.5 endmembers.δ can be also evaluated as δ = O/M − 2. A negative/positive deviation from δ = 0 implies the presence of either oxygen vacancies or oxygen interstitials.Note also, that δ characterizes non-stoichiometry only of a monophase fluorite, while O/M − 2 is also applicable to a poly-phase system.G O 2 is a convenient function to visualize effects of the temperature and/or the partial pressure of O 2 on the chemical potential of O 2 , while the dimensionless quantity log(P O 2 /P 0 ) is more convenient when the temperature and the pressure effects need to be distinguished.Both quantities are used throughout the text.

O 9 + M 3 O 8 →
F3 + M 3 O 8 → M 3 O 8 .Importantly, the F3 phase appearing at O/M = 2.60 differs significantly from the F1 phase occurring within the range of 0 < O/M < 2.25.This F3 phase is very Ln-rich, while its O/M ratio is close to 2.0.Notably, the F2 phase with δ ~ 0.33 is also stable in the doped system at T < 873.The most significant difference relatively to the pure system is the rise of the isotherms within the biphasic M 4 O 9 + M 3 O 8 , F2 + M 3 O 8 and F3 + M 3 O 8 regions to higher oxygen partial pressures.Figures5 and 6investigate the corresponding changes in detail.The positive slope of the isotherms in Fig.4at O/M > 2.25 correlates with the increase in the Ln-content in the M 4 O 9 phase (Fig.5).As the O/M ratio increases, Ln accumulates in M 4 O 9 , while M 3 O 8 takes a negligeable part of the total LnO 2 .The strong partitioning of Ln into M 4 O 9 can be attributed to the stability of the {Ln3 1/4 U 5 1/4 }U 5 1/2 O 9/4 endmember.The assessed standard Gibbs free energy of the {Ln 3

Figure 3 .
Figure 3. Model fit to the experimental data on log(P O 2 /P 0 ) vs. O/M − 2 for pure UO 2 .The experimental data are from Saito 38 .Solid lines correspond to the equilibrium with the hexagonal polymorph of U 3 O 8 .The dashed line corresponds to the orthorhombic α-U 3 O 8 .

Figure 4 .
Figure 4. Model fit to the experimental data on log(P O 2 /P 0 ) vs. O/M − 2 for UO 2 doped with 4.8 mol % of LaO 1.5 .The experimental data are from Stadlbauer et al. 29 .Solid lines correspond to the equilibrium with the hexagonal polymorph of M 3 O 8 .The dashed line corresponds to the orthorhombic α-M 3 O 8 .
Fig. 6).This change occurs because a high total O/M ratio (e.g., O/M ~ 2.6) could only be achieved when the fraction of M 3 O 8 is large.Because M 3 O 8 is nearly free of Ln in the association with M 4 O 9 , the concentration of LnO 1.5 within the minor M 4 O 9 phase increases with O/M and quite rapidly reaches the theoretical limit of z = 0.25.After the M 4 O 9 and F2 phases vanish, fluorite remains as the F3 phase.The content of Ln in the F3 phase rapidly increases to ~ 45 mol %.This implies that the equilibrium oxidation of the doped sample containing 4.8 mol % of LaO 1.5 requires a nine-fold enrichment of the fluorite phase in LnO 1.5 .

Figure 5 .
Figure 5. Predicted evolution of composition of phases in a sample containing 4.8 mol % of LaO 1.5 in the process of equilibrium oxidation at 673 and 873 K. Filled and empty symbols correspond to 873 K and 673 K isotherms, respectively.

Figure 6 .
Figure 6.Fractions of phases in a sample containing 4.8 mol % of LaO 1.5 in the process of equilibrium oxidation at 673 K. Symbols are the same as in Fig. 5.

Figure 7
plots the equilibrium values of a in the system of NdO 1.5 -UO 2 -UO 3 computed at 1123 K and at log P O 2 /P 0 varying in the range [-30, -2].Additionally, the iso- therms of 1373 K and 1673 K are plotted at P O 2 = P 0 .Hyper-, hypo-and strictly stoichiometric states can be dis- tinguished.Flat regions in the predicted variation of a vs. z correspond to two-phase assemblages, where fluorite co-exists with M 3 O 8 or A-Nd 2 O 3 phases.The two-phase co-existence of fluorite and M 4 O 9 is represented as a rectangle.The diagram predicts that M 4 O 9 phase is stable at 1123 K within the interval of −8 < log P O 2 /P 0 < −4 .

Figure 7 .
Figure 7. Variation of the lattice parameter in UO 2 -NdO 1.5 solid solutions predicted from the thermodynamic model.Solid lines are the 1123 K isotherms computed at different values of log(P O 2 /P 0 ) such that the pressure variation approximately covers the range of redox conditions in the data of Wadier45 .Dashed lines (red online) are the 1373 and 1673 K isotherms computed at log P O 2 /P 0 = 0 .These isotherms correspond to the synthesis conditions in the study of Keller & Boroujerdi26 .The other experimental data are from Lee et al.46 , Fukushima et al.47 , Ohmichi et al.48 and Une & Oguma33 . https://doi.org/10.1038/s41598-023-42616-x :F = −(R/2)(yln y + 1 − y ln(1 − y)).