Introduction

Uranium dioxide, UO2, is the most used nuclear fuel [see for example1 or 2. The determination of its thermodynamic properties, required for improving the operation margins of nuclear fuel, has been a challenge for more than 70 years, especially at high temperature. At around 2600 K, heat capacity measurements evidenced a lambda shaped peak3, indicating the existence of a phenomenon similar to a phase transition4. It was initially called Bredig transition by reference to5, in which the authors suggested that UO2 should undergo, in common with certain other fluorite structured materials, such as CaF2 and SrCl26, a diffuse (structural) transition near 0.8 of its absolute melting temperature (equal to 3120 K for UO2). CaF2 and SrCl2 are type II superionic that exhibit a peak in heat capacity Cp at a characteristic temperature T with increasing ionic conductivity. However, little experimental information is available about Bredig transition in UO2 because of the very high temperature at which it occurs. That is why indirect approaches were used to study it. The existence of disorder in UO2 crystalline structure was first reported by Hutchings7, who proposed to explain changes in the intensity of UO2 diffraction peaks by the formation of different types of oxygen interstitial clusters. Later on, some authors considered that UO2 thermodynamic properties could be explained by point defect creation, either polarons8 or oxygen vacancies9, but this would imply a 10% percent concentration of such defects above 2500K10, confirming the idea of grouped defects initially proposed by Hutchings. Collective motion was evidenced by molecular dynamic calculations of UO2 at high temperature as strings of displaced oxygen atoms11,12(see Fig. 4 for an illustration of stringlike motion). The analogy was made with glass forming materials, in which stringlike motion is commonly observed by molecular dynamics13. In parallel to this theoretical approach, new experimental data arose from “total scattering” methods revealing a shortening of the local UO distance, in contradiction with the expansion of UO distance calculated from thermal expansion data in the average crystalline structure14,15,16. This specific feature was first interpreted as a change in of local symmetry for the uranium coordination polyhedron: the cubic coordination polyhedron of uranium atom could change symmetry and become octahedral at the local scale. Two different distortion schemes were proposed to account for a non-cubic local symmetry. The fit of experimental neutron Probability Density Functions led to the choice of Pa-3 local distortion17, while the analysis of atomic positions calculated by molecular dynamics led to a distortion path in two steps first from Fm-3m to P42/nmc and then to Pbcn18. More recently, the short UO distance evidenced by total scattering methods was interpreted as inelastic contributions of a weakly dispersing transverse phonon branch to the equal-time correlation function attributed to a specific low energy phonon19.

In this paper, we aim at going further in the interpretation of UO2 high temperature behavior trying to make a link between low energy vibrations and string like motion. We restricted our analysis to the two specific phonons presented in Fig. 1, X5+ and X2- (the phonon names are taken from Fig. 3 in 19, they can be different in other papers), that correspond to the two local symmetries proposed in 17 and 18. Phonon induces a collective periodic displacement of atoms. With X5+ and X2- phonons, only oxygen displacements exist, uranium atoms staying at rest. With X5+, at any time, the atomic positions fulfill Pa-3 symmetry, whatever the amplitude of the oxygen displacement. With X2-, the atomic positions fulfill P42/nmc symmetry. For the study of these two phonons, we performed empirical potential calculations with CRG potential (from20) because this potential succeeded in reproducing the UO2 thermodynamic properties over a wide range of temperature, and because string like motion was also evidenced using this potential12. We calculated the energy of UO2 unit cell at 0 K for different amplitudes of oxygen collective displacements with Pa-3 and P42/nmc symmetries that gives the potential well of X5+ and X2- phonons respectively. We took temperature into account by changing UO2 cell parameter. These calculations are less accurate than the ones performed using molecular dynamics, in which atomic motions are calculated at a non 0 temperature. Nevertheless, they allow a much more simple interpretation of the calculations because we consider only the phonons we aim at studying, instead of extracting them from all atomic displacements when performing molecular dynamics.

Fig. 1
figure 1

Phonon dispersion curve in UO2 with Fm-3m crystalline structure (a). Atomic displacements associated with X5 + and X2- phonons (b); phonons induce a collective periodic displacement of oxygen atoms: here, the displacements are presented at a given time for the oxygen atoms of the coordination polyhedron of a uranium atom. The uranium coordination polyhedron at rest (light blue cube) is shown inside the UO2 unit cell (c). Uranium and oxygen atoms are in blue and red respectively.

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The potential wells of X5+ and X2- phonons as a function of oxygen displacement are presented on Fig. 2 for cell parameter equal to 5.47 Å, which corresponds to room temperature. X5+ is antisymmetric while X2- is symmetric phonon, which explains why UO2 energy is curve is antisymmetric for X5+ and symmetric for X2-. X2- shows a flat minimum that is enlarged by increasing cell parameter. This is in full accordance with large displacement of oxygen atoms with P42/nmc symmetry that were observed by Molecular dynamics17. Nevertheless, we were not able to find an energetic pathway that would lead to a minimum having Pbcn symmetry. We started from a distorted UO2 cell having oxygen displacements along X2-, and then we minimized energy. In all cases, the system came back in the flat minimum of X2- potential well consistently with21, and we never observed a secondary minimum with Pbcn symmetry. On the contrary, a secondary minimum clearly appears for X5+ phonon on Fig. 2. As discussed in a previous paper19, the correlation length of a regular phonon is short, even more at high temperature, which makes them inappropriate to generate string like motion. On the contrary, non-standard vibration modes with a secondary minimum might be a good candidate.

Fig. 2
figure 2

(a) potential well of X5+ (black) and X2- (red) phonons as a function of oxygen displacements (derived from δ in Table 1 in calculation method section), calculated for 5.47 Å UO2 cell parameter. (b) UO2 cell energy for the first minimum(Fm-3m) indicated by a blue circle on (a), and for the second minimum (Pa-3) indicated by a black circle on a) as a function of UO2 cell parameter.

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Table 1 Oxygen atom position and displacement with X5 + and X2- phonons; for these two modes only oxygen atoms are involved, uranium atoms staying at rest.
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We have two minima for the potential well of X5+ on Fig. 2a) corresponding to two symmetries: Fm-3m for the first minimum with null oxygen displacement, and Pa-3 for the second minimum with + 0.8Å oxygen displacement. The energy of first and second minima is presented on Fig. 2b) as a function of cell parameter. Figure 2b) predicts a phase transition from Fm-3m to Pa-3 symmetry for cell parameter equal to 5.63 Å, equivalent to a temperature near 2500 K (according to Hutchings measurement of UO2 cell parameter6). The phase transition temperature is not accurate because our approach neglects entropic contribution, but it is sufficiently near to the temperature of a phase transition observed as a peak of heat capacity in literature22, so that we assume it is indeed this phase transition. Similar 0K empirical potential calculations of UO2 having different space groups were already performed in literature, which predicted, for example, the Fm-3m to Pnma transition with increasing pressure20. However, to our knowledge, none of them was performed testing Pa-3 space group. The transition Fm-3m to Pa-3 can be interpreted using the first Pauling’s rule, which states that in binary ionic compounds the coordination polyhedron of the cation depends on the ratio: cation over anion ionic radius. At room temperature in UO2, this ratio is equal RU/RO≈ 1/1.36 = 0.735. The theoretical ratio at which a cubic environment become octahedral with Pauling’s first rule is around 0.73, i. e. very near to the one UO2. From that, we can conclude that only a small distortion of UO2 would be sufficient for uranium atoms to switch from cubic to octahedral environment, and this is very likely what is happening with the increase of UO2 cell parameter at high temperature. In the following, we will discuss the relationship between UO2 phase transition and phonons.

For that purpose, we will follow Keen’s approach for PDF interpretation that relies on low energy vibrations rather than on fluctuations of the order parameter per se23. Figure 3 presents the energy of UO2 cell at 0K with a = 5.6 Å deduced from LAMMPS calculations as a function of atomic displacements away from the equilibrium position. The atomic displacements are decomposed as a weighted sum of phonon induced displacements. In Fig. 3, the x direction corresponds to X5+ atomic displacements, consistently with Fig. 2. The y direction corresponds to any other phonon, or a weighed sum of any other phonons. Because phonons are eigenvector of the dynamic matrix, the path of minimum energy going from the absolute minimum to the secondary minimum goes along X5+ displacement (the red line in Fig. 3).

Fig. 3
figure 3

UO2 cell energy calculated for 5.6Å cell parameter for displacements along X5+ phonon (on the left-to-right axis); the other axis in the horizontal plane being any combination of phonons other than X5 + . The red line represents the path from the first to second minimum along X5+ phonon. The purple arrow represents the thermal energy of phonons at around 2000K, which corresponds approximately to a 5.6Å cell parameter.

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This energetic diagram is consistent with a 1st order transition from Fm-3m to Pa-3, and both phases can coexist. The purple arrow of Fig. 3 represents the thermal energy at 2600K that can provide enough energy to an X5+ phonon to bring UO2 from the absolute minimum to the secondary minimum of Fig. 3. As said before, at such high temperature, the correlation length of a phonon is low, a few interatomic distances at maximum. That is why we shall not imagine that a phonon can generate large domains of Pa-3 into an Fm-3m UO2 array. We have to imagine a wave of changing polyhedral configuration as shown on Fig. 4.

Fig. 4
figure 4

Change of Uranium polyhedron coordination at high temperature as a function of phonon propagation along (100) direction (indicated with a black arrow): (a) with X2- phonon, b) with X5+ phonon. The coherence length of the phonon is represented by a black arrow. The cyan and purple disk correspond to UO2 unit cell energy as a fonction of X2- and X5+ propagation respectively. The remaining displaced atoms are underlined by a black ellipse. A black ellipse also underlines a stringlike motion evidenced in a simulation box by molecular dynamic calculations in 13 (c). Stringlike collective atomic motion of O ions in UO2 are represented at T = 2400 K with different colors to discriminate between different string events.

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With a regular harmonic phonon like X2-, only isolated polyhedra with P42/nmc local symmetry can be observed because of the small size of phonon coherence domains. On the contrary, with X5+ anharmonic phonon, chains of distorted polyhedra with Pa-3 local symmetry can be formed in the wake of this phonon, because they can survive as a metastable energetic state in the secondary minimum of Fig. 3. Consequently, the propagation of a X5+ phonon that locally changes the structure from Fm-3m to Pa-3 could produce a collective displacement of oxygen atoms occurring in a short period of time over a small volume, which exactly corresponds a to stringlike motion, as shown on Fig. 4. Therefore, we can interpret string like motion as a consequence of the X5+ energy landscape with two minima at the vicinity of the Fm-3m to Pa-3 phase transition.

This interpretation, which is only justified by calculations for the time being, leads us to consider broader implications than the prototypical UO2 solid. First regarding nuclear fuel, it is of interest to characterize how the Fm-3m to Pa-3 transition occurs in actual nuclear fuel (see 24 for an example about the Bredig transition and its modelling in fuel behavior codes). Second, ionic superconductors of type 2 are said to exhibit no phase transition, however by comparison to UO2, one could look for the existence of phase transition, equivalent to the Fm-3m to Pa-3 in UO2. In the case of Li2O 25 or CeO226, the instability of X2- phonon, calculated by molecular dynamics, could be related to the softening of the same X2- phonon that, in ZrO2, is responsible of the cubic to tetragonal phase transition27. In that sense, stringlike motion could be related the Fm-3m to P42/nmc transition in these ionic superconductors. On a more general point of view, our results question the understanding of first order transition with a local dynamic point of view and the manner by which it is possible to characterize it both experimentally and theoretically. Atomic positions calculated by molecular dynamics are so disordered that is difficult to evidence any phase transition without knowing that it exists. Could stringlike motion be a signature of it? In a similar manner, thermal motion at high temperature makes difficult the detection of additional peaks that are the experimental signature of phase transition using diffraction methods. Could total scattering methods and PDF analysis provide an alternative manner to detect it?

Calculation method

Calculations were performed using LAMMPS code with static positions of the atoms, i.e. at 0K. For each calculation, the uranium atoms kept at the same position while the positions of oxygen atoms were changed at each calculation. Given X5 + and X2- symmetries, all oxygen atoms move with the same distance from their Fm-3m regular site, but with different directions. The displacement of each atom of a UO2 Fm-3m unit cell for each phonon is given in Table 1. At a given value of cell parameter, for each phonon, 200 different positions of oxygen atoms were calculated by changing regularly δ from -0.100 to 0.100, accordingly with Table 1. The calculated value of the cell energy for all these calculations is represented as a function of δ, and hence as a function of the oxygen displacement.