Cation interdiffusion in uranium–plutonium mixed oxide fuels: Where are we now?

The diffusion phenomena in uranium–plutonium mixed oxides U_1−yPu_yO₂ dictate the physicochemical properties of mixed oxides (MOX) nuclear fuel throughout manufacturing, irradiation, and storage. More precisely, it is paramount to estimate the cation interdiffusion insofar as it dovetails with the actinide redistribution during sintering and under irradiation. This paper draws a critical review of the existing experimental data of U and Pu interdiffusion coefficients in MOX fuel.

1 Introduction

The diffusion phenomena in solids dictate their physicochemical properties, such as redox behavior, melting point, recrystallization, creep, sintering, and ionic conductivity, among others. In the nuclear industry, these diffusion properties are of paramount interest since they directly impact the in-pile performances of the fuel and hence the safety of the reactor. For instance, uranium–plutonium U_1−yPu_yO₂ mixed oxides (MOX), with various compositions, are used for decades for nuclear power all over the world (Olander, 2009; Baron et al., 2020; Kato et al., 2020; Dudarev, 2022; Kato and Machida, 2022). During their lifetime, MOX fuel pellets undergo the harshest temperature, atmosphere, and irradiation conditions. More precisely, green compacts are sintered at elevated temperature, usually around 2,000 K, in highly reducing atmosphere (hydrogen-containing gas mixture) (Ramaniah, 1982; Okita et al., 2000; Vauchy et al., 2014a). Under irradiation, because of the concomitant fission reactions and the coolant’s action, a thermal gradient (up to ∼300 K/mm) occurs along the MOX pellet radius and induces a rapid restructuring of the fuel (Bober et al., 1973; Noirot et al., 2008; Maeda et al., 2009a; Maeda et al., 2009b; Van Uffelen et al., 2010; Ishimi et al., 2019; Kato and Greenspan, 2021; Ozawa et al., 2021). During both these high temperature stages, actinide cations migrate to mix up and segregate, respectively. MOX fuels are therefore always subjected to strong chemical gradients (oxygen and cations). Due to their outstanding atomic weight (M_U−Pu ≥ 238 u), these actinides hardly move and need a significant addition of energy to diffuse, hence the extreme sintering temperature.

As their chemical and radiological toxicities are potentially lethal (International Commission on Radiological Protection, 1996; Voeltz, 2000; Rodriguez and Wexler, 2014), Pu-bearing solids need to be handled in dedicated confined environments (glove boxes); thus, they are challenging to study. Only a handful of diffusion coefficients in actinide-bearing oxides are published, anionic and cationic combined. Particularly, the physicochemical processes associated with cation diffusion are remarkably complex, and their scanty migration is hardly measurable. Within this frame, this paper draws a review of the available experimental data on cation interdiffusion coefficients in U–Pu dioxides, since they are the very key for manufacturing and in-pile behaviors.

2 Crystal structure and defect chemistry of actinide dioxides

Actinide dioxides (AnO₂) are known to crystallize in a fluorite structure (CaF₂ type), space group $Fm\overline{3}m$ (#225), where the cations are located in the face-centered cubic lattice (noted f.c.c.) and the oxygen anions in tetrahedral sites (Figure 1) (Fahey et al., 1974).

FIGURE 1

FIGURE 1. AnO₂ fluorite structure (atoms drawn proportionally to the ionic radii of U and O).

Even at room-temperature, this crystal structure can accommodate large deviations from oxygen stoichiometry (O/An = 2) as evidenced in the pure poles UO_2+x (Geønvold and Haraldsen, 1948), PuO_2−x (Gardner et al., 1965), AmO_2−x (Chikalla and Eyring, 1968), CmO_2−x (Mosley, 1972), BkO_2−x (Baybarz, 1968), and CfO_2−x (Baybarz et al., 1972) as well as in the respective solid solutions, the most studied being U_1−yPuyO_2±x Markin and Street, 1967), U_1−yAm_yO_2±x (Bartscher and Sari, 1983), and Pu_1−yAm_yO_2−x (Vauchy et al., 2017).

These deviations from stoichiometry and irradiation defects both induce severe lattice defects and can also enhance atomic diffusion (Kilner et al., 1981; Matzke, 1983a; Ferry et al., 2005; Smirnov and Elmanov, 2016). Due to the large mass of the actinide atoms, cationic vacancies and/or interstitials are unprobeable; thus, only the anion sub-lattice (oxygen) supports the defects (Belle, 1961; Atlas et al., 1966; Matzke and Sørensen, 1981; Matzke, 1987), namely, oxygen vacancies ($V_O^{••}$ and $V_O^{•}$) and interstitials ($O_i^{″}$) for O/An ≠ 2 compositions and electron/hole ($e^{\prime }$/$h^{•}$) pairs in the AnO₂ region (Cristea et al., 2007; Kato et al., 2017a). The migration of oxygen point defects (vacancies and interstitials) is the main mechanism responsible for diffusion in oxide fluorite type structures and more precisely in actinide dioxides (Crank, 1957; Matzke, 1990; Murch, 2001).

3 Interdiffusion vs. self-diffusion

Diffusion in binary substitutional solid solutions is called interdiffusion. This corresponds to the thermally activated atomic transport of species in a chemical potential field as they tend to rearrange to uniformize the molecular distribution in the medium. Interdiffusion then describes the tendency of two materials of different chemical compositions (usually as a diffusion couple) to homogenize as a function of thermodynamic conditions. Figures 2A,B schematically illustrates two examples of diffusion couples, AnO₂/BnO₂ and AnO₂/An_1−yBn_yO₂, respectively (An and Bn having different Z numbers). In both cases, a cationic chemical gradient exists between the two lattices and hence can be defined as interdiffusion.

FIGURE 2

FIGURE 2. Schematic representation of (A) AnO₂/BnO₂ and (B) AnO₂/An_1−yBn_yO₂ interdiffusion couples.

Once the chemical equilibrium is established, i.e., no chemical gradient remains, the diffusion phenomena that take place in such a medium only correspond to the Brownian motion of the constituting atom. This is known as the self-diffusion (Crank, 1957; Murch, 2001; Mehrer, 2007).

Studies on actinide dioxides often report “tracer diffusion” of a given species, which are claimed to be “self-diffusion.” Figure 3 shows a schematic representation of the three experimental cases encountered in the literature for cation “self-diffusion” measurements in AnO₂.

FIGURE 3

FIGURE 3. Schematic representation of (A) isotopic self-diffusion between ^yAnO₂ and ^xAnO₂, (B) tracer layer deposition ^yAn on ^xAnO₂, and (C) tracer layer deposition ^yAn on ^xAn_1−yBn_yO₂.

The first case (Figure 3A) corresponds to contacting two samples of the exact same chemical composition AnO₂ but with different isotopic compositions, one being enriched in a given isotope ^yAn compared with the host material (^xAn). The couple is then annealed, allowing ^yAnO₂ to diffuse in ^xAnO₂, and the ^yAn/^xAn diffusion profile is analyzed. To the best of our knowledge, this is the very definition of measuring self-diffusion. Unfortunately, this type of experimental study is rarely carried out on actinide dioxides (Nagels et al., 1966; Sabioni et al., 1998). Albeit being of prime importance, these results are excluded from the discussion as this review focuses on interdiffusion.

On the other hand, a chemical gradient cannot be excluded in the next two examples of “tracer diffusion” experiments. The associated published data on (U, Pu)O₂ will then be considered in this review, in addition to the “real” interdiffusion measurements.

The second case (Figure 3B) shows the typical experimental procedure proposed in the literature to investigate An self-diffusion in AnO₂ (Auskern and Belle, 1961; Schmitz and Lindner, 1963; Alcock et al., 1966; Yajima et al., 1966; Matzke, 1969; Matzke, 1973; Matzke, 1983b; Glasser-Leme and Matzke, 1983; Ma, 2017). Normally, this technique consists in depositing a thin layer (by evaporation/condensation) of a pure isotope yAn (usually more α-active than ^xAn, e.g., ²³⁸Pu or ²³³U) on a polished surface of a specimen ^xAnO₂ (UO₂, PuO₂, or their solid solution), and the migration of this species is observed in the host lattice using α-spectrometry (Marin and Coniglio, 1966; Hawkins and Alcock, 1968). It is very unlikely that the deposited layer and the substrate material are of the same chemical composition. Indeed, the deposition process is operated in a high vacuum and induces the condensation of a metallic layer on the substrate (Schmitz and Lindner, 1963; Wade, 1971; Wade et al., 1978). An obvious oxygen chemical gradient is then present between the substrate and the layer and may dramatically enhance the diffusion of An in AnO₂. One may accept these data as self-diffusion due to the claimed infinitesimal thickness of the said layer and/or due to its hypothetical oxidation, regardless of the published studies that show it is metallic. However, we believe that the associated results cannot be accepted as pure self-diffusion.

The third case (Figure 3C) and the second case are very similar, except the small, yet important, difference in the chemical composition of the host lattice. Indeed, ^yAn is deposited on the surface of a ^xAn_1−yBn_yO₂ material (An and Bn being different elements). Regrettably, this technique is also widely used to determine “self-diffusion” coefficients of a species An in a host material (Schmitz and Lindner, 1965; Lindner et al., 1967; Riemer and Scherff, 1971; Matzke, 1973; Matzke and Lambert, 1974; Schmitz, Marajofsky; Lambert, 1978; Matzke, 1983a; Matzke, 1983b; Noyau, 2012; Noyau et al., 2012). For some reason, even if one may consider that the second example is suitable for An self-diffusion in AnO₂, this simplification cannot be accepted here. Even in the ideal case of a spontaneously oxidized layer to the same O/An ratio than that of the substrate, the presence of another atom Bn in the cation sub-lattice makes it impossible to accept it as self-diffusion. Neglecting or denying the existence of this chemical gradient may induce severe experimental biases when interpreting the data as pure “self-diffusion.” Indeed, interdiffusion coefficients are of several orders of magnitude larger than that of self-diffusion (Matzke, 1990). When these experimental values of the so-called self-diffusion are used for calculations and/or fuel performance codes, this may be especially problematic.

4 Determination of the interdiffusion coefficients

4.1 Bulk vs. grain boundary diffusion

A polycrystalline material is often considered as a semi-infinite medium composed of adjacent crystals that are separated by grain boundaries. Most of the uranium–plutonium MOX studied are polycrystalline specimens, and the influence of both lattice and grain boundary diffusions needs to be described.

Bulk, or lattice, diffusion refers to the migration of atoms within the volume of a crystal (grain). In the case of interdiffusion measurements, bulk diffusion corresponds to the net mass transport through the surface of the grains. Figure 4 shows an illustration with contacted single crystals with species A diffusing in the B lattice.

FIGURE 4

FIGURE 4. Bulk interdiffusion in two adjacent single crystals of species A and B.

Harrison proposed three types of grain boundary diffusion kinetics in polycrystalline materials (Harrison, 1961). Figure 5 illustrates these kinetics with species A diffusing (considered infinite) in the B lattice.

FIGURE 5

FIGURE 5. Three tyupes of grain boundary diffusion kinetics of species A in B.

The first type (Figure 5A) corresponds to the situation where bulk diffusion is negligible and where a significant grain boundary diffusion occurs. A sharp composition transition is observed between the grain boundaries and the bulk of the grains. This type of diffusion behavior is observed in the first steps of an interdiffusion experiment.

The second type (Figure 5B) corresponds to the situation where the lattice diffusion cannot be considered as negligible anymore. A composition gradient is then established between the grain boundaries and the bulk of the grains. This type of diffusion behavior is observed when the annealing time and/or temperature increases compared to the first type.

The last type (Figure 5C) corresponds to the situation where the grain boundary and bulk diffusion kinetics are similar. Only a slight composition gradient remains between the grain boundaries and the bulk of the grains. Usually, this type of behavior is observed when the diffusion distance is much larger than the grain size and the diffusion fields of the neighboring grains overlap.

In most experiments, the second type of diffusion is observed and is quantified by measuring the isoconcentration contours in adjacent grains along the boundaries (Figure 6).

FIGURE 6

FIGURE 6. Isoconcentration contours in bicrystals with grain boundary normal to the free surface.

Bulk and grain boundary diffusions can be very different in polycrystalline materials, the latter being known to be “pathways” or “shortcuts” for atomic transport (Fisher, 1951; Knorr et al., 1989). Indeed, differences of several orders of magnitude are usually reported between these two types of diffusion resulting from the high level of disorientation of the atoms located along the grain boundaries. In practice, authors rarely differentiate these two effects, and averages are usually calculated in polycrystalline materials. On the other hand, some studies involving single crystals are also reported. Studying such materials allows extracting the bulk diffusion due to the nonexistence of grain boundaries; however, it could also raise questions of preferential penetrations with respect to crystal orientation. This problem is being ignored in polycrystals as a result of the random grain orientation.

For example, Figure 7 shows real EPMA elemental mappings of a diffusion couple A–B (arbitrary gray levels), corresponding to the second case of grain boundary diffusion (Figure 5B).

FIGURE 7

FIGURE 7. Elemental mappings of species A and B in their mutual lattice (arbitrary gray scale).

The grain boundary diffusion can usually be determined by three different means (Le Claire, 1963; Peterson, 1983):

- Measuring the distance of the diffusion apex from the surface (“y” in Figure 6).

- Measuring the angle (“Ø” in Figure 6) between the grain boundary and the tangent to a concentration contour.

- Measuring the amount of diffusing species in slices parallel to the interface plane.

4.2 Experimental techniques

In crystalline materials, the interdiffusion coefficients are usually determined directly via the preparation of diffusion couples and subsequent annealing. Each specimen is polished to obtain a surface roughness appropriate for contact. The samples are then annealed to allow species diffusion, and their concentration is determined as a function of their depth of penetration, usually by electron probe microanalysis (EPMA) (Oishi et al., 1981; Sakka et al., 1982; Dean and Goldstein, 1986; Léchelle et al., 2012) or ion beam analysis [e.g., Rutherford backscattering spectrometry, nuclear reaction analysis or secondary ion mass spectrometry (Ishigaki et al., 1987; Vauchy et al., 2015a; Jeynes and Colaux, 2016)]. The “diffusion profile” is the variation in the elemental concentration with the perpendicular distance from the interface plane (see Figure 7). Other ion beam analysis techniques are also encountered for depth profiling.

4.3 Mathematical approach

Solids submitted to a spatial concentration gradient (herein, chemical gradient) tend to homogenize with time and temperature. The resulting flux of atoms (of the same species) is usually noted as $J$ and is defined by the first Fick’s law given in Eq. 1):

$\mathit{J}=-\mathit{D}\frac{\partial \mathit{C}}{\partial \mathit{x}}(1)$

where $\partial C/\partial x$ is the spatial concentration gradient and $D$ is the diffusion coefficient.

Figure 8 shows a representation of the evolution of composition–distance curves with annealing time tn in an ideal A–B interdiffusion couple.

FIGURE 8

FIGURE 8. Evolution of ideal interdiffusion profiles of species A (dashed) and B (solid) in their mutual lattice with annealing time t_n.

If the atomic transport of species A and B is equal and opposite, the lattice structure remains unchanged by the diffusion process, directly leading to the interdiffusion coefficient $\stackrel{\sim }{D}$. In this sole ideal case, the acquisition of only one of these profiles is necessary to obtain $\stackrel{\sim }{D}$. Although being theoretically not mandatory, repeating the measurement with various annealing times allows reducing uncertainties on the interdiffusion coefficient. The second Fick’s law is used:

$\frac{\partial \mathit{C}}{\partial \mathit{t}}=\mathit{D}\frac{\partial ²\mathit{C}}{\partial \mathit{x}²}(2)$

The Boltzmann–Matano method (Boltzmann, 1894; Matano, 1933) is widely used to calculate the interdiffusion coefficients from the elemental depth profiles (one dimensional) shown in Figure 8. The Matano plane is defined as the abscissa at which the two areas α under the diffusion profile curve are equal (Figure 9).

FIGURE 9

FIGURE 9. Representation of an interdiffusion profile and its interpretation for calculations.

From this representation, Eq. 3 gives the resolution of the diffusion equation:

$\stackrel{\sim }{\mathit{D}}=-\frac{\mathbf{1}}{\mathbf{2}\mathit{t}}\frac{{\int }_{{\mathit{C}}_{\mathbf{1}}}^{{\mathit{C}}_{\mathbf{2}}}\left(\mathit{x}-{\mathit{x}}_{\mathit{M}}\right)\partial \mathit{C}}{{\frac{\partial \mathit{C}}{\partial \mathit{x}}|}_{\mathit{C}}}(3)$

In practice, species A and B rarely have the same diffusion rates, and their interpenetration induces a shift of the lattice planes called the Kirkendall effect (Kirkendall, 1942). The individual diffusion coefficients of A and B ($D_A$ and $D_B$), also called intrinsic diffusion coefficients, correspond to their “net” displacement with respect to their local lattice plane. The mathematical expression of the A–B interdiffusion coefficient is given by the Darken equation (Eq. 4):

$\stackrel{\sim }{\mathit{D}}={\mathit{N}}_{\mathit{A}}{.\mathit{D}}_{\mathit{B}}+{\mathit{N}}_{\mathit{B}}.{\mathit{D}}_{\mathit{A}}(4)$

where $N_X$ and $D_X$ are the molar fraction and intrinsic diffusion coefficient of species X, respectively.

In this same case, an alternative (and more relevant) method consists in measuring both A and B elemental profiles and resolving the second Fick’s law (Eq. 2). The interdiffusion coefficients of the two species are then directly obtained.

5 Uranium–plutonium interdiffusion coefficients

5.1 Reviewed studies

An exhaustive investigation of the published U–Pu interdiffusion coefficients in their dioxide was attempted in this review (Schmitz and Lindner, 1963; Davies and Novak, 1964; Schmitz and Lindner, 1965; Lindner et al., 1967; Theisen and Vollath, 19671967; Riemer and Scherff, 1971; Schmitz, Marajofsky; Chilton and Edwards, 1978; Lambert, 1978; Glasser-Leme and Matzke, 1982; Matzke, 1983b; Glasser-Leme and Matzke, 1984; Verma, 1984; Glasser-Leme, 1985; Jean-Baptiste and Gallet, 1985; Marin, 1988; Mendez; Kutty et al., 1999; Sato et al., 2010; Noyau, 2012; Berzati, 2013). Most of the available data are from the 1970s–1980s, and recent studies are scarce. Table 1 gives the main details about the studies reviewed here.

TABLE 1

TABLE 1. Description of the type of experiment, analysis method, specimens and temperature ranges of the reviewed studies.

5.2 Relation between $\stackrel{\sim }{D}$, pO₂, and T

All the available data are gathered in Figure 10 as an Arrhenius diagram. The color of the data points refers to the analysis techniques used.

FIGURE 10

FIGURE 10. Arrhenius diagram for cation interdiffusion coefficients against the reciprocal of the temperature.

The large scattering in the available data may have resulted from the differences in experimental techniques and analysis procedure, among others. However, most of the authors agree that the cationic composition (Pu content) has only a minor influence on the interdiffusion coefficients. A common trend also emerges, i.e., $\stackrel{\sim }{D}$ increases with temperature, reconfirming that diffusion is thermally activated.

For a given temperature, it can be seen clearly that the interdiffusion coefficients can vary by a factor of 10⁸. Indeed, diffusion in oxides is first governed by temperature but also by the oxygen activity in the surrounding gas mixture. In a previous study, we have shown that even if a composition change is not experimentally noticeable between different conditions (T, pO₂), significant variations in diffusion coefficients are possible (Vauchy et al., 2015a). A more suitable representation is the variation in $\stackrel{\sim }{D}$ as a function of both temperature and oxygen partial pressure pO₂ of the gas (when available). Due to their extremely low (and unrealistic) values, the data from Kutty et al. (1999) were precluded. Also, most of the authors studied the interdiffusion in sintered materials (either coated or coupled), while some investigated the cation migrations during sintering (Theisen and Vollath, 19671967; Verma, 1984; Marin, 1988; Mendez; Berzati, 2013). Since sintering involves physical mechanisms and because both solid solution formation and densification are concomitant processes, the associated interdiffusion coefficients were separated from the others to avoid a direct comparison. Figures 11A,B shows the resulting plots. Figures 11C–I shows the details at different temperatures.

FIGURE 11

FIGURE 11. Variations in interdiffusion coefficients as a function of temperature and oxygen partial pressure obtained (A) from dense samples and (B) after sintering of green specimens. Details of the variation in $\stackrel{\sim }{\mathit{D}}$ in dense materials are given at (C) 1,750 K–1,775 K, (D) 1,850 K–1,875 K, (E) 1,950 K–1,975 K, (F) 2,000 K–2,025 K, (G) 2,118 K–2,123 K, (H) 2,178 K, and (I) 2,223 K, respectively.

Even in this representation, the experimental results on dense materials remain largely scattered (Figure 11A). As a general trend among the same study, interdiffusion coefficients increase with both pO₂ and T (except Chilton et al. and Jean-Baptiste et al.).

Concerning the “sintering” experiments (Figure 11B), the values seem less scattered (10⁻¹⁸–10⁻¹⁴ m².s⁻¹), but the available data are also more restricted. The values of $\stackrel{\sim }{D}$ are larger by a few orders of magnitude than the average of the data shown in Figure 11A (10⁻¹⁹–10⁻¹⁸ m².s⁻¹). Indeed, grain boundaries are particularly active during sintering and greatly contribute to the atomic transport. As highlighted in Section 4.1, grain boundary diffusion is larger than lattice diffusion by several orders of magnitude, making their contribution prevail during the first steps of the sintering process. Once again, $\stackrel{\sim }{D}$ increases with both temperature and oxygen partial pressure. This particular feature can be useful for advanced sintering processes by varying in situ the oxygen partial pressure during sintering of MOX fuel pellets to optimize the formation of solid solution and/or densification (Berzati, 2013; Nakamichi et al., 2020).

5.3 Relation between $\stackrel{\sim }{D}$, O/M ratio, and T

To provide a complete understanding of the interdiffusion phenomena, the O/M ratio of the samples studied in the literature was either tabulated or calculated from the Pu content, temperature, and atmosphere conditions with the relation given in (Hirooka et al., 2022). Figure 12 plots the dependence of $\stackrel{\sim }{D}$ upon temperature and O/M ratio. The “sintering” data were rejected as they should not be directly compared to the other studies.

FIGURE 12

FIGURE 12. Variations in interdiffusion coefficients as a function of (A) O/M ratio of the specimens (tabulated or calculated) and temperature with details at (B) 1,750 K–1,775 K, (C) 1,850 K–1,875 K, (D) 1,950 K–1,975 K, (E) 2,000 K–2,025 K, (F) 2,118 K–2,123 K, (G) 2,178 K, and (H) 2,223 K, respectively.

Regardless of the representation, the literature data still remain largely scattered. One must consider a critical analysis of these data with respect to the experimental procedures. Within this context, the values from Matzke (Matzke, 1983b) and Glasser-Leme (Glasser-Leme and Matzke, 1982; Glasser-Leme and Matzke, 1984) can be considered as doubtful. Indeed, they were carried out on “single crystals” obtained from arc-melted powders, and the melted pool was subsequently cut and polished to obtain a surface suitable for vapor-phased tracer deposition. The problem with this technique (beyond the questions raised in Section 3) resides in the fact that the “crystal” was arbitrary cut, without taking into account its orientation. It is known that the crystal orientation has a strong influence on diffusion properties (Turnbull and Hoffman, 1954; Burriel et al., 2016; Holby, 2019). Therefore, measuring diffusion coefficients without referring to the Miller indices of the associated crystal planes is useless. Furthermore, the “tracer diffusion” experiments, namely, Schmitz (Schmitz and Lindner, 1963; Schmitz and Lindner, 1965; Schmitz, Marajofsky), Lindner (Lindner et al., 1967), Riemer (Riemer and Scherff, 1971), and Lambert (Lambert, 1978), should also be rejected as the nature (composition, thickness, etc.) of the deposited layer is really questionable, and the α-degradation energy method used allows probing only the first atomic layers as the path of α particles in such dense and heavy materials is very restricted. The data obtained by means of this technique cannot be considered as representative of bulk diffusion properties. Unfortunately, only the five investigations, namely, that of Chilton (Chilton and Edwards, 1978), Glasser-Leme (Glasser-Leme, 1985), Jean-Baptiste (Jean-Baptiste and Gallet, 1985), Sato (Sato et al., 2010), and Noyau (Noyau, 2012), are acceptable (Figure 13).

FIGURE 13

FIGURE 13. Variations in the acceptable interdiffusion coefficients from the literature as a function of O/M ratio of the specimens (tabulated or calculated) and temperature.

As the number of data points is extremely small and scattered, it seems unreasonable to make a direct comparison of the associated $\stackrel{\sim }{D}$ values. However, a more general discussion on (cation) diffusion properties in AnO₂ can be proposed.

6 Oxygen/metal ratio, oxygen potential, points defects, clusters, and (inter) diffusion

As already detailed, the driving force for interdiffusion in fluorite structures, hence AnO₂, is the migration of free oxygen vacancies (in AnO_2−x) or interstitials (in AnO_2+x) by the hopping process. However, recent studies seem to suggest that some complex cationic charge-compensation mechanisms can take place in U_1−yPu_yO_2±x (Martin et al., 2022), similarly to what was clearly observed in U_1−yAm_yO_2±x mixed oxides (Epifano et al., 2019). Such a behavior could have an effect on the diffusion mechanisms as the crystal lattice would be distorted due to the difference in ionic size of the cations. As these new results need to be confirmed and because they were never experimentally evidenced at elevated temperatures, we will not further discuss their effects on the cation interdiffusion in MOX fuels.

Thus, in hypostoichiometry, the greater the concentration of vacancies, the larger the interdiffusion coefficient. However, the reality is a little different. Indeed, increasing the number of oxygen vacancies induces a reduction of An(IV) to An(III) to keep electroneutrality. These trivalent cations trap the oxygen vacancies, forming uncharged cluster defects (Anderson, 1971; Ando and Oishi, 1983; Bevan et al., 1986; Nakayama and Martin, 2009; Yoshida et al., 2011; Vinograd, Bukaemskiy, Modolo, Deissmann, Bosbach).

For instance, at a given temperature, a decrease in the oxygen potential of a U_1−yPu_yO₂ MOX leads to a decrease in its oxygen/metal ratio by the partial reduction of Pu(IV)–Pu(III) (Kato et al., 2017b; Hirooka et al., 2022). Due to the attractive Coulomb (electrostatic) interaction, trivalent plutonium atoms trap oxygen vacancies, forming the clusters $P{u}^{\prime }-V_O^{••}-P{u}^{\prime }$ (for double-charged oxygen vacancy) and/or $P{u}^{\prime }-V_O^{•}$ (for single-charged oxygen vacancy) (Manes et al., 1975; Matzke and Sørensen, 1981; Cristea et al., 2007). More complex/extended defects might even be considered (shear planes, micro-domains, etc.) by lowering the O/M ratio (thus pO₂), trapping more and more oxygen vacancies (Catlow and Sørensen, 1981). These traps are a barrier to cation interdiffusion. Experimental evidences of this peculiar behavior were found in U_1−yPu_yO₂ MOX. Indeed, during sintering of UO₂−PuO₂ co-milled compacts, the formation of the solid solution, i.e., cation interdiffusion, is promoted when the oxygen partial pressure of the gas mixture is increased (Mendez; Berzati, 2013; Takeuchi et al., 2011; Vauchy et al., 2016a) and vice versa. This behavior is even more pronounced at near stoichiometric compositions (O/M ≥ 1.99) as a result of a probable dramatic increase in the concentration of free oxygen vacancies (or of a less favorable clustering process) (Norris, 1977).

In hyperstoichiometric (O/M > 2.00) U_1−yPu_yO₂ MOX, the presence of oxygen atoms $O_i^{″}$ in the interstitial position is balanced by the partial oxidation of U(IV)–U(V) to maintain electroneutrality (Brett and Fox, 1966). These atoms in the interstitial position are also known to form clusters (Willis, 1963). To some extent, they can create channels that allow the different species to diffuse with ease. Similarly to hypostoichiometry, a further increase in the deviation from stoichiometry can induce a stagnation (or even a decrease) in the diffusion coefficient due to the formation of extended defects, the most important being di-interstitial clusters (Wang et al., 2014; Brincat et al., 2015; Yu et al., 2015; Caglak et al., 2020).

The Brouwer diagram given in Figure 14A shows the variations in the concentrations of point defects in U_1−yPu_yO₂, at temperature T, as a function of the oxygen partial pressure. For O/M < 2, decreasing pO₂ leads to larger deviations from stoichiometry and thus to a higher concentration in oxygen vacancies. However, these defects (single or double-charged oxygen vacancies, $V_O^{•}$ and $V_O^{••}$) are trapped in clusters, and the fraction of free oxygen vacancies dramatically drops (Cristea et al., 2007) (Figure 14B), inducing a decrease in cation interdiffusion.

FIGURE 14

FIGURE 14. (A) Schematic Brouwer diagram and (B) defects fraction as a function of pO₂ in U_1−yPu_yO_2±x at a given temperature.

This trapping process can explain the experimental observations of enhanced U–Pu interdiffusion in near stoichiometric compositions, as compared to lower O/M ratios.

More generally, the free-point-defects-mediated diffusion mechanism not only impacts cations but is generic for atomic transport, both in hypo- and hyperstoichiometry. Indeed, oxygen chemical diffusion in AnO_2±x was shown to be enhanced at compositions close to O/M = 2.00 (Chereau and Wadier, 1973; Sari, 1978; Bayoǧlu and Lorenzelli, 1979; Bayoǧlu and Lorenzelli, 1984; Stan and Cristea, 2005) as a result of preponderant clustering when the deviation from stoichiometry is increased. This observation remains under discussion among the community as some other studies suggest either an increase or a stagnation in the oxygen chemical diffusion coefficient when the deviation from stoichiometry is increased (Woodley and Gibby, 1973; Kim and Olander, 1981; Kato et al., 2009; Kato et al., 2013; Vauchy et al., 2015b; Watanabe et al., 2015; Watanabe et al., 2017).

7 Diffusion mechanism in AnO_2±x fluorite structure

7.1 Empirical crystallographic approach

In this section, we propose a diffusion mechanism in AnO_2±x based on the crystallographic oxygen defects that the fluorite lattice can accommodate. As the actinide dioxides are known to be ionic crystals, the ionic radii of the constitutive species of the fluorite AnO_2±x structure (herein adapted to U_1−yPu_yO_2±x) are either tabulated from the literature or calculated from the respective values in pure, stoichiometric dioxides UO₂ and PuO₂. Albeit being a very simplistic model, the ions are considered to be contacting hard spheres (sphere packed), since the structure of stoichiometric AnO₂ is the lowest-energy configuration.

Thus, in the fluorite structure of the pure, stoichiometric, AnO₂ dioxide, the O(–II) ionic radius $r_{O\left(–II\right)}$ is equal to 1/4 of the lattice parameter $a$ as the four oxygen atoms are inscribed into the unit cell. The An(IV) ionic radius $r_{An\left(IV\right)}$ can be calculated from the cubic unit cell diagonal, $a\sqrt{3}$. Indeed, in the fluorite structure, the hard sphere model is expressed as Eq. 5.

$\mathit{a}\times \sqrt{\mathbf{3}}=\mathbf{4}\times {\mathit{r}}_{\mathit{O}\left(–\mathit{I}\mathit{I}\right)}+\mathbf{4}{\times \mathit{r}}_{\mathit{A}\mathit{n}\left(\mathit{I}\mathit{V}\right)}(5)$

As the lattice parameter $a$ of the dioxide can be measured using regular X-ray diffraction, giving $r_{O\left(–II\right)}$, the ionic radius of the actinide $r_{An\left(IV\right)}$ can then be easily calculated. Table 2 presents the resulting values in U_1−yPu_yO_2±x. As a reminder, in the fluorite structure, the coordination number of the cations and anions in their “normal” sub-lattice is 8 and 4, respectively. The most stable position of isolated interstitial oxygen ions $O_i^{″}$ is predicted to be (½, ½, ½) in AnO_2+x (Dorado et al., 2011; Middleburgh et al., 2013), so its coordination number is 6 (octahedral site).

TABLE 2

TABLE 2. Ionic radii of the constitutive species of the fluorite U_1−yPu_yO_2±x.

Figure 15 schematically shows the resulting fluorite structure of hypo- and hyperstoichiometric actinide dioxide AnO_2±x.

FIGURE 15

FIGURE 15. Hypo- and hyperstoichiometry lattice defects in AnO_2±x (U_1−yPu_yO_2±x) fluorite structure (atoms drawn proportionally to their ionic radii). The cluster $\mathit{P}{\mathit{u}}^{\prime }-{\mathit{V}}_{\mathit{O}}^{••}-\mathit{P}{\mathit{u}}^{\prime }$ in O/M < 2 and an isolated oxygen interstitial in O/M > 2 are schematically represented. The U(V) ions are represented in their most favorable position (Dorado et al., 2011).

7.1.1 Interdiffusion in hypostoichiometry (O/M < 2)

Hypostoichiometry in AnO₂ is defined as an O/M ratio lower than the reference value 2. In this composition range, the concentration in oxygen vacancies is larger than that of interstitial oxygen atoms, the metal lattice being conserved (see Section 2). Contrary to some common beliefs, the size of oxygen vacancies is smaller than that of the anion. The lattice locally collapses around the vacancy due to the electrostatic interactions (repulsions) between the constitutive ions of the crystal (Marrocchelli et al., 2013; Chatzichristodoulou et al., 2015). As explained before, the cations have to accommodate the charge of the oxygen vacancy and form trivalent ions. This reduction is also accompanied by an increase in the ionic radius of the metal atom (see Table 2), reverberating its effect as a local swelling of the lattice due to steric effects. Usually, the magnitude of the increase in the cation radius is larger than that of the local collapse of the lattice due to the presence of the oxygen vacancy. This competition creates large local distortions in the crystal structure. Macroscopically, the lattice swells proportionally to the magnitude of deviation from stoichiometry (Markin and Street, 1967; Kato and Konashi, 2009; Vauchy et al., 2014b; Vauchy et al., 2017; Tracy et al., 2018).

As explained before, interdiffusion in hypostoichiometric U_1−yPu_yO_2−x is governed by the migration of free oxygen vacancies. As shown in Figure 15, the formation of a doubly charged oxygen vacancy is only possible if at least two plutonium atoms are contiguous. As U_1−yPu_yO_2−x is considered as a solid solution, i.e., that the cations are randomly dispersed in the lattice, not all Pu sites are equivalent in their propensity to form these clusters. Indeed, if doubly charged oxygen vacancies are considered to be formed, the presence of U ions in the first metal shell of Pu atoms tends to stabilize Pu(IV) by decreasing the probability to form an oxygen vacancy, hence of ${Pu}^{\prime }-V_O^{••}-P{u}^{\prime }$. In other words, at least two adjacent Pu(III) atoms are needed to form a neutral tetrahedron with its center being occupied by a doubly charged oxygen vacancy (Figure 15), as confirmed by DFT(+U) calculations (Cheik Njifon, 2018; Talla Noutack, 2019). Herein, the greater the cation distribution homogeneity, the harder the Pu reduction (Vauchy et al., 2015c) and hence the formation of free oxygen vacancies, driving the force of cation interdiffusion in hypostoichiometry.

7.1.2 Interdiffusion in hyperstoichiometry (O/M > 2)

Hyperstoichiometry in AnO₂ is defined as an O/M ratio larger than the reference value 2. In this composition range, the interstitial oxygen atoms become predominant with respect to the concentration in oxygen vacancies, the metal lattice being again conserved (see Section 2). Compared to the atom in its “normal” site (Table 2), these interstitial oxygens have a larger ionic radius. The lattice then locally expands due to the steric hindrance. The presence of such interstitials is balanced by the metal lattice by inducing a partial oxidation of the cations to the pentavalent state. These oxidized ions have a smaller ionic radius than that of the tetravalent ones (Table 2), inducing a local shrinkage of the crystal cell. Again, the ambivalence of these defects creates tremendous local lattice distortions. The contraction induced by the formation of An(V) is greater than the local swelling generated by the presence of the interstitial anion. Macroscopically, the lattice shrinks proportionally to the magnitude of deviation from stoichiometry (Brett and Fox, 1966; Markin and Street, 1967; Sali et al., 2016).

Cation interdiffusion in hyperstoichiometric U_1−yPu_yO_2+x is mediated by the migration of free oxygen interstitials [presumably indirect mechanism (Dorado et al., 2010)]. The clustering effect of interstitial oxygen atoms becomes preponderant when the deviation from stoichiometry is increased. As a result, they become less mobile within the crystal structure, and the cation interdiffusion is then either stabilized or declined.

7.2 Computational approach

Carrying experimental studies on materials containing transuranium elements is difficult and can only be operated in specific, and very limited, nuclear facilities around the world (Vauchy et al., 2016b). Computation is a convenient complementary approach for appraising the diffusion mechanisms that take place in actinide dioxides.

Point defect chemistry is one of the tools used to interpret the diffusion phenomena in the fluorite structure of U_1−yPu_yO_2±x. The formation and migration energies of the crystal defects are either used (when available) or computed to estimate their mobility in the solid, hence providing information on the diffusion of these species. One common representation of the mixed oxide relies on the interconnection of three distinct sub-lattices: [U(III), U(IV), U(V), Pu(III), Pu(IV)]₁[O’(−II), Va]₂[O’(−II), Va]₁ standing for the normal cation, the normal anion and the interstitial anion lattices, respectively. This formalism is used for thermodynamic computations using the CALPHAD method and the TAF-ID database (Guéneau et al., 2011; Guéneau et al., 2021) and more recently for calculating diffusion properties with the cBΩ model (Chroneos et al., 2015; Saltas et al., 2016) and the DICTRA code (Moore et al., 2017; Chakraborty et al., 2020).

Atomistic approaches are also investigated using first-principles calculations based on density functional theory (DFT), often coupled to the Hubbard’s model (DFT + U), or empirical potentials (EPs). Molecular dynamics (MD) simulations can subsequently be carried out for computing the diffusivity of some species in actinide dioxides (Freyss et al., 2005; Dorado et al., 2010; Boyarchenkov et al., 2013; Cooper et al., 2015; Matthews et al., 2019; Wang et al., 2019; Nekrasov et al., 2021; Chen and Kaltsoyannis, 2022; Cooper, 2022).

Although being very useful for interpreting some fundamental properties, these studies focused on self-diffusion and/or on oxygen chemical diffusion phenomena. However, the underlying diffusion mechanisms may possibly be used for interpreting the experimental interdiffusion data.

Eventually, and as a direct engineering application, disposing of reliable experimental cation interdiffusion coefficients can be used for modeling the macroscopic U−Pu homogenization that occur during sintering using a finite elements method (FEM) (Léchelle et al., 2001; Léchelle et al., 2012; Dempowo et al., 2022).

8 Conclusion

Where are we now in the determination of cation interdiffusion in uranium–plutonium mixed oxide fuels? Answering this question remains difficult. Indeed, despite being studied for decades, experimental determinations of U–Pu interdiffusion coefficients in MOX fuel are scarce and highly scattered. The lack of a more systematic investigation of these diffusion properties is clear. A critical review of the literature data unfortunately led to exclude most of the proposed studies as the associated results were doubtful due to experimental approximations and biases, among others. Diffusion being a thermally activated phenomenon, interdiffusion is enhanced, at the first order, by an increase in temperature. Oxygen partial pressure also plays a major role in interdiffusion. In hypostoichiometry, cation diffusion is mediated by free oxygen vacancies, and an excessive decrease in pO₂ can induce severe clustering effects (oxygen vacancy traps), reverberating as a barrier to cation migration. In hyperstoichiometry, oxygen in the interstitial positions greatly enhance diffusion properties until reaching a plateau due to the competition between formation, coalescence, and dissociation of clusters. As a general conclusion, more reliable experiments are needed to properly understand the cation interdiffusion in MOX fuels, either for precisely tailoring sintering or for predicting in-pile behavior. At the light of this critical review of the published data, the preparation of new interdiffusion couples from dense sintered pellets and subsequent EPMA analysis appears to be the most relevant method to obtain the true interdiffusion coefficients of U and Pu in MOX fuels. Finally, the role of americium in the diffusion processes needs to be further discussed as most of the studies simply omit the presence of this daughter element of plutonium.

Author contributions

RV: conceptualization, data collection, formal analysis, writing, and original draft. SH: conceptualization, data collection, and formal analysis. TM: conceptualization, data collection, and formal analysis. MK: conceptualization, funding acquisition.

Acknowledgments

The visualization of crystal structures was performed with VESTA (Momma and Izumi, 2011).

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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