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.

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.

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).

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).

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).

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.

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): J = − D ∂ C ∂ x (1) where ∂ C / ∂ 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.

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 D ∼ . In this sole ideal case, the acquisition of only one of these profiles is necessary to obtain 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: ∂ C ∂ t = D ∂ ² C ∂ 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).

From this representation, Eq. 3 gives the resolution of the diffusion equation: D ∼ = − 1 2 t ∫ C 1 C 2 ( x − x M ) ∂ C ∂ C ∂ x | 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): D ∼ = N A . D B + N B . D 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.

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.

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.

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., 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 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.

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 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, 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).

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 D ∼ upon temperature and O/M ratio. The “sintering” data were rejected as they should not be directly compared to the other studies.

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).

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

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 ( – I I ) 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 A n ( I V ) can be calculated from the cubic unit cell diagonal, a 3 . Indeed, in the fluorite structure, the hard sphere model is expressed as Eq. 5. a × 3 = 4 × r O ( – I I ) + 4 × r A n ( I V ) (5) As the lattice parameter a of the dioxide can be measured using regular X-ray diffraction, giving r O ( – I I ) , the ionic radius of the actinide r A n ( I V ) 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).

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

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).