# Oxygen Self-Diffusion in Fluorite High Entropy Oxides
## 3 Results and Discussion  
In the present work, we studied three general compositions of high-entropy oxides based upon reported experimental synthesis and the availability of oxygen ion potentials to describe their behavior. These are Zr\({}_{(1-x)/3}\)Hf\({}_{(1-x)/3}\)Ce\({}_{(1-x)/3}\)Gd\({}_{x/2}\)Y\({}_{x/2}\)O\({}_{2-x}\) (HEO_A), Zr\({}_{(1-x)/3}\)Hf\({}_{(1-x)/3}\)Ce\({}_{(1-x)/3}\)Y\({}_{x/2}\)La\({}_{x/2}\)O\({}_{2-x}\) (HEO_B) and Zr\({}_{(1-x)/2}\)Hf\({}_{(1-x)/2}\)Y\({}_{x/3}\)La\({}_{x/3}\)Pr\({}_{x/3}\)O\({}_{2-x}\) (HEO_C). Figure 1 shows the calculated values of diffusivity against temperature for the two considered HEOs compared with the calculated values for CeO\({}_{2}\) + 8% Y\({}_{2}\)O\({}_{3}\), C8Y, and ZrO\({}_{2}\) + 8% Y\({}_{2}\)O\({}_{3}\), Z8Y. At these compositions, the oxygen ion diffusivity is generally lower than the ternary oxides. We see generally good agreement with the Arrhenius behavior, and the fitted values of activation energy are in agreement with those reported in the literature. Figure 1 also shows the importance of a statistical approach to the calculations. For a given temperature, there is a scatter in the range of values. For Z8Y, this is quite small but for the HEO compositions, there are significant differences in diffusivity for different configurations within a single composition, and this is evident even with the quite large 6000 atom cells considered here. Top image of Figure 2 shows the ionic diffusivity values at 1000 K for the three HEO compositions and ZrO\({}_{2}\) + Y\({}_{2}\)O\({}_{3}\) for comparison as a function of the total trivalent dopantconcentration, i.e., the sum of the La, Y, Pr and Gd molar fractions. The datapoints for ZY show a rapid increase with Y-doping, a maximum around 10% molar fraction Y (~6% Y\({}_{2}\)O\({}_{3}\)), and a slow decrease in diffusivity. This behavior has been extensively described in the literature as a balance between the creation of charge compensation oxygen vacancies and the trapping of the vacancies through the addition of large densities of trivalent ions. Similar curves are observed for the HEO compositions; however, in these cases, the overall diffusivity is generally lower, particularly for the maximum peak value of diffusivity at around 10% dopant concentration. We explored this behavior in middle and bottom images of Figure 2, where the points show the predicted values of \(D_{0}\) and \(E_{\text{a}}\) for the different compositions. Middle image of Figure 2 indicates that the activation energy for diffusion for ZY decreases to a value of 0.4 eV, comparable with that of oxygen vacancy diffusion in the dilute limit. For the HEOs, however, the activation energy for diffusion tends to be a significantly higher value of around 0.6-0.75 eV, depending upon the composition. The value of \(D_{0}\) calculated from the simulations varies with total aliovalent doping concentration but is relatively unchanged between the different compositions considered. The decrease in the diffusivity in the HEOs as compared with the ZY is therefore due to this increased activation energy, particularly when comparing the data up to ~10% molar dopant concentration. We considered why the HEO compositions exhibited these levels of oxygen ion diffusion, focusing specifically on the reduced diffusion because of the increased activation energy exhibited in top and middle images of Figure 2. Oxygen diffusion is mediated via vacancies moving through the fluorite lattice, a process that, in the doped materials, is hindered both by vacancy-vacancy interactions, Coulomb attraction, and trapping by the trivalent dopant ions. Figure 3a shows an illustration of the fluorite lattice with an example oxygen vacancy in the nearest neighbor position to a Y\({}^{3+}\) ion, and the minimum energy pathway for this cell as the vacancy exchanges places with an adjacent oxygen ion. Figure 3b shows both the energy maxima at the transition point and the finite reaction energy, \(E_{\text{r}}\), as the sites are not symmetrically equivalent.