“…The fractions approach the values for the random distribution of cations (60, 35, and 5% for the Zr–Zr, Sc–Zr, and Sc–Sc pairs, respectively) with increasing temperature, although the distribution is not yet completely random at a sintering temperature of 1627 °C (The situation is quite different in 30 at % Y-doped barium zirconate, for which the fraction of Y–Y pairs is 28%, much higher than that for a statistical random cation distribution at 1600 °C). In step 2, a special representative structure − that matches the averaged 2, 3, and 4-body correlation functions of the AL-RXMC samples from step 1 was generated at each temperature (red symbols in Figure S2). In the final step 3, AL-RXMC sampling of oxygen vacancy and proton configurations at thermal equilibrium were performed at 0, 33, 66, and 100% hydration (100% hydration corresponds to the composition where all oxygen vacancies are filled by oxide ions with a proton concentration of 0.222 per formula unit), while keeping the cation arrangement fixed to the single representative structure determined at 1627 °C in step 2 [To estimate whether equilibrium can be reached within the experimental conditions, we calculated the diffusion lengths of oxide ions from the reported activation energy of 89 kJ/mol and Arrhenius pre-factor of 3.40 × 10 –5 cm 2 /s for Y-doped barium zirconate (Sc-doped barium zirconate has not been reported).…”