A statistical understanding of oxygen vacancies in distorted high-entropy oxides

Technical Summary: A Statistical Understanding of Oxygen Vacancies in Distorted High-Entropy Perovskite Oxides

Problem Statement
High-entropy perovskite oxides (HEPOs) have emerged as promising electrode materials for solid oxide electrolyzer cells (SOECs) due to their potential for high-temperature chemical and structural stability. However, the compositional complexity of HEPOs, particularly those with high-entropy A-sites, makes predicting the formation of oxygen vacancies ($\delta$) challenging. Oxygen vacancies are critical for oxygen ionic conductivity and influence thermal expansion and electrical conductivity. While traditional models rely on the concentration of divalent A-site cations ($X_{2+}$) to predict vacancy formation, these models fail to account for the complex interactions in high-entropy systems where cation size mismatches induce lattice distortions. Existing computational approaches often calculate a single mean vacancy formation energy ($E_v$), which does not consistently align with experimental measurements of vacancy concentrations in disordered materials.

Methodology
The authors employed a combined experimental, computational, and theoretical workflow to investigate oxygen vacancy behavior in 14 perovskite oxides (11 high-entropy and 3 low-entropy) derived from the LSCF ($La_{0.6}Sr_{0.4}Co_{0.2}Fe_{0.8}O_{3-\delta}$) system.

1. Experimental Characterization:

   • Samples were synthesized via solution combustion synthesis, annealed, and ball-milled.
   • Thermogravimetric analysis (TGA) was conducted from 500°C to 1000°C in air to measure changes in oxygen non-stoichiometry ($\Delta\delta_T$).
   • Data was fitted to a defect reaction model to extract formation enthalpy ($\Delta H_f$), formation entropy ($\Delta S_f$), and initial vacancy concentrations ($\delta_0$).
   • Compositional descriptors were calculated, including the divalent cation mole fraction ($X_{2+}$), Goldschmidt tolerance factor ($t$), mixing entropy ($\Delta S_{mix}$), and a newly emphasized A-site ionic radius variance ($\sigma_A$).

2. Atomistic Simulations:

   • High-throughput simulations were performed using a machine-learned universal interatomic potential (MLUIP, Matlantis) on large supercells (1280 atoms).
   • The study sampled hundreds of oxygen sites to calculate the distribution of vacancy formation energies ($g(E_v)$) rather than a single mean value.
   • Structural analysis quantified the variance in B-O bond lengths and angles resulting from A-site size mismatches.

3. Theoretical Framework:

   • A statistical thermodynamic model was derived treating oxygen vacancies as ideal Fermions within a grand canonical ensemble.
   • This model accounts for the variance in vacancy energy ($\sigma_{E_v}$) to derive corrected expressions for $\Delta H_f$ and $\Delta S_f$, moving beyond the assumption of uniform vacancy sites.

Key Contributions and Results

• Identification of $\sigma_A$ as a Critical Descriptor: Experimental data revealed that while $X_{2+}$ sets an upper limit for vacancy concentration, the A-site size variance ($\sigma_A$) significantly influences the temperature dependence of vacancy formation. High-entropy samples with larger $\sigma_A$ exhibited lower formation enthalpies ($\Delta H_f$) and entropies ($\Delta S_f$) compared to low-entropy samples with similar $X_{2+}$.
• Enthalpy-Entropy Compensation: The study observed a strong linear correlation between $\Delta H_f$ and $\Delta S_f$ across the samples. Theoretical analysis attributes this "compensation" to the broadening of the vacancy energy distribution ($\sigma_{E_v}$).
• Statistical Nature of Vacancy Formation: Simulations confirmed that increased $\sigma_A$ induces lattice distortions, broadening the distribution of B-O bond energies. This results in a broader distribution of vacancy formation energies ($g(E_v)$).
  • The "tail" of the distribution (lower energy sites) allows vacancies to form more readily at lower temperatures.
  • The statistical model demonstrates that broadening $g(E_v)$ reduces both $\Delta H_f$ and $\Delta S_f$ compared to a uniform model. Specifically, $\Delta H_f \approx \hat{E}_v - \frac{\sigma_{E_v}^2}{k_B T}$ and $\Delta S_f$ is similarly reduced due to decreased configurational entropy when preferred sites exist.
• Model Validation: The statistical model, which incorporates the variance $\sigma_{E_v}$, accurately predicted the experimentally measured oxygen vacancy concentrations and their temperature dependence (curvature). In contrast, traditional models using only the mean energy ($\hat{E}_v$) failed to differentiate between the linear growth of vacancies in HE samples and the exponential growth in LE samples.
• Quantitative Agreement: The experimentally derived $\sigma_{E_v}$ values, extracted by fitting the statistical model to TGA data, closely matched the values sampled directly from the MLUIP simulations, validating the theoretical approach.

Significance
The paper claims that treating oxygen vacancies statistically is essential for understanding and predicting the behavior of high-entropy perovskite oxides. The primary significance lies in identifying A-site size variance ($\sigma_A$) as a practical "knob" for tuning material properties. By increasing $\sigma_A$, researchers can broaden the vacancy energy distribution, which lowers the formation enthalpy and entropy. This results in oxygen vacancy concentrations that are less sensitive to temperature changes (flatter $\delta$ vs. $T$ curves).

Consequently, properties dependent on vacancy concentration, such as thermal expansion and ionic conductivity, can be tuned to be more stable across operating temperatures. The study provides a guiding principle for selecting cation dopants in complex perovskite oxides, suggesting that structural distortion via cation size mismatch is a viable strategy for optimizing electrode materials for SOECs, extending beyond just high-entropy systems to any oxide with energetically heterogeneous sites.