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Zhang et al. J. Mater. Inf. 2026, 6, 11 Page 11 of 17

Figure 6. Bonding and charge disorder analysis in RE 3 TaO 7 and RE 2 Zr 2 O 7 oxides. (A) Relative bond lengths for RE–O (left) and Zr/Ta–O
(right) polyhedra, where “REO 7 ”, “REO 8 ” and “REO 7/8 ” denote the 7- and 8-fold and mixed-coordination RE–O environments. Insets
illustrate representative polyhedral units; (B) Bond-length disorder degree D b of RE–O (left) and Zr/Ta–O (right) polyhedra, using the
same symbol scheme as in (A); (C) Valence charges of cations RE (left) and anions O (right), derived from Bader charge analysis; (D)
2-
3+
Valence charge disorder degree D e for RE (D e ) and O (D e ), quantifying charge inhomogeneity within coordination polyhedra. RE:
2-
O
3+
RE
Rare-earth.
lower κ of RE TaO oxides compared to zirconates can be partly attributed to the greater structural
3
L
7
complexity and potential for bond length variation in the lower-symmetry C222 structure. Within the
1
RE TaO oxides, smaller RE ions lead to shorter, stronger bonds, but the interplay with bond length
3
7
heterogeneity dictates the overall κ . Additionally, K is also intrinsically linked to the bonding environment.
L
IC
The trend of increasing K with decreasing r in RE TaO oxides correlates with the formation of shorter,
3
IC
RE3+
7
stronger average RE–O bonds, which increases the intrinsic resistance to bond breaking at the crack tip [66,67] ,
as shown in Figure 4. The generally higher toughness of pyrochlore zirconates suggests their specific ordered
structure provides more effective toughening mechanisms compared to the C222 RE TaO oxides.
3
7
1
In Figure 6C and D, electronic structure insights from Bader charge analysis further clarify
structure–property relationships. Most RE ions exhibit Bader charges consistent with a nominal +3 oxidation
state. However, Eu and Yb in the RE TaO system show significantly lower calculated charges, indicative of a
7
3
preference for the divalent (+2) state. As shown in Figure 6D, this deviation introduces local charge
imbalance and strain fields, broadening the bond-length distribution and increasing D , which in turn
e
destabilizes the ordered C222 structure (reflected in higher relative E , as shown in Figure 2). Interestingly,
form
1
such pronounced effects are not observed in the corresponding zirconates, suggesting the more flexible
fluorite-derived framework of the zirconates is better able to accommodate such charge and strain
fluctuations. These mechanisms provide vital guidelines for strategically balancing K and thermal insulation
IC
in RE-based oxide ceramics for advanced TBC applications.
Quantifying structure–property relationships
To establish direct links between atomic-scale descriptors and macroscopic thermomechanical behavior,
quantitative correlation and interpretability analyses were conducted. Figure 7 presents Pearson correlation
coefficient matrices and SHAP feature importance rankings for both RE TaO and RE Zr O oxides. The
3
7
2
7
2
SHAP package was utilized to interpret the ML models effectively . These tools collectively identify the
[68]
most influential parameters controlling κ and K , including the r , V 0, E 0, E , bond energy (E ) for RE–O,
b
L
IC
RE3+
form
Ta–O and Zr–O bonds, and disorder metrics such as bond-length heterogeneity (D ) and Bader charge
b
disorder (D ). Hierarchical clustering within the correlation matrices further highlights groups of interrelated
e
descriptors, as visualized through dendrograms and colored sidebars.