Environmental barrier coatings (EBCs) are crucial for gas turbine technology, protecting SiC/SiC ceramic matrix composites (CMCs) from corrosion by CMAS (CaO-MgO-Al₂O₃-SiO₂) and water vapor at high temperatures. Rare-earth disilicates (RE₂Si₂O₇) are promising EBC materials due to their CMAS/water vapor resistance, low thermal conductivity, and CTE compatibility with CMCs. However, their performance suffers under combined thermal steam, molten CMAS, and thermal stress attacks, partly due to complex phase transformations (α, β, γ, δ, ε, and G) and the β-γ polymorphic transition that can cause cracking or delamination. Multi-component RE₂Si₂O₇ systems offer potential advantages due to high configurational entropy, slow diffusion, lattice distortions, and mixing effects. The challenge is determining if a given composition forms a single-phase RE₂Si₂O₇. Machine learning (ML) has emerged as a powerful tool for materials discovery and prediction, bypassing limitations of linear combinations of descriptors. While ML has been applied to various materials, effective models for β- and γ-(RE)₂Si₂O₇ are scarce. This research aims to develop and visualize an ML model to predict (RE)₂Si₂O₇ phases using relevant (RE)₂Si₂O₇ and RE₂Si₂O₇ characteristics.

Prior research highlights the importance of rare-earth disilicates in EBCs due to their superior resistance to CMAS corrosion and their closely matched thermal expansion coefficients with CMC substrates. However, the complex phase transformations in these materials, particularly the β-γ transition, present significant challenges for designing durable and stable coatings. Studies have shown that the performance of single-component RE₂Si₂O₇ degrades significantly under high-temperature, multi-stress conditions. The use of multi-component systems, high-entropy alloys, and solid solutions has been explored as a strategy to improve phase stability and mitigate these issues. Machine learning (ML) approaches have been successfully employed in predicting the phase formation ability of various material systems, including high-entropy alloys and carbides. However, the application of ML to the design of high-entropy rare-earth disilicates for EBC applications remains relatively unexplored. This gap motivated the current research.

The study utilized a machine learning framework integrating three models: Support Vector Classification (SVC), Artificial Neural Network (ANN), and Random Forest Classification (RFC). The dataset comprised experimental data from published sources, classifying phases as multiple, β, and γ. Seven input features (ionic radius deviation (σᵣ), average RE ionic radius, mass deviation (m), average mass, electronegativity deviation (χ), average electronegativity, and mixing entropy (ΔS)) were selected based on their relevance to phase stability. Feature selection employed Pearson correlation coefficient analysis to eliminate highly correlated features (r > 0.90). Features were then normalized. Model training used random splitting (training/testing ratios of 0.85/0.15, 0.8/0.2, 0.75/0.25) and a grid search to optimize hyperparameters for each model. Model performance was evaluated using validation accuracy, confusion matrices, ROC curves, and AUC scores. The RFC model showed the best performance (validation accuracy of 1.000). To interpret the RFC model, SHAP (Shapley Additive exPlanations) analysis was used to determine feature importance and visualize the influence of each feature on the prediction of pure (RE₁₀.₂₅RE₂₀.₂₅Yb₀.₂₅Lu₀.₂₅)₂Si₂O₇. The trained models predicted the phase formation capability of (RE₁₀.₂₅RE₂₀.₂₅Yb₀.₂₅Lu₀.₂₅)₂Si₂O₇ (RE = La, Ce, Eu, Gd, Tb, Dy, Ho, Er, Tm, Y), which was validated by Density Functional Theory (DFT) calculations. A decision fusion strategy, combining Bayesian theory and majority voting, was applied to optimize the RFC model for predicting single-phase (RE₁₁/₆RE₂₁/₆RE₃₁/₆Gd₁/₆Yb₁/₆Lu₁/₆)₂Si₂O₇ formation. DFT calculations were used to verify the thermodynamic stability of the predicted phases. The mixing Gibbs free energy was calculated to assess phase stability against temperature.

The RFC model achieved a validation accuracy of 1.000 for predicting the phase formation of (RE₁₀.₂₅RE₂₀.₂₅Yb₀.₂₅Lu₀.₂₅)₂Si₂O₇. SHAP analysis revealed that average RE³⁺ ionic radius and ionic radius deviation were the most critical factors influencing phase formation. Smaller ionic radius deviation (σᵢ) favored the formation of pure (RE₁₀.₂₅RE₂₀.₂₅Yb₀.₂₅Lu₀.₂₅)₂Si₂O₇, while larger values led to phase separation. DFT calculations confirmed the thermodynamic stability of 17 predicted pure (RE₁₀.₂₅RE₂₀.₂₅Yb₀.₂₅Lu₀.₂₅)₂Si₂O₇ compounds (3 β, 14 γ). For (RE₁₁/₆RE₂₁/₆RE₃₁/₆Gd₁/₆Yb₁/₆Lu₁/₆)₂Si₂O₇, the decision fusion-enhanced RFC model accurately predicted the formation of 35 single phases (7 β, 28 γ) out of 84 tested compositions. SHAP analysis for this system highlighted the importance of electronegativity deviation (σₓ) in addition to ionic radius-related features. The formation of β phases was associated with σₓ < 0.900 Å, small mass deviation (σₘ), σₓ < 0.046, and m < 478 g·mol⁻¹. The Y element showed the highest frequency in the predicted β-(RE₁₁/₆RE₂₁/₆RE₃₁/₆Gd₁/₆Yb₁/₆Lu₁/₆)₂Si₂O₇ phases. The model's robustness was demonstrated by successfully re-predicting the phase formation of (RE₁₀.₂₅RE₂₀.₂₅Yb₀.₂₅Lu₀.₂₅)₂Si₂O₇ using the decision fusion model.

The high accuracy of the ML models in predicting the phase formation of both quaternary and hexanary rare-earth disilicates demonstrates the effectiveness of this approach for materials design. The SHAP analysis provided valuable insights into the factors governing phase stability, revealing the interplay between ionic radius, mass, and electronegativity. The agreement between ML predictions and DFT calculations validates the models' reliability. The results highlight the importance of controlling ionic radius deviation to achieve single-phase materials. The decision fusion strategy significantly improved the model's performance, especially for the more complex hexanary system. This methodology can accelerate the discovery and development of novel high-entropy rare-earth disilicates with tailored properties for EBC applications. The insights gained from this study contribute to a deeper understanding of the structure-property relationships in these complex materials.

This study successfully employed machine learning and decision fusion to predict the phase formation capability of high-entropy rare-earth disilicates. The developed models accurately predicted the phases of both quaternary and hexanary systems, validated by DFT calculations. SHAP analysis provided insights into critical factors influencing phase stability. This integrated ML-based approach enables the efficient design of advanced EBC materials. Future research could focus on predicting other properties (e.g., thermal conductivity, mechanical strength) using similar ML methods, and experimental synthesis and characterization of the predicted stable high-entropy rare-earth disilicates to validate the predicted properties.

The accuracy of the ML models relies on the quality and quantity of the training data. The current study utilized a dataset primarily from existing literature, which may contain inherent limitations. The DFT calculations were computationally expensive, limiting the number of compositions that could be verified. The model primarily focuses on phase prediction; further investigations are needed to predict other properties, such as thermal conductivity and mechanical strength, of these high-entropy disilicates.