A Generalized Solid Solution Framework for the Gibbs Free Energy Calculation

We propose a generalized solid solution model for calculating configurational contribution to the Gibbs free energy at finite temperatures, incorporating a crystal graph-based on-site energy approach. By leveraging linear graph neural networks, our method unifies pair-based and cluster expansion approaches, enabling broad applicability across crystal structures. Fractional occupation is physically interpreted via mean-field theory, while entropy is modeled using ideal mixing with extended site constraints. To resolve the constants of compositions, we implement three key strategies. First, we employ a softmax-based variable transformation. Second, we introduce a gradient projection method that preserves species composition throughout the optimization process by constraining updates within a subspace that maintains the desired elemental ratios. Finally, a renormalization step is incorporated to correct numerical deviations, ensuring strict adherence to the target composition. We then apply our model to the Mo-Nb-Ta-W quaternary system, achieving an energy model MAE of 1.24 meV. Predicted phase transition temperatures for equal atomic binary alloys align well with expectations, identifying phase separation in MoNb and order-disorder transitions in MoTa, MoW, TaW, NbTa, and NbW. At low temperatures, stable configurations lie below the convex hull of the training data set, demonstrating the model's predictive accuracy. Further analysis of MoNbTaW reveals transition temperatures at 950 and 400 K, with observed asymmetry in Mo/W sublattices. Finally, we extend our approach to ternary phase diagram predictions using Gibbs free energy interpolation and second-derivative analysis, yielding phase diagrams in agreement with optimized atomic configurations.