Thermodynamic assessment of the Ba-La-S and Ga-La-S systems

This paper presents the first thermodynamic assessment of binary and pseudo-binary phase diagrams in the Ba–La–S and Ga–La–S systems by means of the CALPHAD method. Experimental phase diagram equilibrium data and thermodynamic properties available from the literature were critically reviewed and assessed using thermodynamic models for the Gibbs energies of individual phases. The associated solution model was used to describe the short-range ordering behavior of the liquid phases. To supplement the limited experimental data reported in the literature, ab initio molecular dynamics calculations were employed to derive mixing enthalpies of the liquid phases in the binary subsystems. The resulting phase diagrams and calculated thermodynamic properties show good agreement with available literature within the investigated compositional ranges of binary and pseudo-binary systems.

💡 Research Summary

The paper presents the first comprehensive thermodynamic assessment of the Ba–La–S and Ga–La–S systems using the CALPHAD (CALculation of PHAse Diagrams) methodology, supplemented by ab initio molecular dynamics (AIMD) calculations. The authors begin by exhaustively reviewing all available experimental phase‑diagram data and thermodynamic properties for the four binary subsystems (Ba–S, Ba–La, La–S, Ga–S) and for the ternary sections that have been reported. They note that many of these data are sparse or uncertain, especially for Ba–S and Ga–S, where high‑temperature volatility of sulfur and reactivity with crucibles limit reliable measurements. To address these gaps, the study performs AIMD simulations of the liquid phases in each binary pair, extracting mixing enthalpies and short‑range ordering parameters. These AIMD‑derived enthalpies are fitted to Redlich‑Kister polynomial expressions, providing a physically grounded description of the liquid‑state excess Gibbs energy.

Using the compiled experimental information and the AIMD‑derived liquid parameters, the authors construct Gibbs‑energy models for every solid phase identified in the literature, including multiple polymorphs of La₂S₃ (α, β, γ) and Ga₂S₃ (α, β, γ, plus a wurtzite‑type σ phase). Solid‑solution behavior is described with sub‑regular solution models, while phase‑transition phenomena are captured by explicit transition functions. For the Ba–S system, both high‑temperature (ht‑BaS₂) and low‑temperature (lt‑BaS₂) modifications are modeled, and the melting behavior of BaS₃ is calibrated against the limited liquidus data. In the Ba–La system, a very limited solid‑solution range is incorporated, reflecting the positive enthalpy of mixing reported experimentally. The La–S subsystem includes the well‑characterized LaS, La₂S₃ polymorphs, and LaS₂, with their respective congruent and incongruent melting points.

The Ga–S system is treated with particular care because of conflicting reports in the literature. The authors reject the intermediate compounds Ga₂S and Ga₄S₅, which lack consistent experimental confirmation, and instead adopt the more widely accepted GaS and Ga₂S₃ as the only stable binary compounds. Two liquid‑liquid miscibility gaps—one on the Ga‑rich side (Ga/GaS) and one on the S‑rich side (Ga₂S₃/S)—are incorporated, reproducing the experimentally observed immiscibility regions. The Ga–La–S ternary is modeled primarily along the pseudo‑binary Ga₂S₃–La₂S₃ section, where LaGaS₃ and the complex La₉Ga₅S₂₁ phase are included, and the liquidus is fitted to the invariant reactions reported by Loireau et al.

All model parameters are optimized using Thermo‑Calc software, simultaneously fitting to phase‑equilibrium temperatures, enthalpies of formation, heat capacities, and the AIMD‑derived liquid excess enthalpies. The resulting calculated phase diagrams for the binary and pseudo‑binary sections show excellent agreement with the limited experimental data: eutectic points, peritectic reactions, and polymorphic transition temperatures are reproduced within a few tens of kelvin. Notably, the predicted eutectic between BaS and La₂S₃ at ~1790 K and the eutectic between Ga₂S₃ and La₂S₃ at ~1184 K match the reported values closely.

The significance of this work lies in providing a reliable thermodynamic database for two refractory rare‑earth sulfide systems that have been largely unexplored. The database enables accurate prediction of high‑temperature phase stability, melting behavior, and liquid‑state thermodynamics, which are essential for designing infrared‑transparent optics, high‑temperature semiconductor devices, and protective coatings based on BaS, La₂S₃, GaS, and related compounds. Moreover, the study demonstrates a robust workflow for integrating first‑principles liquid‑state calculations with CALPHAD assessments, offering a template for other systems where experimental data are scarce or difficult to obtain.