The design of environmentally friendly solder alloys requires an integrated understanding of thermodynamics, structure–property relationships, and predictive modeling. In this work, the Bi–In–Sn ternary alloy system—a promising lead-free solder candidate—is investigated using a combined CALPHAD (Calculation of Phase Diagrams)–machine learning (ML)–experimental framework. Thermodynamic datasets were generated from FACTSage, encompassing Gibbs free energy, enthalpy of mixing, and activity over wide composition and temperature ranges. Physically meaningful descriptors, including atomic size mismatch (δ), electronegativity difference (χ), valence electron concentration (VEC), and temperature (T), were employed to embed thermodynamic constraints into the ML models. Multiple regression algorithms—gradient boosting, random forests, XGBoost, polynomial regression, LASSO (least absolute shrinkage and selection operator), and neural networks—were systematically trained and evaluated. To rigorously assess model generalization beyond randomly sampled conditions, both conventional random splitting and temperature-based splitting strategies were adopted, enabling explicit evaluation of extrapolative performance toward previously unseen temperature regimes. Ensemble-based models and neural networks demonstrated robust predictive capability under both splitting strategies, achieving coefficients of determination (R2) exceeding 0.999 across training, test, and cross-validation datasets. Experimental validation was performed using drop calorimetry measurements of enthalpy of mixing (767–855 K) and electromotive force (EMF) measurements of indium activity. The close agreement between ML predictions and experimental data confirms the reliability and transferability of the proposed framework. To enhance model interpretability, SHAP (SHapley Additive exPlanations) was employed to quantify descriptor contributions across different data-splitting strategies. The SHAP analysis consistently revealed a physically interpretable descriptor hierarchy (χ > VEC > δ > T), demonstrating that model predictions are governed by chemically and thermodynamically meaningful factors rather than spurious correlations. This integrated CALPHAD–ML–experimental approach provides a generalizable and interpretable surrogate modeling framework that reduces dependence on dense experimental datasets while accelerating the design of lead-free solder alloys with tailored thermodynamic properties. The methodology is readily extendable to other multicomponent alloy systems, including high-entropy alloys and complex solder formulations.

Integrated CALPHAD–Machine Learning–Experimental Framework for Thermodynamic Design of Lead-Free Bi–In–Sn Solder Alloys

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发表日期：

2026年5月14日

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Shanker Kumar,
Mukesh Raushan Kumar

摘要

The design of environmentally friendly solder alloys requires an integrated understanding of thermodynamics, structure–property relationships, and predictive modeling. In this work, the Bi–In–Sn ternary alloy system—a promising lead-free solder candidate—is investigated using a combined CALPHAD (Calculation of Phase Diagrams)–machine learning (ML)–experimental framework. Thermodynamic datasets were generated from FACTSage, encompassing Gibbs free energy, enthalpy of mixing, and activity over wide composition and temperature ranges. Physically meaningful descriptors, including atomic size mismatch (δ), electronegativity difference (χ), valence electron concentration (VEC), and temperature (T), were employed to embed thermodynamic constraints into the ML models. Multiple regression algorithms—gradient boosting, random forests, XGBoost, polynomial regression, LASSO (least absolute shrinkage and selection operator), and neural networks—were systematically trained and evaluated. To rigorously assess model generalization beyond randomly sampled conditions, both conventional random splitting and temperature-based splitting strategies were adopted, enabling explicit evaluation of extrapolative performance toward previously unseen temperature regimes. Ensemble-based models and neural networks demonstrated robust predictive capability under both splitting strategies, achieving coefficients of determination (R²) exceeding 0.999 across training, test, and cross-validation datasets. Experimental validation was performed using drop calorimetry measurements of enthalpy of mixing (767–855 K) and electromotive force (EMF) measurements of indium activity. The close agreement between ML predictions and experimental data confirms the reliability and transferability of the proposed framework. To enhance model interpretability, SHAP (SHapley Additive exPlanations) was employed to quantify descriptor contributions across different data-splitting strategies. The SHAP analysis consistently revealed a physically interpretable descriptor hierarchy (χ > VEC > δ > T), demonstrating that model predictions are governed by chemically and thermodynamically meaningful factors rather than spurious correlations. This integrated CALPHAD–ML–experimental approach provides a generalizable and interpretable surrogate modeling framework that reduces dependence on dense experimental datasets while accelerating the design of lead-free solder alloys with tailored thermodynamic properties. The methodology is readily extendable to other multicomponent alloy systems, including high-entropy alloys and complex solder formulations.