Predicting mixing enthalpy of liquid alloys with artificial neural networks

H.-L. Chen¹*, Q. Chen¹

1 Thermo-Calc Software AB

*hailin@thermocalc.com

The mixing enthalpy of liquid alloys is a thermodynamic property that plays a fundamental role in the thermodynamic construction of phase diagrams, understanding solidification behavior, alloy design, and process optimization. The semi-empirical Miedema model has been used to predict the mixing enthalpy of liquid alloys for about 50 years [1-4], but its accuracy remains to be improved. This work aims to more accurately predict the mixing enthalpy of liquid alloys by using artificial neural networks (ANNs). Elemental properties were selected from the Magpie Python module [5] and used to build features for training ANNs. The importance of each feature was evaluated with an algorithm similar to that in the ELI5 library [6]. Nine elemental properties were found relevant to the mixing enthalpy, such as electronegativity, group number, V (atomic volume), $n_{ws}^{1/3}$ (, where $n_{ws}$ is the electron density at the boundary of a Wigner-Seitz cell), ∅ (work function), etc. It is worth mentioning that the latter three properties are parameters in the Miedema model. Moreover, compound features of the properties were derived based on the Miedema formula [4]. They helped to simplify the architecture of ANNs and made it possible to train reliable models based on shallow neural networks.

Data were generated on mixing enthalpy of liquid alloys for 1073 binary systems using our thermodynamic databases. Typically, a system has been assessed several times with the CALPHAD approach [7] based on experimental phase equilibria, thermochemic measurements, and possible theoretical values. Different versions of assessments can be slightly or significantly different from each other. At the very beginning, a simple criterion was used to evaluate the reliability of the datasets for each system. Systems were considered reliable if they were assessed more than once with a small deviation between the assessments. For such a system, the dataset close to the mean value was used for training. According to this criterion, 220 binaries were initially selected for feature engineering/selection and optimizing the model architecture. Once models with decent performances were derived, they were utilized to make predictions for all systems. The reliability of a system could then be evaluated by comparing the expected and predicted values. A collection of a new set of reliable systems were then used for tuning hyperparameters and training new models. The new models were expected to generalize better and predict more accurately. Therefore, more reliable systems could be identified and included in the training. After several iterations, the optimal model performance was able to be reached.

Eventually, 853 (about 80 %) of the 1073 binary systems were used for training and validating the final models. 100 models were trained with different folds of training and validation sets. The ensemble of 100 models has been evaluated to have an average validation score (R² score) of 0.96 and outperforms the best model in the ensemble in terms of generalizability. It will be demonstrated that the resulting ANNs models are much more reliable than the semi-empirical Miedema model for predicting the mixing enthalpy of liquid alloys.

References

[1] A.R. Miedema. Philips Tech. Rev., 1973, 33, 149-160.

[2] F.R. de Boer, et al. Cohesion in Metals. Transition Metal Alloys, 1988.

[3] R.F. Zhang, et al. Computer Phys. Commun., 2016, 209, 58-69.

[4] Cai Li, et al. Phys. B, 2022, 627, 413540.

[5] L. Ward, et al. npj Comput. Mate., 2016, 2, 16028.

[6] https://eli5.readthedocs.io/en/latest/blackbox/permutation_importance.html

[7] H.L. Lukas, S.G. Fries, B. Sundman, Computational thermodynamics: the Calphad method, New York, 2007.