Uncertainty Quantification of First Principles Computational Phase Diagram Predictions of Li-Si System Via Bayesian Sampling

Abstract

Within the field of computational materials discovery, the calculation of phase diagrams plays a key role. Uncertainty quantification for these phase diagram predictions enables a quantitative metric of confidence for guiding design in computational materials engineering. In this work, an assessment of the CALPHAD method trained on only density functional theory (DFT) data is performed for the Li-Si binary system as a case study. with applications to the modeling of Si as an anode for Li-ion batteries. Using a parameter sampling approach based on the Bayesian Error Estimation Functional (BEEF-vdW) exchange-correlation. By using built-in ensemble of functionals from BEEF-vdW, the uncertainties of the Gibbs Free Energy fitting parameters are obtained and can be propagated to the resulting phase diagram. To find the best fitting form of the CALPHAD model, we implement a model selection step using the Bayesian Information Criterion (BIC) applied to a specific phase and specific temperature range. Applying the best selected CALPHAD model from the DFT calculation, to other sampled BEEF functionals, an ensemble of CALPHAD models is generated leading to an ensemble of phase diagram predictions. The resulting phase diagrams are then compiled into a single-phase diagram representing the most probable phase predicted as well as a quantitative metric of confidence for the prediction. This treatment of uncertainty resulting from DFT provides a rigorous way to ensure the correlated errors of DFT is accounted for in the estimation of uncertainty. For the uncertainty analysis of the single-phase diagram of the Li-Si system, we explore three different methods using BEEF as three kinds of samplers with various assumptions of statistical independence: independent points of phases, independent pairs of phases, and independent convex hulls of phases. We find that each method of propagating the uncertainty can lead to different phases being identified as stable on the phase diagram. For example, the phase Li 4.11 Si at 300K is predicted to be stable by all functionals using the second and third method, but only 15% of functionals predict it to be stable using the first method. From the phase diagram, we have also determined intercalation voltages for lithiated silicon. We see that the same phase can have a distribution of predicted voltages depending on the functional. In combination, we can generate a better understanding of the phase transitions and voltage profile to make a more analysis-informed prediction for experiments and the performance of Si-anodes within batteries.

MeltNet: Predicting alloy melting temperature by machine learning

“…To quantify the uncertainty associated with such choice, the ensemble approach was employed here, which has been applied successfully in other fields of computational materials science, e.g., the Bayesian Error Estimation Functional (BEEF) where an ensemble of density functionals are used to estimate the exchange-correlation errors 24 . Especially, the ensemble approach has been applied in thermochemical properties 25 and phase diagrams 26,27 . In the present work, the training dataset was sampled randomly for 100 times forming 100 subsets, and each subset contains 75% of the total training data.…”