Sensitivity estimation for calculated phase equilibria

Richard Otis,Zi‐Kui Liu,Brandon Bocklund

doi:10.1557/s43578-020-00073-6

Richard Otis, Zi‐Kui Liu + Show 1 more

Open Access

https://doi.org/10.1557/s43578-020-00073-6

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Abstract

The development of a consistent framework for Calphad model sensitivity is necessary for the rational reduction of uncertainty via new models and experiments. In the present work, a sensitivity theory for Calphad was developed, and a closed‐form expression for the log‐likelihood gradient and Hessian of a multi‐phase equilibrium measurement was presented. The inherent locality of the defined sensitivity metric was mitigated through the use of Monte Carlo averaging. A case study of the Cr–Ni system was used to demonstrate visualizations and analyses enabled by the developed theory. Criteria based on the classical Cramér–Rao bound were shown to be a useful diagnostic in assessing the accuracy of parameter covariance estimates from Markov Chain Monte Carlo. The developed sensitivity framework was applied to estimate the statistical value of phase equilibria measurements in comparison with thermochemical measurements, with implications for Calphad model uncertainty reduction.

Highlights

Calphad-based thermodynamic models are routinely used to probe the phase stability in multicomponent systems
Computational efficiency and the ability to incorporate experimental measurements, atomistic simulations, and expert intuition in a semi-empirical fashion have led to the broad adoption of the Calphad approach, but it is only in recent years that serious attention has been paid to uncertainty quantification (UQ) of the model predictions
Clear definitions must be given to all observation types, including multi-phase equilibrium information, commonly referred to as “phase diagram data.”

Summary

Introduction

Calphad-based thermodynamic models are routinely used to probe the phase stability in multicomponent systems. If a candidate model mis-predicts the presence of a phase γ, an experimental observation of α/β equilibrium under the same conditions defines a log-likelihood that is locally a function of the parameters of all three phases. The approach taken in this work was to define an observation in terms of each measured phase region, such that each dataset consisted of multiple “observations,” all assumed statistically independent.

Results

Conclusion