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In this work, a long-established but sparsely documented method of obtaining semi-analytic derivatives of thermodynamic properties with respect to equilibrium conditions is briefly reviewed and rigorously derived. This procedure is then leveraged to construct general forms of derivatives of the residual driving force, a metric for measuring phase stability used in CALPHAD model optimization, with respect to overall system and individual phase compositions. Applied examples – calculating heat capacity in the Al-Fe system, thermodynamic factors in the Nb-V-W system, and residual driving force derivatives in the Ni-Ti system – demonstrate the versatility, accuracy, and extensibility of this method. Using the developed method, residual driving force gradients can be applied directly in CALPHAD model optimizers, as well as in materials design frameworks, to identify regions of phase stability with an efficient, gradient-based approach. 

The Cluster expansion (CE) is a powerful method for representing the energetics of alloys from a fit to first principles energies. However, many common fitting methods are computationally demanding and do not provide the guarantee that the system’s ground states are preserved. This paper demonstrates the use of an efficient implementation of a Bayesian algorithm for cluster expansion construction that ensures all the input structural energies are fitted perfectly while reducing computational cost. The method incorporates an active learning scheme that searches for new optimal structures to include in the fit. As performance tests, we calculate the phase diagram of the Fe-Ir system and study the short range order in an equimolar MoNbTaVW system. The new method has been integrated into the Alloy Theoretic Automated Toolkit (ATAT).