Data workflow to incorporate thermodynamic energies from Calphad databases into grand-potential-based phase-field models

Abstract

In order to approximate Gibbs energy functions, a semi-automated framework is introduced for binary and ternary material systems, using Calphad databases. To generate Gibbs energy formulations by means of second-order polynomials, the framework includes a precise approach. Furthermore, an optional extensional step enables the modeling of systems in which a direct generation leads to the unsatisfactory results in the representation of the thermodynamics. Furthermore, an optional extensional step enables the modeling of systems, in which a direct generation leads to the unsatisfactory results, when representing the thermodynamics. Within this extension, the commonly generated functions are modified to satisfy the equilibrium conditions in the observed material systems, leading to a better correlation with thermodynamic databases. The generated Gibbs energy formulations are verified by recalculating the equilibrium concentrations of the phases and rebuilding the phase diagrams in the considered concentration and temperature ranges, prior to the simulation studies. For all comparisons, a close match is achieved between the results and the Calphad databases. As practical examples of the method, phase-field simulation studies for the directional solidification of the binary – and the ternary – eutectic systems are performed. Good agreements between the simulation results and the reported theoretical and experimental studies from literature are found, which indicates the applicability of the presented approaches. Graphical Abstract

Rotating Directional Solidification of Ternary Eutectic Microstructures in Bi-In-Sn: A Phase-Field Study

“…Each curve is constructed on the basis of the measured amounts of the resultant undercoolings at the different lamellar spacings. The general shapes of the curves clearly show a minimum undercooling-spacing points at and correlate well with the velocity dependent behavior expected from the Jackson–Hunt theory, which has also been observed in simulation studies of other material systems [ 33 , 49 ], in the literature. For each solidification velocity, the spacing at which the individual minimum of each curves occurs is labeled with .…”

“…From these new Gibbs energy formulations, other thermodynamic properties, such as the chemical potentials and the grand potentials, can be calculated [ 49 ]. In the course of this endeavor, the general procedure introduced in [ 33 ], for the approximation of the Gibbs energies of the ternary eutectic material systems, is used in this work. This approximation procedure contains the following five essential steps: In the first step, the equilibrium concentrations for the given working temperatures below the melting point are determined for all involved phases, using the Calphad databases.…”

“…In the last step, the resultant functions are validated by measuring the maximum and average deviations between the Calphad data and the approximated data, as well as by rebuilding the equilibrium conditions of the phases. It has to be mentioned that the used method of [ 33 ] is limited to binary and ternary systems without stoichiometric phases. As the observed system fulfills these conditions, this method is selected for the modeling.…”

“…All grand potentials depend on the phase-field vector, the chemical potentials and the temperature T and are stored in the vector . Together with the concentrations and the chemical potentials , the grand potentials can be derived from Calphad databases, by using the general method introduced in [ 33 ]. In this method, the volume and the pressure are assumed to be constants, which ensures the thermodynamic consistency of the system.…”

“…Then, the thermodynamic modeling of the system is shown, using the method from Noubary et al. [ 33 ] and the Calphad database from Witusiewicz et al. [ 29 ].…”