A comparison of Redlich-Kister polynomial and cubic spline representations of the chemical potential in phase field computations

TL;DR

This paper compares Redlich-Kister polynomial and cubic spline representations of free energy functions in phase field models, showing splines offer better accuracy and computational efficiency for complex thermodynamic data.

Contribution

It demonstrates that cubic spline functions can effectively replace Redlich-Kister polynomials for representing free energies, improving fit quality and computational speed in phase field simulations.

Findings

Spline functions provide better fits to fluctuating free energy data.

Using splines speeds up phase field computations by nearly tenfold.

Splines require fewer degrees of freedom than polynomials for accurate representation.

Abstract

Free energies play a central role in many descriptions of equilibrium and non-equilibrium properties of solids. Continuum partial differential equations (PDEs) of atomic transport, phase transformations and mechanics often rely on first and second derivatives of a free energy function. The stability, accuracy and robustness of numerical methods to solve these PDEs are sensitive to the particular functional representations of the free energy. In this communication we investigate the influence of different representations of thermodynamic data on phase field computations of diffusion and two-phase reactions in the solid state. First-principles statistical mechanics methods were used to generate realistic free energy data for HCP titanium with interstitially dissolved oxygen. While Redlich-Kister polynomials have formed the mainstay of thermodynamic descriptions of multi-component solids, they require high order terms to fit oscillations in chemical potentials around phase transitions. Here we demonstrate that high fidelity fits to rapidly fluctuating free energy functions are obtained with spline functions. Spline functions that are many degrees lower than Redlich-Kister polynomials provide equal or superior fits to chemical potential data and, when used in phase field computations, result in solution times approaching an order of magnitude speed up relative to the use of Redlich-Kister polynomials.