A comparison of RedlichKister polynomial and cubic spline representations of the chemical potential in phase field computations
Free energies play a central role in many descriptions of equilibrium and nonequilibrium 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 twophase reactions in the solid state. Firstprinciples statistical mechanics methods were used to generate realistic free energy data for HCP titanium with interstitially dissolved oxygen. While RedlichKister polynomials have formed the mainstay of thermodynamic descriptions of multicomponent 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. As a result, spline functions that are many degrees lower than RedlichKister 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 RedlichKister polynomials.
 Authors:

^{[1]};
^{[2]};
^{[1]};
^{[2]};
^{[1]};
^{[2]};
^{[1]}
 Univ. of Michigan, Ann Arbor, MI (United States)
 Univ. of California, Santa Barbara, CA (United States)
 Publication Date:
 Grant/Contract Number:
 SC0008637
 Type:
 Accepted Manuscript
 Journal Name:
 Computational Materials Science
 Additional Journal Information:
 Journal Volume: 128; Journal ID: ISSN 09270256
 Publisher:
 Elsevier
 Research Org:
 Univ. of Michigan, Ann Arbor, MI (United States)
 Sponsoring Org:
 USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC22)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; 36 MATERIALS SCIENCE; free energy; spinodal decomposition; phase transformation
 OSTI Identifier:
 1332699
Teichert, Gregory H., Gunda, N. S. Harsha, Rudraraju, Shiva, Natarajan, Anirudh Raju, Puchala, Brian, Van der Ven, Anton, and Garikipati, Krishna. A comparison of RedlichKister polynomial and cubic spline representations of the chemical potential in phase field computations. United States: N. p.,
Web. doi:10.1016/j.commatsci.2016.11.024.
Teichert, Gregory H., Gunda, N. S. Harsha, Rudraraju, Shiva, Natarajan, Anirudh Raju, Puchala, Brian, Van der Ven, Anton, & Garikipati, Krishna. A comparison of RedlichKister polynomial and cubic spline representations of the chemical potential in phase field computations. United States. doi:10.1016/j.commatsci.2016.11.024.
Teichert, Gregory H., Gunda, N. S. Harsha, Rudraraju, Shiva, Natarajan, Anirudh Raju, Puchala, Brian, Van der Ven, Anton, and Garikipati, Krishna. 2016.
"A comparison of RedlichKister polynomial and cubic spline representations of the chemical potential in phase field computations". United States.
doi:10.1016/j.commatsci.2016.11.024. https://www.osti.gov/servlets/purl/1332699.
@article{osti_1332699,
title = {A comparison of RedlichKister polynomial and cubic spline representations of the chemical potential in phase field computations},
author = {Teichert, Gregory H. and Gunda, N. S. Harsha and Rudraraju, Shiva and Natarajan, Anirudh Raju and Puchala, Brian and Van der Ven, Anton and Garikipati, Krishna},
abstractNote = {Free energies play a central role in many descriptions of equilibrium and nonequilibrium 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 twophase reactions in the solid state. Firstprinciples statistical mechanics methods were used to generate realistic free energy data for HCP titanium with interstitially dissolved oxygen. While RedlichKister polynomials have formed the mainstay of thermodynamic descriptions of multicomponent 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. As a result, spline functions that are many degrees lower than RedlichKister 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 RedlichKister polynomials.},
doi = {10.1016/j.commatsci.2016.11.024},
journal = {Computational Materials Science},
number = ,
volume = 128,
place = {United States},
year = {2016},
month = {12}
}