Spherical Gaussians: An intuitive method for creating complex anisotropies in interface energies for the phase field method


In most practical applications across various fields, the phase-field transitions are mainly described using the phase field method (PFM). This method is generally efficient and has been extensively used to study and simulate material properties. For example, it has been used to simulate void formation, corrosion and solid phase transformations, among other phenomena. Consequently, PFM has emerged as a promising method for predicting the interface evolution of different materials because it can be easily coupled with other parameters such as magnetic fields. Moreover, PFM-based models can be used in the quantitative material analysis provided the properties of the material in question are available, making it one of the most suitable methods for meso-scale simulation.

Despite the prospective effectiveness and usefulness of PFM, it fails to effectively model materials with complex anisotropies, the interface energies that depend on the direction, in the interface energies. To date, there are few attempts to overcome the limitations of PFM. For instance, the two-dimensional (2D) interfaces approach can model materials with many dendrites but requires that the directions responsible for the dendrite growth have equal energy intensity same as that of the energy surface. This limits its practical applications. Similarly, the 3D anisotropy technique is more effective but only limited to material anisotropies like face-centered cubic. Unfortunately, most materials with more complex anisotropies do not fit the existing anisotropy models. Therefore, developing more reliable and robust methods for modeling complex anisotropies in interface energies is highly desirable.

PFM requires continuous functions to control the interface energies between different phases properly. Equipped with this knowledge, scientists from the Pacific Northwest National Laboratory, Professor Jacob Bair (also affiliated with Oklahoma State University), Mr. Nikhil Deshmukh and Dr. David Abrecht developed a spherical Gaussians method to add more complex directional anisotropies to PFMs. In their approach, spherical Gaussians, a set of continuous functions, were used to describe the interface energies. Using native spherical Gaussian properties to create interface energies, this technique was applied to produce a range of crystal geometry arrays such as cubes, rods, cubic dendrites and hexagonal rods to demonstrate its feasibility. The work is currently published in the research journal, Computational Materials Science.

The research team showed that spherical Gaussians are intuitive and easy to understand. They demonstrated great potential to add more complex anisotropies to PFMs. This could be partially attributed to the flexibility of the spherical Gaussians as they could be placed in any direction and still produce complex anisotropy with differing minima. Moreover, the high flexibility enabled the construction of more complex directional anisotropies. The addition of anisotropy showed a remarkable difference in material growth. Furthermore, for materials with low anisotropy, the dendrite morphology was significantly influenced by the parameters responsible for the sharpness of the Gaussians.

In summary, Pacific Northwest National Laboratory scientists reported the creation of a variety of complex anisotropies for interface energies using a simple phase field model. The observed difference in material growth indicated the physical method’s significance for anisotropy addition. Unlike sinusoidal and algebraic functions, spherical Gaussians came out as a more general approach for constructing complex directional anisotropies. In a statement to Advances in Engineering, Professor Bair explained that the new method will expand the quantitative capability of PFMs desirable for complex materials and modeling other parameters influencing the morphology and evolution of materials.

Spherical Gaussians: An intuitive method for creating complex anisotropies in interface energies for the phase field method - Advances in Engineering

About the author

Jacob L. Bair was raised on a farm near Ephrata, WA. He earned a B.S. degree in Mechanical Engineering with a second major in French from Washington State University in December of 2012, then his Ph.D. in Materials Science and Engineering from Missouri University of Science and Technology in December of 2016.

His research has been on experimental and computational studies of the microstructural evolution of materials using scanning electron microscopy with Electron Backscatter Diffraction, Phase Field Modeling, and Molecular Dynamics. Topics he has researched include hydrides in Zr fuel rod claddings, grain boundary migration in face centered cubic materials, twinning in nickel, phase field modeling of UC formation in a liquid U melt, and Pu Oxalate precipitation. He joined the faculty of the Mechanical and Aerospace Engineering Department at Oklahoma State University in August of 2020 and currently serves there as an Assistant Professor.


Bair, J., Deshmukh, N., & Abrecht, D. (2021). Spherical Gaussians: An intuitive method for creating complex anisotropies in interface energies for the phase field methodComputational Materials Science, 188, 110126.

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