## Phase Field Benchmark Problems Overview

*Extended essay outlining the motivation for phase field benchmark problems*

##### by Olle Heinonen

Many important processes in materials microstructural evolution, such
as coarsening, solidification, polycrystalline grain evolution,
magnetic and ferroelectric domain formation and motion, occur on
mesoscale length and time scales. The mesoscale is “in between,” in this
case in between atomistic and macroscopic length and time
scales. There are important conceptual distinctions, as well as
modeling approaches, between atomistic, meso-, and macroscale
processes. Atomistic scale modeling uses atoms and electrons as
central entities, and the structure and dynamics are governed either
by quantum mechanics, classical mechanics, or hybrid semiclassical
approaches. The relevant length and time scales are typically
Ångströms and femtoseconds (for electrons) or picoseconds (for atoms).
At the macroscale, on the other hand, materials properties are
averaged over many domains (*e.g.*, compositional, structural, magnetic,
or ferroelectric domains). Central entities at the macroscale are
usually materials properties that describe a response to some
macroscopic external forces, e.g., conductivity, elastic
properties, or susceptibilities. The responses are given in terms of
macroscopic constitutive equations. The relevant length and time
scales range from micrometers and milli- or microseconds and up.

The mesoscale, our area of concern, comprises length scales larger than several unit cells but small enough to resolve microstructural entities such as crystalline grains or ferromagnetic/ferroelectric domains that are usually averaged over in macroscale modeling. The length scales typically span nanometers to micrometers (or larger), and time-scales span nanoseconds to milliseconds or microseconds (or longer). Conceptually, there is a key distinction between time evolution in mesoscale modeling compared to atomistic modeling. Mesoscale dynamics is dissipative, so that forces give rise to velocities, as opposed to acceleration in atomistic modeling (although hydrodynamic and dissipative dynamics can be coupled).

There are two general approaches to mesoscale modeling. One approach models interfaces as sharp boundaries of one lower dimension than the structural domains they separate, such as crystalline grains or domains. This approach can be very efficient when simple microstructural geometries are simulated. However, tracking interfaces with complex geometries (e.g., dendritic growth) and topology changes, such as the merging and splitting of particles, is challenging. The other mesoscale modeling approach uses diffuse (finite width) interfaces. This method can easily track complex interface geometries and topological changes. This benefit is not without cost, as the interface, typically of the order of a nanometer, has to be resolved, yielding considerably more computationally intense calculations.

One way to implement diffuse interfaces is using phase fields. These are smooth and continuous fields that describe local microstructure. For example, a two-phase system can be described by a single phase field that takes the value 0 in one phase and 1 in the other and that smoothly interpolates between these values at a phase boundary (the actual values of the phase field are essentially irrelevant as long as they are different in different phases). If there are many variants in a system, for example crystalline orientations in crystalline grains, there can be a number of phase fields, each one representing a particular variant. The phase field method (and other, associated, order parameter-based approaches) has been used to study dendritic growth, spinodal decomposition, grain growth, and ferroelectric domain formation, to name a few phenomena.

As phase field modeling has gained popularity, a variety of codes have emerged. Some of them are community-based codes, such as MOOSE, FEnICs, or DUNE, and some are proprietary or in-house. With this variety of codes and numerical implementations, there is a concomitant need for benchmark problems that can be used to assess, validate, and verify codes. In that respect, phase field modeling is at a point similar to micromagnetic modeling in the late 1990s and early 2000s. With the emergence of a number of micromagnetic codes, the community self-organized and developed a number of Micromagnetic Standard Problems. These problems are non-trivial to solve analytically and test different aspects of both the physics being modeled as well as numerical methods implemented in micromagnetic codes, but were still designed to be not too computationally demanding. The National Institute of Standards and Technology has played a coordinating role in the development and management of these problems, and also hosts a website for the Micromagnetic Standard Problems, as well as solutions submitted by the community. The Micromagnetic Standard Problems were extremely useful to the community in the development of codes such as OOMMF, Mumax, and Magpar. It should be noted that the Micromagnetic Standard Problems continue to evolve as new physics, such as spin torque and Dzyaloshinksii-Moriya interactions, is added to the micromagnetic canon.

Inspired by the Micromagnetic Standard Problems, the Center for Hierarchical Materials Design (CHiMaD) is partnering with NIST and the phase field community to develop a set of benchmark problems for the phase field modeling and development community. These problems are vetted by the community in workshops and Hackathons, and then posted on this open-access website.

These phase field benchmark problems are designed to exhibit several key features. The problems are nontrivial and exhibit differing degrees of computational complexity but are not prohibitively computationally demanding. The outputs are defined to be easily comparable between different codes. The problems are also constructed to test a simple, targeted aspect of either the numerical implementation or the physics.

The first set of benchmark problems, BM1 and BM2, involve diffusion of a solute and grain growth. Technically, they use the Cahn-Hilliard equation for a conserved order parameter, and coupled Cahn-Hilliard and Allen-Cahn equations for conserved and non-conserved order parameters. Successfully solving the models demonstrates that the fundamentals of a modeling framework are sound.

The second set of benchmark problems, BM3 and BM4, involve coupling phase transformations with additional physics, namely, Fourier’s heat equation and Hooke’s Law. BM3, which is dendritic growth in solidification from an undercooled liquid, also focuses on how a solver can address the very different length scales that arise in the problem.

The third set of benchmark problems, BM5 and BM6, extend to additional physics less commonly seen in phase field problems. Specifically, BM5 involves modeling Stokes’ flow equations and BM6 couples diffusion to the Poisson equation for electrostatic charge.

The fourth set of benchmark problems, BM7, directly tests the implemented discretizations of space and time using the Method of Manufactured Solutions applied to the Allen-Cahn equation.

For details of any or all of these benchmark problems, please refer to the list of benchmarks problems.