On topological defects in two-dimensional orientation-field models for grain growth

TL;DR

This paper investigates topological defects in 2D orientation-field models for grain growth, revealing their implications and proposing new formulations to eliminate these defects by modifying the order parameter space.

Contribution

The paper introduces two new model formulations that make the order parameter space simply-connected, thereby removing topological defects in 2D grain growth simulations.

Findings

Topological defects cause singularities and numerical issues.

New models eliminate topological defects by changing the order parameter space.

Singularities are avoided, improving simulation stability.

Abstract

Standard two-dimensional orientation-field based phase-field models rely on a continuous scalar field to represent crystallographic orientation. The corresponding order parameter space is the unit circle, which is not simply-connected. This topological property has important consequences for the resulting multi-grain structures: (i) trijunctions may be singular; (ii) for each pair of grains, there exist two different grain boundary solutions that cannot continuously transform to one another; (iii) if both solutions appear along a grain boundary, a topologically stable, singular point defect must exist between them. While (i) can, (ii) and therefore (iii) cannot be interpreted in the classical picture of grain boundaries. In addition, singularities cause difficulties, such as lattice pinning in numerical simulations. To overcome these problems, we propose two new formulations of the model. The first is based on a 3-component unit vector field, while in the second we utilise a 2-component vector field with an additional potential. In both cases, the additional degree of freedom introduced make the order parameter space simply-connected, which removes the topological stability of these defects.