Topological defects in two-dimensional orientation-field models for grain growth

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 multigrain 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 be interpreted in the classical picture of grain boundaries, (ii) and therefore (iii) cannot. In addition, singularities cause difficulties, such as lattice pinning in numerical simulations. To overcome these problems, we propose two formulations of the model. The first is based on a three-component unit vector field, while in the second we utilize a two-component vector field with an additional potential. In both cases, the additional degree of freedom introduced makes the order parameter space simply connected, which removes the topological stability of these defects.

Efficient and quantitative phase field simulations of polycrystalline solidification using a vector order parameter

“…∆ := (T l − T )/((1 − k e )|m e l |c 0 ) = 0, or ∆ = 0.55. In all cases, solid-liquid ordering, represented by R, develops a typical cusp-like morphology in the grain boundary region consistent with earlier polycrystalline PF models, such as in orientation field models [7]. Moreover, the solid-solid grain boundary ordering decreases as the misorientation increases, approaching a "wet" grain boundary at high misorientation, as expected.…”

“…While this form reproduces some features of solidification and coarsening qualitatively, this R−θ representation suffers from lattice pinning due to topological defects, analogous to those seen in the orientation PF models [7]. Furthermore, it is unclear how to implement thermodynamically consistent noise for the θ field in our vector order parameter model.…”

“…We will simulate phase field dynamics using Eqs. ( 8) and (7). Furthermore, in order to achieve quantitative solidification results in the limit of a diffuse interface, we will incorporate an anti-trapping flux in Eq.…”

“…( 6); secondly, we carry out the variational derivatives as specified in the first of Eqs. (7); thirdly, we simplify the chemical driving force analogously to Ref. [13].…”

“…|∇θ| gradient term in the free energy. Additionally, these models have been shown to exhibit unphysical topological defects, that have recently been, at least partially, addressed in 2D simulations [7]. Finally, the extension to 3D of this class of models is not immediately straightforward, although some schemes have been explored [8].…”