Modeling grain boundaries using a phase-field technique

doi:10.1016/S0022-0248(99)00856-8

Journal of Crystal Growth

Volume 211, Issues 1–4, 1 April 2000, Pages 18-20

https://doi.org/10.1016/S0022-0248(99)00856-8 Get rights and content

Abstract

We propose a two-dimensional phase-field model of grain boundary dynamics. One-dimensional analytical solutions for a stable grain boundary in a bicrystal are obtained, and equilibrium energies are computed. By comparison with microscopic models of dislocation walls, insights into the physical accuracy of this model can be obtained. Indeed, for a particular choice of functional dependencies in the model, the grain boundary energy takes the same analytic form as the microscopic (dislocation) model of Read and Shockley (Phys. Rev. 78 (1950) 275).

Introduction

Previously, we presented a new model [1], [2] and phase-field simulations of the simultaneous processes of solidification, impingement, and coarsening of arbitrarily oriented crystals. In earlier models of this phenomenon, a finite number of crystalline orientations are allowed with respect to the fixed coordinate reference frame. Morin et al. [3], and Lusk [4] constructed a free energy density having N minima by introducing a rotational (orientation) variable in the homogeneous free energy. Chen and Yang [5] and Steinbach et al. [6] assigned N-order parameters to the N allowed orientations. In these approaches, the free energy density depends on the orientation of the crystal measured in the fixed frame — a property which is not physical.

In Refs. [1], [2] the model introduced was invariant under rotations of the reference frame. A similarly motivated approach has also been developed by Lusk [7]. Herein, we present a completely general formulation of a new class of models which allow for accurate, physical modeling of grain boundary dynamics, with particular attention paid to low-angle boundaries.

Section snippets

Development of the model

We focus on a completely solid two-dimensional system with grain boundaries present. The model parameters η and θ represent a coarse-grained measure of the degree of crystalline order and the crystalline orientation, respectively. We restrict 0⩽η⩽1 and −π⩽θ⩽π. The variable θ is an indicator of the mean orientation of the crystalline subregion (in 2D). Note, that when symmetrically equivalent crystalline directions exist, θ may not be uniquely defined. However, the point group symmetry

Comparisons with dislocation models of grain boundaries

The energies of low-misorientation tilt grain boundaries have been approximated by summing the energy of distribution of edge dislocations [10]. Here, we discuss a method for reproducing the same energetic dependence on misorientation. Using the above results we may compute the free interface energy of a bicrystal grain boundary to be $γ=sχ(η_{0}) Δ θ+2α∫^{1}_{η_{0}} 2f(η) d η.$ If we choose f=a²(1−η)²/2 (a very plausible choice), and a somewhat unusual choice for χ of $χ(η)=−2 ln (1−η),$ then we find $γ=s Δ θ 1− ln s Δ θ αa$ holds.

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