A phase field model with arbitrary misorientation dependence of grain boundary energy

Abstract

Grain growth in polycrystals is often simulated using orientation-field models, which employ a field to represent the local orientation of the crystal lattice. These models can be challenging to represent a realistic misorientation dependence of grain boundary free energy. We prove that existing orientation-field models, in general, cannot reproduce a decrease in the grain boundary free energy with a increasing misorientation angle, demonstrating a significant limitation of previous models in applications to polycrystalline materials. To overcome this limitation, we propose a modification to the Kobayashi-Warren-Carter model for grain growth wherein the coefficients of the free-energy functional become functions of the misorientation between the grains, which is a non-local quantity. Due to this modification, an arbitrary dependence of the grain boundary free energy on the misorientation can be embedded in the model. We propose calculating the non-local misorientation by interpolating the orientation field at a fixed distance in both directions along the local grain boundary normal vector. The capabilities of the model are demonstrated by introduction of a sharp cusp to the misorientation dependent grain boundary free energy. Finally, we propose an extension of the model to three dimensions.

Figures (4)

• Figure 1: (a) Grain boundary energy function including a cusp at $\Delta\theta=\pi/2$ used in the present results. (b) Coefficient of the linear $\nabla\theta$ term as a function of nonlocal misorientation, $s(\Delta\theta)$, which results from Eq. \ref{['eq_kwc_s_deltatheta']} given the grain boundary energy function shown in (a).
• Figure 2: Equilibrium profiles for varying misorientation with $s(\Delta\theta)$ given by Eq. (\ref{['eq_kwc_s_deltatheta']}) with $\epsilon=0.5$, for (a) $\phi$ and (b) $\theta$. The colors described in the legend on (b) apply to both subfigures.
• Figure 3: Grain boundary energy as a function of misorientation calculated from simulations of planar grain boundaries in 1D, compared to the input analytical function for $\gamma_{GB}(\Delta\theta)$ (Eq. (\ref{['eq_simulation_gamma']})).
• Figure 4: Grain boundary mobility as a function of misorientation, calculated from simulations of shrinking circular grains embedded in a matrix. Each point corresponds to a simulation; the line is to guide the eye.