Grain boundary energy models and boundary splitting

Abstract

Models of grain boundary energy are essential for predicting the behavior of polycrystalline materials. Typical models represent the minimum boundary energy as a function of macroscopic boundary parameters. An energy model may allow for boundary dissociation, i.e., for a further reduction of the overall energy by splitting a boundary into two boundaries parallel to the original one. Such splitting is prevented by constraining the energy model with inequalities opposite to the boundary wetting condition. The inequalities are applicable only to triplets of boundaries that match the assumed geometric configuration. Relationships connecting the parameters of such boundaries are derived, implications of the inequalities that prevent boundary splitting are considered, and an example energy model is shown to allow boundary decomposition. Knowing whether a given energy model permits boundary dissociation and which boundaries can be affected is important for evaluating its performance in polycrystal simulations.

Figures (4)

• Figure 1: Schematic illustration of decomposition of boundary $\mathbf{B}_a \simeq (M_a,\mathbf{n}_a)$ into the pair of boundaries $(\mathbf{B}_{b_1},\mathbf{B}_{b_2})$, where $\mathbf{B}_{b_i} \simeq (M_{b_i},\mathbf{n}_{b_i})$. The symbols $g_k$ ($k=0,1,2$) denote grain orientations.
• Figure 2: Slopes of the BRK energy function for Ni near cusps at $\mathbf{B}_C = \mathbf{B}_1$ (a) and $\mathbf{B}_C = \mathbf{B}_4$ (b) compared to corresponding slopes of the cusp at $\mathbf{B}_0$. The misorientation of $\mathbf{B}_{\epsilon}$ is $M_{\epsilon} = R \left(\mathbf{k}_{\epsilon}, \theta_{\epsilon} \right)$. The directions of $\mathbf{k}_{\epsilon}$ are $[\overline{2}\, 0\, 1]$ in (a) and $[1\, 1\, 2]$ in (b). Values of remaining parameters are determined by the geometric compatibility conditions.
• Figure 3: Wettability parameter $w(\mathbf{B}_C,M_{\epsilon})$ versus the misorientation angle $\theta_{\epsilon}$ calculated using the BRK model for Ni. The misorientation of $\mathbf{B}_{\epsilon}$ is $M_{\epsilon} = R \left(\mathbf{k}_{\epsilon}, \theta_{\epsilon} \right)$. The directions of $\mathbf{k}_{\epsilon}$ are the same as in Fig. \ref{['Fig_slopes_BRK_FA']}. The discontinuity of $w(\mathbf{B}_1,M_{\epsilon})$ at $\theta_{\epsilon} \approx 1.68^{\circ}$ results from a discontinuity present in the BRK model.
• Figure 4: Wettability parameter $w(\mathbf{B}_C,M_{\epsilon})$ versus the misorientation axis $\mathbf{k}_{\epsilon}$ of $M_{\epsilon}$ based on the BRK model for Ni. The angle of $M_{\epsilon}$ was set to $5^{\circ}$. Results for the cusps at the twin boundary $\mathbf{B}_C = \mathbf{B}_1$ (a,b) and at $\Sigma 11$ boundary $\mathbf{B}_C = \mathbf{B}_4$ (c,d). Figures (a,c) and (b,d) show stereographic projections of the upper and lower hemishperes from the poles $[0 \, 0 \, \overline{1}]$ and $[0 \, 0 \, 1]$, respectively.