Probability densities of disorientation angles among randomly oriented grains in tricrystals

Abstract

Orientation relationships of grains in tricrystals are discussed. Disorientation angles among grains are evaluated and a constrained condition of these around a triple junction is shown. Probability densities of the disorientation angles are obtained for randomly oriented grains by numerical calculations. A disorientation-angle diagram for tricrystal is proposed to characterize the orientation relationships of grains around the triple junctions.

Appendix

A grain orientation can be describe by using a rotation matrix ${\varvec{T}}$ with respect to a reference coordinate system. This ${\varvec{T}}$ is a 3 × 3 orthogonal matrix whose determinant is 1. Any orientation is expressed by a product of appropriate three rotation matrices which cause rotations around the coordinate axes. Three rotation angles around the coordinate axes are called the Euler angles. The rotation matrix ${\varvec{T}}$ with the three the Euler angles $0\le {\theta }_{\text{e}}\le \pi$, $0\le {\phi }_{\text{e}}<2\pi$ and $0\le {\psi }_{\text{e}}<2\pi$ given by the following equation is an example of the rotation matrix giving any orientation:

$${\varvec{T}}=\left(\begin{array}{ccc}\mathrm{cos}{\phi }_{\text{e}}& -\mathrm{sin}{\phi }_{\text{e}}& 0\\ \mathrm{sin}{\phi }_{\text{e}}& \mathrm{cos}{\phi }_{\text{e}}& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}{\theta }_{\text{e}}& -\mathrm{sin}{\theta }_{\text{e}}\\ 0& \mathrm{sin}{\theta }_{\text{e}}& \mathrm{cos}{\theta }_{\text{e}}\end{array}\right)\left(\begin{array}{ccc}\mathrm{cos}{\psi }_{\text{e}}& -\mathrm{sin}{\psi }_{\text{e}}& 0\\ \mathrm{sin}{\psi }_{\text{e}}& \mathrm{cos}{\psi }_{\text{e}}& 0\\ 0& 0& 1\end{array}\right).$$

(11)

Describing the Euler angles ${\theta }_{\text{e}}$, ${\phi }_{\text{e}}$ and ${\psi }_{\text{e}}$ as a function of random numbers appropriately, we can obtain ${\varvec{T}}$ giving random orientations. When ${M}_{i}\left(i=\mathrm{1,2} or 3\right)$ are random numbers in the range of 0 to 1 with uniform distribution over that range, Eq. (11) with the Euler angles given by

$${\theta }_{\text{e}}={\mathrm{cos}}^{-1}\left(2{M}_{1}-1\right),$$

(12a)

$${\phi }_{\text{e}}=2{M}_{2}\pi$$

(12b)

and

$${\psi }_{\text{e}}=2{M}_{3}\pi$$

(12c)

can generate three dimensionally random orientations with uniform distribution. Note that ${\theta }_{\text{e}}$ is not proportional to ${M}_{1}$. The reason is that the probability density $\rho \left({\theta }_{\text{e}}\right)$ of ${\theta }_{\text{e}}$ to generate the random orientations is proportional to $\mathrm{sin}{\theta }_{\text{e}}$ [18]. The inverse cosine function in the right-hand-side of Eq. (12a) comes from $d\mathrm{sin}{\theta }_{\text{e}}/d{\theta }_{\mathrm{e}}=\mathrm{cos}{\theta }_{\mathrm{e}}$.

To obtain the disorientation angle $\theta$ between cubic grains, it is necessary to include the crystal symmetry. Here, we explain the procedure by considering two cubic Grains 1 and 2. When orientations of Grains 1 and 2 are given by the rotation matrices ${{\varvec{T}}}_{1}$ and ${{\varvec{T}}}_{2}$ with respect to a reference coordinate system, respectively, the rotation matrix ${{\varvec{R}}}_{12}$ from Grains 1 to 2 described by using the frame of Grain 1 is written as [19]

$${\varvec{R}}_{{{12}}} = {\varvec{T}}_{{1}}^{{\text{T}}} \,{\varvec{T}}_{{2}} ,$$

(13)

where ${{\varvec{T}}}_{1}^{\mathrm{T}}$ is the transpose of ${{\varvec{T}}}_{1}$. The cubic structure is invariant under 24 proper symmetrical rotations and these are (a) identity element or no rotation, (b) rotations of $90^\circ$, $180^\circ$ or $270^\circ$ about the three < 100 > axes, (c) rotations of $180^\circ$ about the six < 110 > axes and (d) rotations of $120^\circ$ or $240^\circ$ about the four < 111 > axes [2]. Writing ${{\varvec{S}}}_{\text{i}}$ (i = 1, 2, …, 24) as the matrix representations of these symmetrical rotations, the disorientation angle ${\theta }_{12}$ between Grains 1 and 2 is w

$${\theta }_{12}={\mathrm{cos}}^{-1}\left\{\left[\underset{i}{\mathrm{max}}{\text{Tr}}\left({{\varvec{R}}}_{12}\boldsymbol{ }{{\varvec{S}}}_{\text{i}}\right)-1\right]/2\right\},$$

(14)

where ${\text{Tr}}\left({{\varvec{R}}}_{12}\boldsymbol{ }{{\varvec{S}}}_{\text{i}}\right)$ is the trace of the matrix ${{\varvec{R}}}_{12}\boldsymbol{ }{{\varvec{S}}}_{\text{i}}$.