Twist, tilt, and symmetric grain boundaries in hexagonal materials

Krzysztof Glowinski,Adam Morawiec

doi:10.1007/s10853-013-7958-5

Abstract

Grain boundaries of characteristic geometry, e.g., twist, tilt, and symmetric boundaries are often used as reference boundaries in analyses of boundary networks in polycrystalline materials. This article deals with the issue of proper identification of characteristic boundaries in the case of materials with hexagonal D6h symmetry. To identify all boundaries of characteristic types, both analytical calculations and numerical searches are used. The first approach provides exact parameters of the characteristic boundaries, whereas the second one gives boundaries which can be classified as characteristic if some tolerance is allowed. In both methods, all symmetrically equivalent boundary representations are taken into consideration. The obtained sets of twist, tilt, symmetric, and 180°-tilt boundaries are presented in the form of two-dimensional maps containing stereographic projections of the corresponding boundary plane normals for selected grain misorientations. These diagrams facilitate interpretation of experimental distributions of grain boundaries; with the representation used, they can be directly linked to experimental distributions. Examples of such diagrams for lattice parameter ratios c/a of \(\sqrt{5/2}\) and \(\sqrt{20/21}\) are presented. They are compared to example boundary distributions in Ti alloy and distributions of WC/WC boundaries in WC–Co composites available in the literature.

Highlights

A wide range of properties of polycrystalline materials is influenced by intergranular boundaries
Grain boundaries of characteristic geometry, e.g., twist, tilt, and symmetric boundaries are often used as reference boundaries in analyses of boundary networks in polycrystalline materials
In the case of D6h symmetry, geometry of a boundary is fully described by a misorientation M and Miller-Bravais indices (h1 k1 i1 l1) of the boundary plane expressed in the reference frame of the first grain [11]

Summary

Introduction

A wide range of properties of polycrystalline materials is influenced by intergranular boundaries. The most basic attribute of a boundary is its ‘‘macroscopic’’ geometry, which can be described by a misorientation between neighboring grains and boundary plane indices, i.e., by five independent parameters [1]. A thorough understanding of macroscopic features of boundaries is essential for more comprehensive boundary studies at the atomic scale. Distributions of grain boundaries with respect to their macroscopic parameters are studied (e.g., [4,5,6]). These distributions indicate which boundary geometries are preferred, and which are underrepresented. The question arises whether maxima in the distributions correspond to boundaries of special geometries, e.g., to twist, tilt, or symmetric boundaries. In the presence of crystal symmetries, every geometric configuration may have multiple equivalent representations; it is crucial to take them all into consideration to recognize boundary types correctly

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Methods

Conclusion