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In this paper we apply the concept of the Clifford torus and the derived square torus maps to the study of disorientations in microstructures. First, we interpret the Clifford torus in terms of the more commonly used orientation representations (Rodrigues-Frank vectors, 3D stereographic vectors, and homochoric vectors) and show representations of the torus in those spaces. This leads to a simple graphical interpretation of the generation and meaning of the square torus maps. Then we apply this approach to the study of disorientations in polycrystalline materials (CSL boundaries in grain boundary engineered Nickel) as well as intervariant boundaries in martensitic and bainitic steels. We show that pre-computed theoretical square torus maps can be used to determine population fractions of specific boundaries. 

This paper introduces a new 2D representation of the orientation distribution function for an arbitrary material texture. The approach is based on the isometric square torus mapping of the Clifford torus, which allows for points on the unit quaternion hypersphere (each corresponding to a 3D orientation) to be represented in a periodic 2D square map. The combination of three such orthogonal mappings into a single RGB (red–green–blue) image provides a compact periodic representation of any set of orientations. Square torus representations of five different orientation sampling methods are compared and analyzed in terms of the Rieszsenergies that quantify the uniformity of the samplings. The effect of crystallographic symmetry on the square torus map is analyzed in terms of the Rodrigues fundamental zones for the rotational symmetry groups. The paper concludes with example representations of important texture components in cubic and hexagonal materials. The new RGB representation provides a convenient and compact way of generating training data for the automated analysis of material textures by means of neural networks. 

Electron backscatter diffraction (EBSD) is a powerful tool for determining the orientations of near-surface grains in engineering materials. However, many ceramics present challenges for routine EBSD data collection and indexing due to small grain sizes, high crack densities, beam and charge sensitivities, low crystal symmetries, and pseudo-symmetric pattern variants. Micro-cracked monoclinic hafnia, tetragonal hafnon, and hafnia/hafnon composites exhibit all such features, and are used in the present work to show the efficacy of a novel workflow based on a direct detecting EBSD sensor and a state-of-the-art pattern indexing approach. At 5 and 10 keV primary beam energies (where beam-induced damage and surface charge accumulation are minimal), the direct electron detector produces superior diffraction patterns with 10x lower doses compared to a phosphor-coupled indirect detector. Further, pseudo-symmetric variant-related indexing errors from a Hough-based approach (which account for at least 4%-14% of map areas) are easily resolved by dictionary indexing. In short, the workflow unlocks fundamentally new opportunities to characterize materials historically unsuited for EBSD. 

We investigate the coarsening dynamics of the three-phase eutectic Al-Ag2Al-Al2Cu at 723 K via in situ transmission X-ray nano-tomography. Unlike previous investigations that compared observations between different samples annealed for different times, our three-dimensional measurement shows at nanoscale resolution the microstructural changes occurring in the same field-of-view, enabling new insight on the capillary-driven evolution of a ladder-like pattern. With the aid of a new reconstruction algorithm and machine learning segmentation, we trace the interfaces of the eutectic and observe significant structural changes within 4 hr. of aging. Even though the average length-scales of the eutectic solids follow a temporal power law, the microstructure is not self-similar. Instead, it evolves (in part) through the coalescence of neighboring Ag2Al solids at the expense of the intervening Al2Cu. By combining our X-ray data with electron diffraction to identify the common planes at the interphase boundaries, we show that coalescence leads to a decrease in lattice misfit, and hence, interfacial energy. At longer times, the interphase boundaries with low misfit compete for surface area, resulting in a ‘locking’ of the interfacial shape. 

Hydride precipitation in zirconium alloys leads to embrittlement, making it essential to understand their prevalence and stability in the microstructure. Dictionary indexing of Kikuchi patterns, along with orientation relationship analysis and x-ray diffraction, confirmed the presence of both delta and gamma hydride phases in Zircaloy-4. Both phases were found to be stable in recrystallised zirconium, with the gamma phase exhibiting a distinct orientation relationship with the matrix. Delta hydride morphology and orientation were influenced by local stresses, resulting in a change in orientation during precipitation. By analysing the orientation relationships, the evolution of hydride phases could be visualised, providing insights into the room temperature stability of both delta and gamma hydrides.