This Award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
This project seeks to develop a new mathematical framework for the description of crystal orientations, as well as misorientations between crystals. The approach is built upon four-dimensional vectors called quaternions; a quaternion can represent the axis and angle of a rotation in a simple way, and distributions of quaternions can be used to describe populations of crystal orientations (i.e., texture information), as well as populations of grain boundary misorientations. The intellectual merit of the present project lies in three technical objectives. The first of these is the development of the mathematical tools required to express orientation distributions in the quaternion framework, inclusive of relevant crystal and sample symmetries. For the case of grain boundaries dual symmetries reflective of the two adjacent crystallites are simultaneously applied. The second technical objective of the project is to provide a sound mathematical connection between the new quaternion framework and the more conventional Euler angle-based framework in wide use in the field of texture analysis today; this connection permits forward-translation of decades of prior research into the new framework. The final technical objective of the project is to apply the new quaternion-based framework to address some pressing problems in the crystallography of grain boundaries and grain boundary junctions in polycrystalline materials; in particular the local correlations in grain boundary misorientations induced by triple junctions and quadruple nodes are assessed mathematically using this framework. The project also has significant broader impacts for the scientific community. In addition to student training in materials microstructures, texture analysis, and advanced characterization techniques, the project develops open-source software that is freely disseminated on the Internet. The software renders all of the mathematical manipulations developed under the project accessible to the community in the form of simple algorithmic tools. These tools permit researchers to evaluate crystallographic texture, intuitively represent grain boundary statistics, and map grain boundary misorientations overlaid upon conventional micrographs.
NON-TECHNICAL SUMMARY:
The discipline of materials science and engineering has reached a point where it is possible to design a material at the level of assembling individual crystals into a solid. Among the important considerations in materials design are the orientations of individual crystals in the material, and the interfaces or boundaries between crystals in the structure. The specific crystal orientations and interface structures formed in a material dictate to a very large extent the material?s ability to resist damage, to exhibit high strength and toughness, and to display functional properties. The present project develops the mathematical tools necessary to design materials microstructure, with specific emphasis on crystal orientations and misorientations between crystals. The project specifically develops methods based on so-called quaternions, four-dimensional vectors that can describe a crystal orientation. The use of quaternions in the field of material microstructure greatly simplifies the description of many important problems, and is conducive to the development of fast algorithms to characterize, analyze, and design new materials microstructures. The present project, in addition to developing these mathematical tools, also seeks a broader impact by disseminating computer algorithms free for use by the entire scientific community. These computer codes allow researchers to more intuitively visualize materials structure, and to perform the calculations required to optimize them as well.
