A continuous and one-to-one coloring scheme for misorientations

Grain boundaries and their networks have profound influence over properties and structure evolution in every class of polycrystalline materials. Despite recent advances in characterization techniques, there remain fundamental problems in representing grain boundary network information; existing methods neglect the full complexity of misorientation information and often rely on boundary classification schemes of dubious physical significance. This situation has arisen in part because grain boundary misorientations have no known mapping to a simple Euclidean space; conventional wisdom suggests that the misorientation space is equivalent to the rotation space, which is known to require five variables for a continuous one-to-one mapping. In this paper, we show that, contrary to this expectation, the misorientation spaces for homophase misorientations for the 432 point group can indeed be mapped to three-dimensional Euclidean space. With this advance, we show that grain boundary networks can now be "colored", with every color uniquely reflecting the full misorientation information of every boundary in the network.