Peter Joachim Konijnenberg

Consistent representations of and conversions between 3D rotations

In materials science the orientation of a crystal lattice is described by means of a rotation relative to an external reference frame. A number of rotation representations are in use, including Euler angles, rotation matrices, unit quaternions, Rodrigues–Frank vectors and homochoric vectors. Each representation has distinct advantages and disadvantages with respect to the ease of use for calculations and data visualization. It is therefore convenient to be able to easily convert from one representation to another. However, historically, each representation has been implemented using a set of often tacit conventions; separate research groups would implement different sets of conventions, thereby making the comparison of methods and results difficult and confusing. This tutorial article aims to resolve these ambiguities and provide a consistent set of conventions and conversions between common rotational representations, complete with worked examples and a discussion of the trade-offs necessary to resolve all ambiguities. Additionally, an open source Fortran-90 library of conversion routines for the different representations is made available to the community.

Advanced Methods and Tools for Reconstruction and Analysis of Grain Boundaries from 3D-EBSD Data Sets

We report the recent development of a 3D orientation data post-processing software, which we refer to as QUBE. Amongst other functionalities, it offers the possibility to specify the spatial and orientational distribution of boundary normals. We describe a method to reconstruct a voxel-accurate and smooth 3D boundary triangle mesh by algorithmic means. A proof of concept is given by a benchmark on a generic dataset and we demonstrate a first result with the description of selected grain boundaries in an Fe-28%Ni sample.

Electron backscatter diffraction (EBSD) techniques for studying phase transformations in steels

Abstract: The electron backscatter diffraction (EBSD) technique and EBSD-based orientation microscopy (ORM) are very powerful techniques for the comprehensive characterization of crystalline microstructures in the scanning electron microscope. ORM delivers accurate and quantitative data about the morphology and crystallography of a material, e.g. shape, size and neighbourhood relations of crystals, crystallographic phases, and crystallographic texture and misorientation distribution. By combining two-dimensional EBSD-based ORM with serial sectioning the technique can be extended to a three-dimensional technique. In addition to the morphological description of microstructures, this technique allows the comprehensive crystallographic characterization of grain boundaries and the quantification of deformation structures in terms of densities of geometrically necessary dislocations. The present text gives an overview on EBSD applications to the study of phase transformations in steels and briefly describes future developments of the technique.

Analysis of 3D-EBSD Datasets Obtained by FIB Tomography

3D-EBSD represents a destructive and crystallographic tomographic method. It comprises a repeated sectioning and subsequent surface characterization by Electron Backscatter Diffraction (EBSD). Several sectioning alternatives exist, most widespread though, is an integrated in-situ sectioning technique in which an SEM based focused ion beam (FIB) is applied to generate accurately spaced serial sections.

Advanced analysis of 3D EBSD data obtained from FIB-EBSD tomography

The combination of serial sectioning with a focused ion beam (FIB) and EBSD-based orientation microscopy in a combined FIB-SEM constitutes, for certain applications, an ideal technique for 3D microstructure characterization. The FIB sectioning delivers accurately spaced serial sections while EBSD-based orientation mapping delivers quantitative materials characterization ideally suited for microstructure reconstruction. The spatial 3D EBSD data are measured as unconnected volume pixels (“voxels”). In order to group these voxels into crystallites the measured sections have to be aligned such that subsequent sections fit together in an optimum manner. Alignment of complete sections is accomplished using cross-correlation techniques. Additionally, we have implemented a simple Monte-Carlo Potts-model algorithm which effectively cleans up the microstructure while conserving most of the sub-structure inside of the grains. Voxel-based description of microstructures can be used for characterization of stereological data, as grain size and shape, volume fraction of different phases etc., see an example in figure 1. Note, however, that many of these data can also be obtained from 2-dimensional measurements using stereological correlations. More complicated parameters like gradients of crystal rotations can be used to obtain a measure for the density of geometrically necessary dislocations in the sense of Nye ([1],[2]), for example. 3D EBSD data open the unique possibility to describe the 5-parameter crystallographic nature of interfaces consisting of the misorientation across the boundary (3 parameters) and the crystallographic orientation of one of the interface normals (2 parameters), see e.g. [3]. For boundary reconstruction we have employed the Marching Tetrahedra algorithm [4] which solves a number of problems involved in the originally proposed Marching Cube algorithm [5]. The as-meshed surface structure is subsequently smoothed using a strategy [6] which is inspired by computer models for grain growth simulation as implemented in a vertex model by Barrales [7]. The described algorithms have been employed for the analysis of selected grain boundaries in an Fe 28% Ni alloy. The microstructure of this material consists almost entirely of lenticular martensite, Fig. 2a. Three adjoining grains are isolated from the microstructure, Fig. 2b. The boundary character is described by plotting, in one pole figure, the rotation axis and the position of the grain boundary normal, figs. 2c and d. The boundary character is tilt when boundary normal and rotation axis are perpendicular and twist when they are parallel. It is interesting to note that in the present microstructure grain boundaries with a [110] 55° misorientation are all twist boundaries, fig. 2 (c) while boundaries with a [323] 53° misorientation almost all have tilt character, fig 2 (d).