D.M. Collins

Measurement of probability distributions for internal stresses in dislocated crystals

Here, we analyse residual stress distributions obtained from various crystal systems using high resolution electron backscatter diffraction (EBSD) measurements. Histograms showing stress probability distributions exhibit tails extending to very high stress levels. We demonstrate that these extreme stress values are consistent with the functional form that should be expected for dislocated crystals. Analysis initially developed by Groma and co-workers for X-ray line profile analysis and based on the so-called “restricted second moment of the probability distribution” can be used to estimate the total dislocation density. The generality of the results are illustrated by application to three quite different systems, namely, face centred cubic Cu deformed in uniaxial tension, a body centred cubic steel deformed to larger strain by cold rolling, and hexagonal InAlN layers grown on misfitting sapphire and silicon carbide substrates.

The contrasting roles of creep and stress relaxation in the time-dependent deformation during in-situ cooling of a nickel-base single crystal superalloy

Time dependent plastic deformation in a single crystal nickel-base superalloy during cooling from casting relevant temperatures has been studied using a combination of in-situ neutron diffraction, transmission electron microscopy and modelling. Visco-plastic deformation during cooling was found to be dependent on the stress and constraints imposed to component contraction during cooling, which mechanistically comprises creep and stress relaxation. Creep results in progressive work hardening with dislocations shearing the γ′ precipitates, a high dislocation density in the γ channels and near the γ/γ′ interface and precipitate shearing. When macroscopic contraction is restricted, relaxation dominates. This leads to work softening from a decreased dislocation density and the presence of long segment stacking faults in γ phase. Changes in lattice strains occur to a similar magnitude in both the γ and γ′ phases during stress relaxation, while in creep there is no clear monotonic trend in lattice strain in the γ phase, but only a marginal increase in the γ′ precipitates. Using a visco-plastic law derived from in-situ experiments, the experimentally measured and calculated stresses during cooling show a good agreement when creep predominates. However, when stress relaxation dominates accounting for the decrease in dislocation density during cooling is essential.

Applications of multivariate statistical methods and simulation libraries to analysis of electron backscatter diffraction and transmission Kikuchi diffraction datasets

Multivariate statistical methods are widely used throughout the sciences, including microscopy, however, their utilisation for analysis of electron backscatter diffraction (EBSD) data has not been adequately explored. The basic aim of most EBSD analysis is to segment the spatial domain to reveal and quantify the microstructure, and links this to knowledge of the crystallography (e.g. crystal phase, orientation) within each segmented region. Two analysis strategies have been explored; principal component analysis (PCA) and k-means clustering. The intensity at individual (binned) pixels on the detector were used as the variables defining the multidimensional space in which each pattern in the map generates a single discrete point. PCA analysis alone did not work well but rotating factors to the VARIMAX solution did. K-means clustering also successfully segmented the data but was computational more expensive. The characteristic patterns produced by either VARIMAX or k-means clustering enhance weak patterns, remove pattern overlap, and allow subtle effects from polarity to be distinguished. Combining multivariate statistical analysis (MSA) approaches with template matching to simulation libraries can significantly reduce computational demand as the number of patterns to be matched is drastically reduced. Both template matching and MSA approaches may augment existing analysis methods but will not replace them in the majority of applications.

Applications of Multivariate Statistical Methods to Analysis of Electron Backscatter Diffraction and Transmission Kikuchi Diffraction Datasets

Multivariate statistical analysis (MSA) is widely used across science disciplines including microscopy, however, to date its application to electron backscattered diffraction (EBSD) data has been surprisingly limited. Brewer, Kotula, and Michael [1] demonstrated that MSA could be applied to EBSD data but this appears to be the only previous publication in the area. MSA are typically used for data reduction, identifying inter-related variables, and clusters in datasets. In applying MSA to EBSD and microstructural analysis the key benefits to pursue are its use in (i) segmenting the spatial domain into distinct similar regions (i.e. grains or sub-grains), and (ii) producing representative lower noise patterns associated with these domains to aid pattern indexing. MSA is unlikely to ever replace Hough-based analysis of EBSD patterns but may augment it by improving analysis of low quality data sets, or differentiating finer details not routinely detected though the Hough/Radon transform. We have used factor analysis within Matlab to implement MSA and present some illustrative examples below.

Analysis of Dislocation Densities using High Resolution Electron Backscatter Diffraction

Cross-correlation analysis of electron backscatter diffraction (EBSD) patterns allows measurement of elastic strain and lattice rotation variations at a sensitivity of ~±10 [1]. Cross-correlation is used to measure shifts between sub-regions of test and reference patterns and simple geometry allows the elastic strains and lattice rotations to be calculated from the measured dispersion of pattern shifts. In deformed metals the lattice rotations are often significantly larger than the elastic strains and in these situations a pattern ‘remapping’ approach has proved necessary to avoid artefacts in the strain fields [2]. One area of application for EBSD has been the determination of dislocation densities. The most explored route for quantifying the dislocation density has been to use the relationship, described by Nye [3] between the geometrically necessary dislocation (GND) density and the lattice curvature. In Nye’s analysis the excess density of dislocations in the crystal is directly related to the gradient of the lattice rotation field that induced. Unfortunately solution of the reverse problem of finding dislocation densities from the measured rotation gradients often does not have a unique solution. This situation is exacerbated by the fact that only 6 of the nine possible rotation gradient terms can be established from EBSD on a single section. Despite these issues, a lower bound estimate of the total dislocation density can be made (though the densities of particular dislocation types are ambiguous). Figure 1 shows an example map of GND density distribution within a Cu polycrystal deformed to 10% tensile strain. Of course the GND density is only a fraction of the total dislocation density because any dipoles or multipoles between the measurement points cause no measureable rotation gradient. This leads to differences in the GND density as the step size is varied. Figure 1 shows the GND density recovered in the same region but with the rotation gradients calculated using three different length scales. This is shown in a more quantitative way in figure 2 which shows how the average GND density recorded in this map reduces as the effective step size is made progressively larger [4]. To estimate the total dislocation density a new approach has been recently proposed [5]. This borrows from peak profile analysis used in X-ray and neutron diffraction assessment of dislocation density. From the EBSD data the variations of stress from the mean value in each grain can be calculated and used to form a plot showing the probability of obtaining a given stress level. In all dislocated crystal we have analyzed the probability has the form of a central Gaussian-like part but has tails showing higher probabilities at the high stress levels with the probability following ܲ(ߪ) = ܣ ߪ ⁄ . The form of these tails is consistent with the high stresses being generated by the localized stresses close to isolated dislocation cores, and the magnitude of the proportionality constant A can be used to determine the dislocation density. As the tails correspond to low probability, where experimental data tends to be somewhat noisy, rather than fitting the data directly we follow Groma’s method [5] developed for analysis of X-ray diffraction peak intensities of using the restricted second moment V2 of the probability ଶܸ(ߪ) = න ܲ(ߪ) ߪ ଶ dߪ ାఙ