, N, where N is the total number of voxels on the grid !, the assumptions can be expressed mathematically as ~ Yang et al., 2007, 2008, 2010 a, 2010 b !. It also assumes the total attenuation of an X-ray beam by a voxel is the sum over the attenuations by the individual compositional materials in the voxel. The model assumes that the total volume of a voxel is the sum of volumes of individual compositional materials: void, strontium chromate, and titania. On each voxel, there are three numerical channels storing volume fraction values for void, strontium chromate, and titania, and another two numerical channels storing X-ray attenuation coefficients obtained at two X-ray beam-energy spectra as described in the last section. The DCM numerical model is implemented on a simple cubic grid of voxels. Nevertheless, the DCM method can include partially occupied voxels in its model so that this issue does not present an obstacle to DCM modeling of the compositional microstructure. Similar datasets acquired with larger particle sizes did not show the same disparity. The resolution in this case is circa 2–3 m m, and because many of the particles are only a few microns across and in many cases appear to have a plate-like morphology, it is likely that the combination of resolution limitations and voxels being only partially occupied by a particle may cause the attenuation value disparity. This resolution can be affected by a number of factors including the point-spread function of the imaging system, thermal drift during data collection, and mechanical instabilities of the rotation axis. The second and most likely reason is because the particles are small in one or more dimensions relative to the resolution of the tomo- graphic reconstructions. One is that the particles them- selves may be microporous and therefore have lower density than that expected for the bulk material. There are two possible reasons for this observation. The final reconstructions showed attenuation values lower than expected for the particles. The 3D reconstructions for the Ti-target and Fe-target data were aligned in 3D using an Affine transformation function in the Avizo software package so that they could be used for DCM analysis ~ see Fig. A 3D reconstruction was then generated from each of the datasets. Prior to tomographic reconstruction the views were processed by a phase-retrieval algorithm ~ Paganin et al., 2002 !, which converts a phase-contrast image into a form suitable for tomographic reconstruction with a conventional CT algorithm. This takes the form of an edge enhancement of features in the images. The images produced by the XuM exhibit phase- contrast features due to the small X-ray source size of 0.1 m m. The third dataset using the Cr target still had to be collected separately from the other two however, pairwise alignment of views against the Fe dataset prior to reconstruction minimized displacement between this and the other two datasets. As they were acquired in parallel, any minor artifacts arising due to thermal drift over time were the same in both datasets, and the only differences between them were largely confined to the desired differences due to the different X-ray spectra. The pair of datasets using the Ti and Fe targets was acquired in this way with 720 views and a 0.5 8 angular step between each view. This minimizes the effect that any thermal drift might have in creating differences between the two datasets. For a given rotation position of the sample, an image is collected first using one X-ray target, and then using the other, before rotating the sample to acquire the next pair of views. The mod- ification enabled two such datasets to be acquired in parallel by deflecting the electron beam between the two X-ray targets during data collection ~ Fig. Tomographic data consist of a series of views, or X-ray images, of a sample acquired with the sample in different rotation positions around a ~ usually ! vertical axis. The XuM was subsequently modified to reduce misalignment artifacts, in particular misalignment between datasets. In particular, changes and thermal drift over time meant that the two different datasets suffered from different misalignment artifacts that negatively affected their usability for constraining the DCM. Analysis on this data was only possible after extensive use of in-house software ~ Mayo et al., 2007 ! to correct for misalignment. This creates difficul- ties because the two or more micro-CT datasets being used as DCM constraints must be aligned very accurately in three dimensions. In this case the two datasets suffered from misalignment artifacts, both within each dataset and between the two datasets. first attempts to collect data for analysis using this system were made by collecting each dataset one after the other, changing targets in between.
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