Three-dimensional plastic response in polycrystalline copper via near-field high-energy X-ray diffraction microscopyThis article forms part of a special issue dedicated to advanced diffraction imaging methods of materials, which will be published as a virtual special issue of the journal in 2013.

Li, S.F.; Lind, J.; Hefferan, C.M.; Pokharel, R.; Lienert, U.; Rollett, A.D.; Suter, R.M.

doi:10.1107/S0021889812039519

Volume 45
Part 6
Pages 1098-1108
December 2012

Received 23 May 2012
Accepted 17 September 2012
Online 25 October 2012

Three-dimensional plastic response in polycrystalline copper via near-field high-energy X-ray diffraction microscopy¹

S. F. Li,^a,^b ^* J. Lind,^b C. M. Hefferan,^b R. Pokharel,^b U. Lienert,^c A. D. Rollett ^b and R. M. Suter ^b

^aLawrence Livermore National Laboratory, 7000 East Avenue, Livermore, CA 94550, USA,^bCarnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, USA, and ^cAdvanced Photon Source, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439, USA
Correspondence e-mail: li31@llnl.gov

The evolution of the crystallographic orientation field in a polycrystalline sample of copper is mapped in three dimensions as tensile strain is applied. Using forward-modeling analysis of high-energy X-ray diffraction microscopy data collected at the Advanced Photon Source, the ability to track intragranular orientation variations is demonstrated on an 2 µm length scale with 0.1° orientation precision. Lattice rotations within grains are tracked between states with 1° precision. Detailed analysis is presented for a sample cross section before and after 6% strain. The voxel-based (0.625 µm triangular mesh) reconstructed structure is used to calculate kernel-averaged misorientation maps, which exhibit complex patterns. Simulated scattering from the reconstructed orientation field is shown to reproduce complex scattering patterns generated by the defected microstructure. Spatial variation of a goodness-of-fit or confidence metric associated with the optimized orientation field indicates regions of relatively high or low orientational disorder. An alignment procedure is used to match sample cross sections in the different strain states. The data and analysis methods point toward the ability to perform detailed comparisons between polycrystal plasticity computational model predictions and experimental observations of macroscopic volumes of material.

Keywords: three-dimensional plastic response; polycrystalline copper; near-field high-energy X-ray diffraction microscopy; diffraction imaging.

1. Introduction

Structural materials are subject to complex mechanical loading under a wide variety of conditions. The ability of a material to sustain loads over time depends on many factors, but damage initiation and accumulation are often life limiting. Because loading alters the structure inside the bulk, it is critical to be able to measure three-dimensional responses. It is also increasingly clear that each individual grain is strongly influenced by its neighbors, which in turn means that understanding evolution under plastic deformation requires knowledge of the polycrystal structure in all three dimensions (Zeghadin et al., 2007). With spatially resolved, in situ measurements, comparison with and testing of computational models becomes possible, offering a route to a more robust understanding of materials responses and to the predictive capability that underlies, for example, the integrated computational materials engineering concept (Committee on Integrated Computational Materials Engineering, National Research Council, 2008).

Ideally, structure measurements at the meso-scale would characterize crystallographic orientations, defect content (or damage) giving rise to orientation gradients and local elastic strain fields (from which stresses can be computed). It remains challenging to combine all these quantities in a nondestructive spatially resolved measurement. Here, we demonstrate the capability to track crystallographic orientations in deformed microstructures on subgrain length scales. The measurements use near-field high-energy X-ray diffraction microscopy (nf-HEDM) to map orientations at 2 µm length scales and with 0.1° orientation resolution. Our measurement combines nf-HEDM with absorption tomography, which tracks the sample shape and, in late stages of ductile failure, can detect the formation of voids.

The ability to map orientation fields (a term meant to imply spatial resolution at subgrain length scales) inside bulk material is currently being developed by several groups. The following very brief review gives some context for the capabilities demonstrated below. The first detailed point-by-point measurements at surfaces became practical in the mid-1990s with the advent of automated electron backscatter diffraction (EBSD) microscopy (Adams, 1997). Automated serial sectioning systems now allow the generation of large three-dimensional data sets (MacSleyne et al., 2009; Rowenhorst et al., 2010). However, the restriction to surface sensitivity limits the ability to track sample evolution since the measured volume is destroyed in the sectioning process. Further, extraction of lattice curvature and geometrically necessary dislocation content is incomplete because of the missing third dimension (Sun et al., 2000; Pantleon, 2008). Surface relaxation limits conclusions about intrinsic bulk dislocation structures (Ren et al., 1998; Schwartz et al., 2009 $[Schwartz, A. J., Kumar, M., Adams, B. L. & Field, D. P. (2009). Editors. Electron Backscatter Diffraction in Materials Science. New York: Springer.]$ ; Field et al., 2010b) as well as inter- and intragranular strain distributions (Field et al., 2010a) (lattice elastic strain is a recent addition to the EBSD measurement repertoire; Britton & Wilkinson, 2012; Wright et al., 2011).

Advances in synchrotron X-ray techniques such as differential aperture X-ray microscopy (DAXM) (Larson et al., 2002; Barabash et al., 2010; Levine et al., 2006; Budai et al., 2003), three-dimensional X-ray diffraction (3DXRD) (Aydiner et al., 2009; Poulsen, 2004 $[Poulsen, H. F. (2004). Three-Dimensional X-ray Diffraction Microscopy. Berlin, Heidelberg: Springer Verlag.]$ ; Poulsen et al., 2001), near- and far-field HEDM (nf-HEDM and ff-HEDM) (Suter et al., 2006; Lienert et al., 2009, 2011; Bernier et al., 2011; Edmiston et al., 2011) and diffraction contrast tomography (DCT) (Ludwig et al., 2008, 2009; Johnson et al., 2008; King et al., 2010) have led to the possibility of measuring orientation and strain states inside polycrystalline samples without the need for sectioning. This nondestructive capability opens the door to direct studies of materials evolution. DAXM, which uses a polychromatic sub-micrometre focused beam with an energy range up to 20 keV, yields detailed maps that include point-by-point strain sensitivity as well as the orientation field (Barabash et al., 2010; Levine et al., 2006; Budai et al., 2003). As a result of the sub-micrometre beam size, the spatial resolution from DAXM orientation maps generally tends to be superior to that obtained using the monochromatic rotating-crystal techniques; however, the use of relatively low energy X-rays limits the thickness of sample that can be interrogated (e.g. <50 µm in copper). In contrast, the high-energy (>50 keV) monochromatic beam techniques (3DXRD, HEDM and DCT) can probe millimetre-sized samples containing elements covering much of the periodic table, and they multiplex data collection from many grains by using area detectors in combination with either a line-focused or a rectangular beam to illuminate a quasi-planar section or an entire volume of material, respectively. Sample rotation is required to achieve a set of Bragg conditions for each grain and resolution is typically limited by the detector and beam-focusing optics to a few micrometres. The usual analysis techniques used for 3DXRD (Pantleon et al., 2004; Poulsen et al., 2003; Aydiner et al., 2009; Oddershede et al., 2011), far-field HEDM (Lienert et al., 2009; Bernier et al., 2011; Edmiston et al., 2011) and DCT (Ludwig et al., 2008; Johnson et al., 2008; King et al., 2010) are performed on a grain-by-grain basis through identification of compact diffraction spots that are associated with individual crystalline grains (Schmidt et al., 2003; Lienert et al., 2009; Aydiner et al., 2009; Bernier et al., 2011; Edmiston et al., 2011).

Since many engineering materials are laden with defects, there is strong motivation to resolve orientation and strain variations within grains. Substantial progress in this direction has been made. For example, in 3DXRD, broadening of individual scattering peaks due to orientation distributions has been analyzed (Winther, 2008; Poulsen et al., 2003), and in ff-HEDM, grain orientation distribution functions have been extracted on a grain-averaged basis (Bernier & Miller, 2006; Miller et al., 2008; Bernier et al., 2008). Very far field HEDM measurements (Lienert et al., 2011; Jakobsen et al., 2006) have been used to resolve subgrain structures in a region of reciprocal space associated with a single peak. The reticulography extension of DCT (King et al., 2010), and a modified version of the GrainSweeper software (Borthwick et al., 2012) have successfully extracted intragranular orientation variations.

Further, with the use of prior assumptions and a probabilistic algorithm, Rodek et al. (2007) have shown promising results in improving reconstructions that are originally based on back-projection methods. Voxel-based nf-HEDM forward-modeling reconstructions that independently optimize orientations in each voxel with no a priori assumption of grain structure have recently been shown to be sensitive to complex intragranular structure (Li, 2011; Hefferan et al., 2012). As seen below, this allows in situ tracking of deformation structures in rather strongly defected samples.

A distinguishing feature of the present measurement is that, instead of using a point-by-point measurement as in EBSD or DAXM, a wide beam is used to illuminate a thin sample cross section or layer of material. The data collection is done in parallel for each layer studied. This offers the advantage of faster collection times and reduced data volumes for comparable sample space coverage or access to dramatically increased sample volumes. However, this multiplexing also raises the question of overlap on the detector images of diffraction signals originating in different parts of the sample and, thus, the uniqueness of the interpretation of signals and the resultant reconstructions of complex orientation fields. Further, it is well known that, in deformed copper, dislocation cells form that, depending on the extent of deformation, can have sizes that are smaller than the spatial resolution of the measurement (Hughes, 2002; Hughes et al., 2003; Levine et al., 2006). So there are two questions: are the obtained orientations in each voxel in some definable sense correct (to within some uncertainty) and what do they represent? One aim of this work is to test the efficacy of the entire procedure, from overcoming experimental geometrical constraints to performing adequate reconstructions. It is shown that the generated reconstructions strongly, although imperfectly, reflect the observed diffraction signals and contain ordered structural features characterized by subdegree orientation differences; these features extend over distances up to the grain size and have boundaries that are smooth to within the experimental spatial resolution (2 µm). In §4, we will return to the interpretational questions posed above, after we present the experimental and data treatment procedures in §2 and results in §3.

2. Measurement methods

2.1. Sample and experimental setup

Measurements were performed at the Advanced Photon Source beamline 1-ID, which is a dedicated high-energy X-ray facility (Lienert et al., 2011; Suter et al., 2006). We collected both nf-HEDM and absorption tomography data sets in a succession of sample strain states. For nf-HEDM orientation mapping, we used a 64 keV monochromatic line-focused X-ray beam of roughly 3 µm height to illuminate a cross section of the sample. A volume measurement is obtained by measuring successive cross sections, translating the sample vertically in 4 µm steps. At each such sample position, Bragg scattering was imaged using an optically coupled 2k × 2k CCD camera that was focused on a thin scintillation screen (Martin & Koch, 2006); the effective detector pixel size was 1.47 µm. Images were collected as the sample rotated (about the normal to the X-ray beam plane) through 180 successive $[\delta\omega = 1]$ ° integration intervals. For most layer measurements, this procedure was repeated at two rotation-axis-to-detector distances (6.4 and 8.4 mm). For a small subset, a third distance (10.4 mm) was included to aid in the precise determination of the experimental geometry needed for reconstructions of the microstructure (the layer used in illustrations below used two detector distances). For absorption tomography measurements, the X-ray focusing lenses are removed so that a 300 µm-high beam illuminates the gauge section of the wire (referred to as full-field imaging). Three-dimensional tomographic contours (as in Fig. 1b) trace the outline of the sample in each strain state. Near failure, these contours also make it possible to identify void formation in the narrowest region of the necked wire (Weck et al., 2008).

Figure 1
(a) Side-view schematic of the tensile loading system and experimental geometry. X-rays (thick black line) are incident from the right and a diffracted beam is shown propagating to detector screens on the left at distances L₁ and L₂ from the [omega]

rotation axis. The sample (box enclosed by Macor) is positioned on this axis. A beamblock (black) attenuates the undiffracted beam but allows passage of upward-directed diffracted beams from the gray sample gauge volume. The load-bearing elements are the Macor ceramic cylinder, the aluminium plate and the vertical standoffs. The entire load stage is mounted on XYZ translation stages which, in turn, are mounted on an air-bearing rotation stage ( [omega]

) (Suter et al., 2006

[Suter, R. M., Hennessy, D., Xiao, C. & Lienert, U. (2006). Rev. Sci. Instrum. 77, 123905.]

). (b) An expanded view of the sample and ceramic holder. Set screws clamp the ends of the wire. The necked sample cross section is imaged on the right via in situ full-field tomography. The arrow indicates the approximate location of a cross section for which detailed analysis is presented.

The starting material for the sample was 1 mm-diameter high-purity (99.9999%) copper wire. Using an electro-polishing treatment, the diameter of a section 1.2 mm in length was reduced to 0.5 mm. A gauge section 0.3 mm in length was further electro-polished to 0.2 mm to form the experimental volume. The resulting sample, shown in Fig. 1(b), has a tapered neck, which focuses the strain on the experimentally probed volume. The smoothly varying cross section minimizes local stress concentrations that would be found at sharp corners. Pulling on this sample geometry amounts to pulling on a roughly cylindrically symmetric notched specimen. To remove residual damage the wire was annealed in forming gas (95% N₂, 5% H₂) for 30 min at 673 K. However, it was found in this case that handling the sample during mounting resulted in significant damage, as will be evident below.

A miniature uniaxial loading system was used to apply tensile strain in constant macro-strain steps. As shown in Fig. 1(a), the system consists of an X-ray-transparent hollow ceramic tube, a calibrated force sensor or load cell, and a stepper motor. The copper wire is fixed at the top of the tube, passes through its length and is attached to the load cell below. The gauge section is placed close to the center of the ceramic tube. The cylindrically symmetric design allows diffraction imaging with the near-field detector as close as 6 mm from the sample. The entire loading system, mounted on a precision rotation stage, can be rotated over the 180° necessary for high-spatial-resolution mapping of the crystallographic orientation field (Lienert et al., 2011; Poulsen, 2004 $[Poulsen, H. F. (2004). Three-Dimensional X-ray Diffraction Microscopy. Berlin, Heidelberg: Springer Verlag.]$ ; Suter et al., 2006).

The force-extension curve shown in Fig. 2 shows the work hardening typical of pure copper (Christodoulou et al., 1986). The peak force of 6.5 N applied to the 0.2 mm-diameter narrowest section of the wire indicates a stress of 210 MPa, which corresponds to 12% strain (Christodoulou et al., 1986). The strain elsewhere is reduced by the increased cross-sectional area. The macroscopic average strain in the gauge section is calculated from changes in length based on the positions of corners marking the transition between the 500 and 250 µm-diameter electro-polished regions, which can be seen in Fig. 2. We estimate the average strain at the peak of the force-extension curve to be $[6\pm 2]$ %.

Figure 2
Load-cell readings as a function of the displacement of the tension-applying translation stage. Extension is plotted instead of strain because the gauge cross section, and thus strain, are variable. Light-gray dots indicate states where HEDM imaging was performed, but only data from the first three such points are discussed here.

nf-HEDM measurements spanned 200 µm along the wire length and were centered on the narrowest part of the neck. After load has been applied, the sample was allowed to relax for 30-60 min before data collection began. The sample was then held at constant displacement for up to 24 h during measurements. The load reading was stable over this time, indicating that rotation and translation stage movements have minimal effect on the sample state. The measurements thus correspond to `stop-action' states of the sample microstructure.

While HEDM data were collected in the five states shown in the force-extension curve of Fig. 2, we report here on the first three states, labeled S0, S1 and S2, and, specifically, we show the evolution of microstructure in one cross section [indicated by the arrow in Fig. 1(b)] that is tracked across the three states. The strain in the measured cross section should be close to the average (as calculated above) because the cross section is of intermediate dimension. We thus estimate the strains in the measured cross section to be 0, <1 and 6% in S0, S1 and S2, respectively. Note that, between S0 and S1, the sample and apparatus simply become taut and no significant deformation is expected. Comparison of reconstructions of these states provides a convenient check of several aspects of the analysis, including alignment and repeatability. S2 is near the top of the force-extension curve and exhibits significant damage accumulation. Further damage accumulation in higher-strain states and closer to the neck center makes orientation reconstructions increasingly difficult. We therefore use the reference layer in states S0, S1 and S2 to demonstrate the nature of the analysis that becomes possible with spatially resolved data sets as well as to illustrate some of the difficulties.

2.2. Orientation reconstructions

Microstructure reconstructions are generated using forward-modeling software that runs on high-performance computing systems (Li, 2011; Suter et al., 2006). The reconstruction code performs a simulation of the experiment while adjusting orientations in each of many discretized mesh elements or voxels. The reconstructions presented below simulate scattering from equilateral triangular elements of 0.625 µm side length. Effects of this discretization and the two-dimensional approximation are discussed in §4. Each voxel orientation is optimized by maximizing the overlap of simulated diffraction signals with observed experimental intensity patterns; each voxel is optimized independently. There are approximately 3×10⁵ voxels per layer; each layer required roughly 12 h while using 2048 cores. A goodness-of-fit or confidence parameter, $[{\cal C}]$ , is computed in each voxel as the fraction of simulated Bragg peaks that overlay experimentally observed intensity at more than one detector distance. Note that the crystallographic orientation in each voxel is determined independently by comparison of all of its Bragg scattering (100 to 150 peaks over the experimental range used) with the entire experimental data set. $[\cal C]$ is large only when many intensity features in the data set follow the precession of the position of the voxel as it rotates with [omega] and generates diffracted beams at a discrete set of [omega] positions.

For undeformed materials and with carefully tuned experimental geometry parameters, $[{\cal C}\simeq 1]$ over most map areas (with expected reductions near grain boundaries and triple lines due to decreased intensity and noise in the precise shapes of diffraction spots). Orientation and position resolution were found to be 0.1° and 2 µm, respectively (Hefferan et al., 2009; Li, 2011; Lienert et al., 2011). Note that standard EBSD systems yield 1° orientation noise but have significantly better spatial resolution ( [less-than or equal to] 0.1 µm). Reduction in $[{\cal C}]$ is expected as diffraction signals weaken and broaden because of damage accumulation (as discussed below). However, given that roughly 50-75 diffraction peaks from a voxel overlap experimental scattering even for $[{\cal C} = 0.5]$ , the reconstruction is fairly robust against this reduction in signal-to-noise ratio.

In deformed materials, diffracted intensity is spread into arcs in the detector space as a result of orientation gradients and defect content within grains. Elastic strains in copper are small (of order 10^-4), which means that associated peak shifts in the near-field measurement are negligible. The intensity arcs are seen in the experimental detector space as the broadening of diffraction spots into azimuthal arcs (the [eta] direction) and onto multiple contiguous detector images (the [omega] direction). This broadening eventually results in detector images with severely overlapping peaks that make analysis difficult (Jakobsen et al., 2006; Lienert et al., 2009; Poulsen, 2004 $[Poulsen, H. F. (2004). Three-Dimensional X-ray Diffraction Microscopy. Berlin, Heidelberg: Springer Verlag.]$ ). Note that lattice rotation about a particular crystallographic axis, say <uvw>, generates a specific characteristic broadening profile. Scattering through reciprocal lattice vectors close to the <uvw> direction remains relatively sharp and, therefore, intense, while scattering near the perpendicular plane will be maximally broadened and weakened. With a finite experimental dynamic range, one expects to lose some scattering in the background while other scattering remains visible. It is reasonable then that deformed microstructures, prior to the point of severe overlap, can be reconstructed but with reduced average values of $[{\cal C}]$ ; in this paper, we demonstrate the degree to which this is practical in the current experimental context.

The reconstruction code uses as input a binary form of the detector images. Raw images are median filtered and background subtracted, after which peaks are identified and thresholded. Above-threshold pixels are saved. Despite the simplicity of this procedure, we show below that the reconstructed microstructures reproduce rather well the complex intensity patterns that originate from deformed grains. This is because, within a given spatially resolved peak, intensity variations are dominated by geometry. A sample cross-sectional region of uniform orientation will be projected onto the detector along the scattered wavevector, $[{\bf k}_{\rm f} = {\bf k}_{\rm i}+{\bf G}_{{hkl}}]$ , where $[{\bf k}_{\rm i}]$ is the incident-beam wavevector and $[{\bf G}_{{hkl}}]$ is a reciprocal lattice vector at the Bragg condition. For example, consider the case in which the projection of $[{\bf G}_{{hkl}}]$ onto the X-ray beam plane has only a component along the propagation axis ( $[\eta = 0]$ in conventional notation; Poulsen, 2004 $[Poulsen, H. F. (2004). Three-Dimensional X-ray Diffraction Microscopy. Berlin, Heidelberg: Springer Verlag.]$ ; Suter et al., 2006). In this case, a subvolume in sample space, $[V_{\rm p}\simeq \gamma\gamma\gamma/\tan 2\theta]$ , will project onto a given pixel, where [gamma] is the effective detector pixel size and the large dimension is that along the incident-beam direction (one of the factors of [gamma] would be replaced by the beam height if that height were smaller than [gamma] ). The intensity at this pixel is proportional to V_p. Intensities near boundaries are reduced in part because this volume spans the boundary and only a fraction contributes. Similarly, a lack of uniform orientation within V_p may spread intensity to different pixels and different detector images. Of course, in this case neighboring sample volumes may well contribute intensity to this pixel.

In our voxel-based reconstructions, the geometry described above implies the same situation: many neighboring voxels will generate scattering that strikes the same pixel. Simply counting the number of strikes gives an estimate of geometrically determined intensity variation. This assumes that voxel dimensions are not substantially larger than [gamma] . Orientation variations in the sample on a length scale smaller than the voxel size (which are expected in substantially deformed materials) will be averaged out, so it cannot be expected that the simulation will perfectly match measured patterns.

To demonstrate the above arguments, Fig. 3 compares observed intensity patterns with simulated intensities (the number of simulation voxels contributing to each detector pixel) in a region of overlap. A similar degree of overlap is observed over a large fraction of the measurement space. The simulated intensity originates from a region of 100×50 µm and extends over 9° in [omega] and up to 10° in [eta] . Diagonal streaking along the [eta] direction (shown by the arcs in the figure) reflects mosaic broadening, whereas the displacements of features are due to different positions of origin in the sample. The degree to which the simulated intensity reproduces intensity variations may seem surprising given that the binary data set includes all the pixels with above 25 counts on this scale. However, each simulated peak (connected regions of lit pixels) is a result of optimization of tens of voxels. The orientation of each voxel is estimated by optimizing overlaps between measured and simulated peaks projected on the detector; e.g. maximizing the number of pixel-overlapping experimental data for each simulated diffraction peak. Further, the combination of the Bragg condition with sensitivity to scattering origin implies that the diffraction peaks must lie on $[2\theta]$ rings that are centered on a moving diffraction origin. Thus, only a small set of diffraction intensities are geometrically compatible with each voxel, which leads to the relative consistency between the simulated and measured diffracted peaks.

Figure 3
(a)-(c) Scattering from a local region of the sample in strain state S1. The gray scale shows experimentally observed intensities in a small sub-region of the detector in three successive [omega]

intervals, with the black arcs indicating the azimuthal direction. The detector position is L₁ = 6.372 mm downstream from the rotation axis. Simulated patterns are shown as colored contours in arbitrary units, ranging from blue as the minimum to red as the maximum. This 200×200 pixel region is <1% of the detector area. The axis labels correspond to pixel number.

Beyond the strong correlation of experimental and simulated intensities, there is substantial diagonal streaking in the experimental data that is not captured by the simulation. This implies that the reconstruction tends to underestimate local orientation spreads as a result of the sharp Bragg condition used in each voxel. Note also that the simulated scattering at the bottom of the detector region does not overlap visible measured intensity. This scattering, which originates from a distinct sample region from that above, is again generated by an orientation that matches many observed scattering peaks; it may be that the scattering at this location (generated by a specific $[{\bf G}_{{hkl}}]$ ) has been weakened to the point that it is not visible or detectable in our measurement - the fact of its absence implies that information in the experimental data set has not been extracted by the reconstruction.

2.3. Strain state alignment

As the sample is strained by pulling on one end, rigid-body motion occurs and material flows because of plastic deformation. While tomographic data are used to assure that the same material volume is spanned by the measurement in each state, layers measured in one state will not, in general, correspond perfectly to those measured in another. Further, the complex sample geometry implies that the deformation is inhomogeneous along the 200 µm length of the necked section of the wire. In order to enable voxel-to-voxel comparisons between states, we perform an alignment optimization. For a particular measured layer in one state, we make comparisons with several layers in the same region of the wire in an adjacent strain state. Assuming that not all grains rotate in the same direction, the alignment is optimized by minimizing the total misorientation between the reconstructed orientation maps. This is done by calculating the following:

$[\min\limits_{{{\bf T}\in{\tt R}^{3},{\bf O}\in {\rm SO}(3)}}\left(\textstyle\sum\limits _{{i\in{\rm Voxel}}}{{d\left\{q({\bf x}_{i}),q^{{\prime}}[{\bf O}({{\bf x}_{i}}+{\bf T})]\right\}} / {N}}\right).\eqno(1)]$

$[ q({\bf x})]$ is the orientation at location $[{\bf x}]$ , d(q₁,q₂) measures the misorientation angle between q₁ and q₂, and N is the total number of voxels used for the alignment. The summation is over all voxels with reconstruction confidence above a threshold $[{\cal C}_{\rm t} = 0.4]$ in both maps {i.e. both q(x_i) and $[q[{\bf O}({\bf x}+{\bf T})]]$ exist and have well defined orientations}. $[{\bf O}]$ is a rigid-body rotation and $[{\bf T}]$ a translation that spans both in-plane and interlayer displacements. Monte Carlo variation of $[{\bf O}]$ and $[{\bf T}]$ finds the minimum average point-to-point misorientation.

3. Results

Fig. 4 shows reconstructions of the optimally aligned layer measurements in the three strain states. Comparison of the orientation maps shows similar overall grain structures on a color scale that spans the cubic fundamental zone of inequivalent orientations. The maps in S0 and S1 are consistent with no structure evolution (also consistent with Fig. 2) but with slight variations in boundary positions due to inexact matching of layers because of the discrete 4 µm measurement spacing and the slight sample rotation implied by straightening as the wire was pulled taut (while the alignment procedure aligns grain orientations, the line-focused X-ray beam illuminates planar sections that are tilted relative to those of the S0 state). The degree of structure matching is a good validation of the alignment procedure (for a more quantitative comparison see Fig. 5 below). While essentially all pre-existing orientations also appear in S2, taken near the top of the force-extension curve, some boundaries have now shifted significantly as a result of plastic deformation and alignment errors. Since the specimen is deformed anisotropically, the alignment between states S1 and S2 can only be viewed as approximate.

Figure 4
(a)-(c) Reconstructed orientation maps for the reference layer in states S0, S1 and S2. The color maps scale red, green and blue contributions so as to span the cubic fundamental zone of orientations in Rodrigues space. (d)-(f) Maps of the confidence parameter, $[{\cal C}]$ , corresponding to (a)-(c). (g)-(i) Kernel-averaged misorientation maps corresponding to (a)-(c). Superimposed lines indicate orientation discontinuities of [greater-than or equal to]

2°. The gray scale bar is in degrees. In all images, only voxels with $[{\cal C}\,\gt\, 0.3]$ are plotted.

The similarity of the maps of the confidence parameter, $[{\cal C}]$ , in Figs. 4(d) and 4(e) is also consistent with the structure being unmodified between S0 and S1. These two independent data sets yield the same high- and low-confidence regions. As discussed above, reduction of $[{\cal C}]$ along grain boundaries is expected. Here, however, entire regions have $[{\cal C}\leq 0.4]$ . These regions evidently contain sufficient orientational disorder to broaden and weaken scattering. Thus, the confidence maps in Figs. 4(d) and 4(e) indicate that damage was accumulated even as the delicate sample was mounted in the experimental apparatus. In Fig. 4(f), it is seen that additional weakening of the scattering has occurred with strain application. A valley of low confidence/heavy deformation has developed, extending from the lower left to the top of the image. Small regions (shown in white) along this valley are sufficiently damaged that no qualifying orientation could be identified. Qualitatively similar behavior in the same region is observed in neighboring layers.

3.1. Spatially resolved lattice rotation

Lattice orientation changes are computed on a point-to-point basis using the aligned data sets. Figs. 5(a) and 5(b) show rotation angles for S0 $[\rightarrow]$ S1 and S1 $[\rightarrow]$ S2, respectively. A strong confidence limit is used to assure that comparisons are made only between points with well defined orientation in both states. Thus, the `valley of low confidence' in Fig. 4(c) is not included here. As expected, little change is observed between states S0 and S1, with most of the map indicating <2° rotations. Dark-red regions (10-15° rotations) occur along grain boundaries reflecting residual misalignment and/or layer mismatch. The histogram for S0 $[\rightarrow]$ S1 in Fig. 6 shows a strong peak near 1°, indicating the alignment precision. Note that 1° is a factor of ten larger than the orientation noise, and this peak is probably due to the influence on the alignment function (§2.3) of imperfectly matched voxels [i.e. the dark-red ones in Fig. 5(a)]. Thus the alignment error dominates the orientation measurement error.

Figure 5
(a) S0 to S1 and (b) S1 to S2 lattice rotation angle maps. The color scale is in degrees and the horizontal scale bar indicates 100 µm. Only lattice rotations [less-than or equal to]

15° are shown for voxels in which both states have $[{\cal C}\,\gt \,0.4]$ . Voxels with $[{\cal C}\,\lt\, 0.4]$ are shown in white. (c) Inverse pole figure showing the rotation of the tensile axis, in the stereographic triangle, in local crystallographic coordinates, between states S1 and S2. Blue arrows show grain-averaged motions where `grains' are defined as connected regions with no orientation discontinuities [greater-than or equal to]

2°; these are roughly the regions of distinct color in Figs. 4

(b) and 4

(c). The radii of the blue circles indicate the relative areas of the respective grain cross sections; only the largest 20 cross sections are shown. Red arrows indicate motions of randomly selected individual voxels matched across states. (d) A selection of voxel-based arrows as in (c) plotted on the rotation angle map of (b) (here in gray scale). Arrow tails are placed at the relevant voxel positions.

Figure 6
Distribution of point-to-point intragranular lattice rotation angles between strain states. The `unoptimized' S1 $[\rightarrow]$ S2 histogram shows the effect of sample alignment optimization.

With S0 $[\rightarrow]$ S1 providing a `null result' calibration, we proceed to focus attention on responses in the S1 $[\rightarrow]$ S2 strain step. Fig. 5(b) shows that, between states S1 and S2, most measured voxels exhibit well resolved lattice rotations larger than 2°. Comparison with the grain positions and shapes in Fig. 4 indicates that, as expected, different grains rotate by different amounts and also that different regions within grains have distinct rotation angles. The distribution of lattice rotation angles seen in Fig. 6 for this case shows a broad range from 3 to 10° with peaks being associated with different average rotations of distinct grains. The dashed curve shows the apparent distribution before optimization of the sample orientation so as to give a rough bound on the uncertainty. The dotted curve should be considered as close to a lower bound since it corresponds to the minimum global misorientation.

The inverse pole figure in Fig. 5(c) gives an indication of the direction of rotation by plotting the position, in local crystal coordinates, of the tensile axis before and after strain application (Taylor, 1938). Substantial differences are observed between grain-averaged rotations and those based on individual voxels, and the voxel rotations indicate substantial variations in local regions of the stereographic triangle (especially the lower central region). Fig. 5(d) shows the fully spatially resolved rotations, including the magnitude (gray scale) and the direction (arrows) of rotations. While the rotations are defined for each voxel, for clarity only a small selection of arrows are shown. It is clear that averaging over grains, as occurs in measurements without subgrain resolution (Margulies et al., 2001; Oddershede et al., 2011; Poulsen et al., 2003; Winther et al., 2002, 2004), loses significant information about the dispersion of rotations within grains. Different regions of large grains rotate by different amounts and in different directions; this is the expected signature of grain break up. Intragranular dispersion is responsible for the broadening of diffraction peaks along Debye-Scherrer rings that is commonly observed (Lienert et al., 2009; Poulsen, 2004 $[Poulsen, H. F. (2004). Three-Dimensional X-ray Diffraction Microscopy. Berlin, Heidelberg: Springer Verlag.]$ ). Thus, at the estimated strain level of S2 ( $[6\pm 2]$ %) a far-field measurement from individual grains would observe arcs in reciprocal space that are rotated relative to the unstrained state by 3-10° and that are broadened to a few degrees in both the [eta] and the [omega] directions. Our results are consistent with statistical analysis of both rotation magnitudes and peak broadening of bulk grains (Poulsen et al., 2003; Winther et al., 2004). Spatially resolved rotation dispersion has recently been observed, via DAXM, in nickel as a function of distance from a sample surface with, evidently, quite complex behavior (Barabash et al., 2010). As a result of the complex geometry of our sample and the fact that most grains intersect the surface, a quantitative comparison with deformation models requires the use of numerical simulations, such as finite element (Ritz et al., 2010; Roters et al., 2010; Wong & Dawson, 2010) or image-based methods (Lebensohn, 2001).

3.2. Damage quantification

To quantify microstructure evolution and damage accumulation, we have calculated both the kernel-averaged misorientation and the geometrically necessary dislocation densities (Arsenlis & Parks, 1999; Mishra et al., 2009; Sun et al., 2000). Given the orientation field $[q({\bf x})]$ , the kernel-averaged misorientation, $[K({\bf x})]$ , is defined to be

$[K({\bf x}) = {{\sum\limits _{i}w_{i}({{\bf x}_{i}},{\bf x})d[q({{\bf x}_{i}}),q({\bf x})]}\over {\sum\limits _{i}w_{i}({{\bf x}_{i}},{\bf x})}},\eqno(2)]$

where $[q({\bf x})]$ and $[d(q,q^{{\prime}})]$ are defined as in (1) and the summation is performed over the N nearest neighbor points of $[{\bf x}]$ . The weighting factor, $[w({\bf x}^{{\prime}},{\bf x})]$ , is defined as

$[ w({\bf x}^{{\prime}}, {\bf x}) = \bigg\{\matrix{ {{1} / {|{\bf x}^{{\prime}}-{\bf x}|}}& {\rm if }\,\,d[q({\bf x}),q({\bf x}^{{\prime}})]\leq\theta _{\rm t}\cr 0 \hfill &{\rm otherwise.} \hfill }\eqno(3)]$

The threshold angle, $[\theta _{\rm t}]$ , is set so as to avoid inclusion of points that cross high-angle grain boundaries. The resolution of K is controlled by selecting the number of nearest neighbors to be consistent with the resolution of the measurement. In our case, this is limited to 1.5 µm because of the detector effective pixel size. It should be noted that the traditional definition of K in the EBSD literature does not include the weighting factor, w, which is used to produce a metric that is insensitive to voxel size and that can include a roughly isotropic region around each voxel. Here we determine orientations on a triangular mesh in two dimensions and we include the 12 side- and vertex-sharing triangles in computing K. $[\theta _{\rm t}]$ is set to 5°.

Kernel-averaged misorientation maps are shown in Figs. 4(g)-4(i). While both high- and low-angle boundaries change only slightly across the strain states, small features of 10 µm are seen to vary significantly. Islands of low K occur, indicating that deformation accumulation is non-uniform, as expected. In some cases, line segments of high K extend across grain boundaries, indicating shear banding (Kocks et al., 2000) and illustrating that deformation in a given grain is affected by deformation in neighboring grains (e.g. Field & Alankar, 2011).

Since we have measured a three-dimensional orientation field, the Nye tensor of geometrically necessary dislocation content is, in principle, computable. We demonstrate this capability but show that the result is substantially noisier than the scalar quantity, K. The Nye tensor is given by $[\alpha _{{ij}} = \varepsilon _{{ilk}}(e_{{jl,k}}+g_{{jl,k}})]$ , where $[g_{{ij}} = \Omega _{{ij}}+\delta _{{ij}}]$ , with $[\Omega _{{ij}}]$ being the orientation matrix (for full definitions and details, see Sun et al., 2000). Using the procedure outlined by Arsenlis & Parks (1999), the geometrically necessary dislocation densities, $[\rho _{{\rm GND}}]$ , can be computed by first rewriting the Nye tensor as sums of outer products of Burger's vectors and line directions, $[\alpha _{{ij}} = \textstyle\sum _{n}\rho _{n}b^{{(n)}}_{i}l^{{(n)}}_{j}]$ , and finding the set of $[\kappa = \{[b^{{(n)}}_{i},\; l_{i}^{{(n)}}]\}]$ such that $[\rho _{{\rm GND}} = \textstyle\sum _{n}\rho _{n}]$ is minimized, either in the L¹ or L² sense. Since the decomposition in the set [kappa] is non-unique, the individual dislocation types that result from the optimization are unphysical, but $[\rho _{{\rm GND}}]$ is a valid estimate provided that the set chosen samples the space of dislocation types in a reasonable way (Arsenlis & Parks, 1999).

A Nye tensor and its corresponding L²-minimized geometrically necessary dislocation density was computed for each sample point as illustrated in Fig. 7(a). The dislocation density distributions in the three strain states are shown in Fig. 7(b). A distinct shift in the cumulative distribution of dislocation density toward the larger values is seen between strain states S1 and S2, as expected, because of damage accumulation. A shift with noticeably smaller magnitude is seen between states S0 and S1, which gives an idea of the uncertainty in the GND estimate. The dislocation structure appears noisier than the K plots seen in Figs. 4(g)-4(i). Because numerical derivatives are produced using finite differences, noise in the orientation map is amplified in both cases. However, Nye tensor components are related to particular orientation derivatives in particular directions, whereas K is an average of the scalar misorientation angle alone. For this reason, one expects K to be a more stable scalar quantifier of damage accumulation although it also contains substantially less physical information. Furthermore, both quantities are computed from orientations resolved at the micrometre scale, whereas it is well known that additional substructure exists at shorter length scales. While K is simply a characterization of the measured field, one must be careful to interpret the Nye tensor or $[\rho _{{\rm GND}}]$ as a lower bound on the actual dislocation density present in the sample.

Figure 7
(a) Map of geometrically necessary dislocation density in the S1 state. The gray scale saturates at 10¹⁴ m^-2. Corresponding maps in S0 and S2 are qualitatively similar. (b) The cumulative geometrically necessary dislocation density distribution in each of the three strain states.

4. Discussion

The reconstructions shown in Fig. 4 are based on several approximations. The simulated incident X-ray beam is treated as truly two dimensional in that scattering patterns are generated by projecting two-dimensional discretized elements in sample space onto the detector. In reality, the incident beam is several micrometres (thus, several detector pixels) in extent. In a case where the beam illuminates a vertical region of uniform orientation, each observed diffraction peak will be extended in the vertical direction on the detector. Since the process of orientation reconstruction for each discrete sample element is done by implicitly finding the best fit orientation $[{\bf M}]$ for the set of $[{\bf G}_{{hkl}}]$ measured experimentally by the diffraction signal, the vertical extension of a diffraction peak due to beam height is manifested as the uncertainty in the measured $[{\bf G}_{{hkl}}]$ for a specific direction in the reciprocal space. The fact that the experimentally measured peaks are spread across the reciprocal space reduces the systematic bias that would otherwise dominate the measurement; furthermore, the resulting uncertainty in the estimated orientation $[{\bf M}]$ can be characterized as random. By assuming that the uncertainty induced by beam height follows a Gaussian distribution, the uncertainty in the reconstructed orientations can be crudely estimated by $[\sigma _{{\rm bh}}(M_{{ij}})\simeq{{\delta} / {(\ell{N^{1/2}})}}+{\rm c.t.}]$ , where $[\sigma _{{\rm bh}}(M_{{ij}})]$ is the uncertainty in the elements of the orientation matrix, $[{{\delta} /{\ell}}]$ is the angle subtended by the full width at half-maximum of the beam height at the voxel-to-detector distance l and c.t. represents the terms arising from correlated error statistics. At 100 diffraction peaks, the uniform sampling of the reciprocal space leads us to expect the correlated terms to be near zero, and a detector geometry of 6 µm FWHM and $[7\leq\ell\leq 9]$ mm results in a negligible $[\sigma _{{\rm bh}}(M_{{ij}})]$ contribution. However, when the number of diffraction peaks fitted is small, the uncertainty statistics become correlated owing to biased sampling of the reciprocal space; hence the uncertainty would be directionally dependent. In this case, the estimate above becomes inaccurate because of the contribution from the correlated terms.

Even though the sample space is discretized by 0.625 µm-side-length voxels, the minimum resolvable feature size is still dictated by the effective pixel size of 1.47 µm. The subpixel resolution discretization enhances the reconstruction quality by eliminating aliasing effects natural to pixelized detectors. Similar to the case of orientation reconstruction, pixel aliasing is an error with preferential direction in the detector space, but the resulting uncertainty is distributed into varying directions in the sample space because of sample rotation in the imaging method. However, subpixel-sized regions are not expected to be readily resolvable because of their small scattering volume, which results in diffraction peaks that are below our detection limits. As a result, voxels may be assumed to be single valued in the reconstruction. Imaging of subvolume features would require techniques coupled to more sensitive imaging systems, as used, for example, by Pantleon et al. (2004). Nonetheless, if scattering from the region delineated by the voxel indeed results in multiple orientations, the orientation with the largest volume fraction will dominate the diffraction signal. While it is possible for the current algorithm to present multiple solutions for each voxel, the orientation producing the strongest diffraction signal is presented instead.

Finally, while tomography was used to track the overall sample shape as shown in Fig. 1, for the single-layer analysis presented here, we have not trimmed the HEDM-determined shapes so that reconstruction uncertainties can be seen clearly. Note that the strong contrasts amongst diffraction signals from neighboring regions with multiple orientations contribute to the spatial resolving power of the grain boundary; the voxel orientation with the best confidence is necessarily the optimal solution. However, the same is not true for the sample exterior boundaries of the sample; the lack of diffraction signal outside of the sample requires orientation determination to be based purely on thresholding of the confidence function. In our experiment, an appropriate threshold level is chosen so that the reconstructed exterior surface is consistent with the tomographic contour at the outer rim of the relevant layer (blue in Figs. 4d-f) in the confidence map images.

5. Conclusions

A significant body of work is developing that uses high-energy X-rays to probe micro-mechanical responses of single grains inside bulk polycrystalline materials. In situ measurements under tensile deformation have monitored grain rotations (Barabash et al., 2010; Margulies et al., 2001; Poulsen et al., 2003; Winther, 2008, 2002, 2004) and the development of lattice strain tensors (Efstathiou et al., 2010; Lienert et al., 2009; Oddershede et al., 2011). Demonstrations have used aluminium, copper and titanium alloys, but the methods are applicable to a broad range of materials. Until recently, these measurements isolated diffraction signals from individual grains but did not place them in space or determine their microstructural neighborhoods. Recent work from Risø (Oddershede et al., 2011) determined positions of apparent centers of mass for 10³ grains and measured strain tensors as well as rotation responses.

The current work, while preliminary, demonstrates the new capability of spatially resolving the development of orientational disorder within individual grains whose shapes and neighborhoods are quantitatively tracked as strain is applied. Micrometre-scale spatial resolution allows determination of the orientation gradients that underlie the well known broadening of Bragg peaks along Debye-Scherrer cones. Specific orientation gradients give rise to specific broadening patterns and the forward-modeling reconstruction tracks these patterns. Rather than estimating intragranular orientation dispersion from individual peak smearing (Poulsen et al., 2003; Winther et al., 2004), we obtain spatially resolved orientation variations, which give access to full orientation distributions for individual grains. Voxel-to-voxel orientation differences can be used to compute local kernel-averaged misorientations, and even the populations of specific classes of dislocations that generate orientation variations can be computed. The ability to track such quantities as deformation proceeds provides a unique opportunity for testing computational models. The forward-modeling approach ultimately lends itself to an intimate interplay between models of evolution and raw diffraction data sets.

All the experimental and analysis methodologies discussed here are applicable to a variety of other materials processing environments. Thermal evolution of defected (Hefferan et al., 2012) and well ordered (Reed et al., 2012) grains is being studied, as are defected structures associated with fatigue, fracture and shock deformation.

Acknowledgements

This research was supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under award DESC0002001. Use of the Advanced Photon Source was supported by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, under contract No. DE-AC02-06CH11357. Research was also supported in part by the National Science Foundation through XSEDE resources provided by Texas Advanced Computing Center under grant No. DMR080072. SFL gratefully acknowledges funding under Laboratory Directed Research and Development program LDRD 10-ERD-053. This work was performed in part under the auspices of the US Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344 (LLNL-JRNL-557578).

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