3.1.1. Elastic and plastic deformation in bent Si plates

The dislocation density tensor, _ij, introduced originally by Nye (1953) for rotations [and generalized later to include elastic strains; see Kröner (1981)], provides a quantitative spatially localized measure of the GND density required to accommodate lattice curvature and elastic strain in deformed materials (Needlemann & Sevillano, 2003, and references therein). Defined mathematically (El-Azab, 2000) in terms of the difference between the local lattice curvature (i.e. three-dimensional rotation gradients) and the curl of the local elastic strain (i.e. three-dimensional strain gradients), GND densities are given numerically by the norm of the dislocation tensor divided by the Burgers vector (Larson et al., 2008, 2007). For dislocation-free elastic bending of brittle materials, the lattice curvature is accommodated entirely by elastic strains, while in elastically soft materials bending deformation is accommodated largely by the introduction of dislocations due to low yield strengths. With the ability to measure both local lattice rotations and elastic strains nondestructively, 3DXM is ideally suited for GND determinations. The first demonstration of nondestructive three-dimensional measurements of GND densities in terms of the dislocation tensor with micrometre resolution was performed on deformed thin (42 and 25 µm thickness) silicon plates using 3DXM (Larson et al., 2008). Using the geometries shown in Fig. 6, micrometre-spatial-resolution 3DXM measurements of local orientations and strains were made in the yz planes of an elastically bent Si plate and a similar, but plastically deformed, Si plate (Larson et al., 2008). GND densities were determined in the Si plate bent elastically to a radius of 5 mm (Fig. 6a) and in the similarly bent plate (Fig. 6b) after annealing to 973 K (i.e. plastically deformed by heating above the brittle-to-ductile transformation temperature under the bending stress). The spatially resolved dislocation tensor components, _ij, plotted in Fig. 6 were determined directly from the gradients of the rotations and strains measured by 3DXM, which were in turn extracted by numerical differentiation of the spatially resolved orientation and strain measurements (Larson et al., 2008).

Figure 6 Photographs of a 42 µm-thick silicon plate bent elastically to a 5 mm radius (a) and a 25 µm-thick silicon plate bent plastically by heating to 973 K under bending stress (b). The left side of the figure shows micrometre-resolution spatially resolved measurements of dislocation tensor components for (a) the elastically bent silicon plate and (b) the plastically bent silicon plate. Within statistical uncertainties, the elastically bent case (a) has no GNDs. The plastically bent case (b) indicates strong bands (red) of GNDs in the xy component and smaller density distributions (blue and red) in the xz component. Weaker but nonzero densities are visible in the remaining components as well. The dashed lines in the xy component of (b) correspond to two of the {111} plane traces, indicating that the plastic deformation is concentrated largely on these slip directions. The full-scale magnitudes of the GNDs in the color scale correspond to = 0.95 × 109 cm-2. Adapted from Larson et al. (2008) with permission.

As expected for a brittle material, no significant densities of GNDs were found in the elastically bent Si sample after removing measurement noise. However, the sample that was bent plastically 373 K above the brittle-to-ductile transformation temperature contained strong inhomogeneously distributed bands of GNDs lying along {111} slip-plane traces in the _xy component and somewhat lower GND densities in the _xz component, along with activity to a lesser degree in other components. The presence of low-density GNDs in several components suggests the presence of more complex local slip activity and, to some extent, the inherent presence of uncertainties in extracting numerical derivatives from local orientation measurements.

This nondestructive and quantitative GND density measurement capability of 3DXM has strong implications for fundamental investigations of deformation on mesoscopic length scales. It is possible to perform direct quantitative tests of crystal plasticity and discrete dislocation dynamics simulations of deformation. We note that GND densities are a function of local lattice curvature and, hence, are inherently a function of the spatial step size of the probing measurements. Accordingly, direct and absolute comparisons of measurements of GND densities with GND simulations require coarse graining of calculations with a resolution that matches that of the curvature and strain experimental measurements.

3.1.2. Deformation correlations in tensile-deformed Cu single crystals

The inhomogeneous and multiple-length-scale fluctuating nature of deformation in ductile materials suggests that statistical as well as local analyses of three-dimensional deformation measurements will be required in order to make quantitative comparisons of experiments with theory and simulations. Pang et al. (2010) performed the first three-dimensional spatial correlation analysis on deformation-induced misorientations using micrometre-resolution 3DXM deformation measurements on single-crystal Cu crystals deformed 8% in tension along the [123], [111] and [001] directions. The angular misorientations showed self-affine behavior for length scales (correlation lengths) of 70 µm for the single-slip [123] orientation compared with only 15-20 µm for the multi-slip [111] and [001] orientations. The significantly longer correlation length for the easy-glide single-slip direction indicates the disruptive impact of multiple-slip-plane deformation for face-centered-cubic materials. The spatial correlation analysis of the misorientations further provided a measure of the so-called Hurst exponent (0 < H < 1) (Zaiser & Moretti, 2005), characterizing whether the spatial correlations within the sample are positive (H > 0.5), negative (H < 0.5) or random (H = 0.5, uncorrelated). Interestingly, values of H = 0.72 were found at the sample surface for each strain direction; this is similar to the results simulated by Zaiser & Moretti (2005) for surface roughness in deformed crystalline materials. On the other hand, 10 µm below the surface of deformed Cu, H was found to decrease smoothly to H 0.7, 0.47 and 0.32 for the [123], [111] and [001] directions, respectively. The positive rotation correlations for single-slip deformation and anticorrelations for the strong multi-slip [001] direction are understandable qualitatively in terms of the intersection of slip on active slip planes in the bulk and the (geometrically limited) absence of slip intersections at the surface. The quantitative nature of the measurements and the near-surface spatial dependence of the correlations provide an opportunity for quantitative theoretical predictions and simulation of deformation and dislocation patterning on mesoscopic length scales.

3.1.3. Dislocation structures in heavily deformed metals

The formation and evolution of dislocation structures (patterning, dislocation cell structures) at the subgrain length scale is one of the most important aspects of the deformation process in ductile metals. Although it is generally appreciated that patterning arises from the collective interactions of dislocations (Hahner & Zaiser, 1999; Argon & Haasen, 1993; Thomson & Levine, 1998; Levine et al., 2006), it is not yet possible to predict the evolution of dislocation distributions and the resulting local stresses. Such understanding is a crucial foundation for developing trustworthy models for material deformation.

Stresses in materials cannot be measured directly. However, since SM-3DXM can measure absolute lattice parameters, comparison with the unstrained lattice spacing provides a direct measure of the elastic strain, which can in turn be converted to a stress using the stiffness tensor for the material. The first proof-of-principle demonstration that SM-3DXM can be used to measure local elastic strains was performed by Yang et al. (2003) on a cylindrically (elastic) bent thin plate of silicon. The first use of this technique to measure elastic strains within the dislocation microstructure of plastically deformed metal specimens was by Levine et al. (2006) on two unloaded single-crystal Cu samples that had been deformed in tension and compression (each) by 25% along the <001> axes. Using the axial 006 reflection, the average axial elastic strain was measured for a number of different dislocation cell interiors from both samples. The cell interiors of the tensile-deformed specimen were found to be in compression and those in the compression-deformed specimen were in tension. This counterintuitive result confirmed the long-standing composite model of plastic deformation that had been proposed by Mughrabi (1983) more than 20 years earlier. Equally importantly, these first spatially resolved measurements showed an unpredicted dramatic variation of strains from cell to cell within the sample.

More recently, Levine et al. (2011) combined submicrometre-spatial-resolution SM-3DXM with diffracted-beam masking of selected detector pixels to make (several hundred) direct measurements of axial elastic strain in individual dislocation cell walls and their adjacent cell interiors in heavily deformed (28% strain in compression) single-crystal copper. Converting to axial [001] stresses using the elastic modulus in the [001] direction, E₀₀₁, these data are combined in Fig. 7 to show the volume-normalized stress distribution functions for dislocation cell walls and cell interiors. These spatially resolved measurements showed broad asymmetric distributions of dipolar stresses that are markedly displaced from one another. Comparison with detailed deformation models demonstrated that the distribution of local stresses is consistent with random trapping of mobile dislocations by dislocation walls.

Figure 7 Axial stresses within individual dislocation cell walls and cell interiors in copper single crystals deformed in compression by 28%. Adapted from Levine et al. (2011) with permission.

These SM-3DXM measurements of elastic strains in deformed metals have already answered numerous long-standing fundamental questions about metals deformation, as discussed in a recent comprehensive contextual review by Kassner et al. (2012). Extending these results to larger length scales requires consideration of the interaction of dislocations with interfaces such as grain boundaries in polycrystalline materials. Such studies could then address real-world engineering problems. An example of how polychromatic 3DXM has provided new insights on damage nucleation at this larger length scale was published recently by Wang et al. (2011). In that work, the role of twinning on damage nucleation was studied in commercial purity titanium strained to 1.5%. Using 3DXM measurements, the dislocation activity associated with forming T2 twins was investigated by quantitative analysis of the streaked shapes of various Bragg peaks within the spatially resolved Laue diffraction patterns.

Another important capability of 3DXM is that of assessing volume-averaged diffraction measurements. It should be remembered that the vast majority of X-ray diffraction measurements performed around the world have spatial resolutions larger than the microstructural features that are being explored. Thus, critical assumptions are often implicit in interpretations of these measurements. In a recent study, Levine et al. (2012) used SM-3DXM to directly test long-standing assumptions that are used to analyze volume-averaged X-ray line profiles from deformed metals (Ungár et al., 1984; Mughrabi et al., 1986). Submicrometre-resolution diffraction data obtained from the dislocation microstructures responsible for the volume-averaged profiles showed that, without additional constraints, the traditional volume-averaged analyses can produce ambiguous results.

3.1.4. Thermal grain growth in polycrystalline aluminium

One of the long-standing issues in technological processing of structural materials is that of understanding and predicting the three-dimensional process of recrystallization and grain growth on mesoscopic length scales following rolling or other deformation steps. Accordingly, the possibility of nondestructively monitoring the thermodynamics-driven evolution of polycrystalline grains with submicrometre spatial resolution was one of the driving forces for the development of polychromatic 3DXM. Fig. 8 shows 3DXM measurements of grain evolution in a 10 × 10 × 100 µm volume of Al after 1 h annealing steps between 623 and 633 K (Budai et al., 2008). Clearly, the grain-coarsening process for this 473 K hot-rolled 1000 series polycrystalline Al with 5 µm grain size involves processes with broad crystal growth fronts overtaking smaller grains in addition to local grain boundary orientation and local grain boundary curvature effects. Analysis procedures for three-dimensional point-to-point micrometre- or submicrometre-resolution data in terms of grain boundary orientations, boundary types and coincidence-site orientations have been developed, as have software routines to assess local drivers such as dislocation density and intragranular defects that can impact grain boundary motion. Unfortunately, as shown in the 633 K anneal panel, the impingement of fast growing grains from outside the original volume scanned by 3DXM limits the ability to model or understand such processes because of lack of knowledge about the initial source of these fast growing grains. Procedures to increase the speed of such three-dimensional measurements through faster detectors and optimized analysis routines have now been implemented at the APS, and significantly larger volumes are being analyzed. This will provide databases appropriate for direct predictive computational analyses (see Rollett et al., 2007; Olmsted et al., 2009).

Figure 8 False color plots of crystal grains in polycrystalline Al. The three-dimensional picture shows the volume of polycrystalline Al investigated with polychromatic 3DXM. The three slices below are false color plots of the grain sizes on a particular slice of the three-dimensional volume after isochronal annealing at 623, 628 and 633 K for 1 h between measurements. Adapted from Budai et al. (2008) with permission.

3.1.5. Deformation under micro-indentations

One of the important aspects of 3DXM is the ability to make local misorientation and strain measurements with high spatial resolution. However, the heterogeneous nature of deformation in large bulk samples provides a dilemma, with the need to choose between detailed measurements in a small selected volume or less comprehensive measurements averaging over larger volumes. Micro-indentations provide an inherent confinement of deformation within mesoscopic volumes and so provide an alternative option, albeit with spatially varying deformation conditions. Polychromatic 3DXM measurements with micrometre resolution under conical and Berkovich (pyramidal) indents (Yang, Larson, Pharr et al., 2004) demonstrated the confined nature of the deformation field below indents and the ability to perform measurements of the local rotations with high angular precision, and also that the deformation fields below both conical and pyramidal indents possess a hierarchy of local rotations and counter-rotations. Depending on the trajectory of the microbeam through the middle or the periphery of the indents or whether the microbeam trajectory passed directly under the indentation, the microbeam Laue diffraction patterns displayed complex rotational patterns with strong evidence for inhomogeneous dislocation patterning and cell structure characteristic of deformed copper, as discussed above for uniaxially deformed copper (Levine et al., 2006, 2011).

3.1.6. Mechanical properties of metal-matrix Mo-NiAl composites

The spatially resolved nature of 3DXM provides a natural capability for investigating mechanical properties in complex materials such as composites. To investigate the mechanical properties of 1 µm-diameter reinforcing Mo fibers in an Mo-NiAl metal-matrix composite, Barabash et al. (2011) used SM-3DXM at the APS to probe (see Fig. 9) the axial strain distributions in surface-normal-embedded Mo fibers and in Mo fibers exposed by 5 µm by selectively etching the matrix. Depth-resolved lattice parameter measurements were made using the 006 reflection along the [001] axial direction of the Mo fiber, as illustrated in the schematic at the top of Fig. 9. Etching the matrix to expose Mo fibers provided an opportunity to measure both the unstressed Mo [001] axial lattice parameter (in the exposed part of the fibers) and the width of the transition region as the fibers enter the matrix. This provided a test of the strength of the interface to maintain the -1% differential thermal expansion strain that was generated as the composite cooled from the melt. Using similar measurements on polished but not etched Mo-NiAl surfaces provided a second test of the interface integrity transition depth. The results of measurements for both cases are plotted in Fig. 9. Making use of the geometry, including the 45° angle of the beam with the sample surface and the fibers, the inter-fiber separations in the composite, the coincidence (along the microbeam) of gaps in matrix Bragg reflections with the appearance of fiber Bragg reflections and the position of the onset of matrix Bragg reflections at the surface, Barabash et al. (2011) extracted an effective interface strength of 180 MPa and a near-surface slip-zone distance of 5 µm by fitting to a mechanical model. This slip-zone distance corresponds approximately to the distance over which the fiber strain falls from -1% to zero in Fig. 9.

Figure 9 The upper schematic drawing illustrates the geometry for scanned monochromatic 3DXM measurements of strain normal to the NiAl matrix face and along the axis of the Mo fibers in an Mo-NiAl metal-matrix composite. The plot below shows the strain in the exposed fibers (square symbols, red line) as a function of depth from the top of the exposed fibers (zero) to 25 µm deep into the matrix. The gray circles and black line correspond to similar measurements made on a similar sample without etching to expose fibers above the matrix surface. The lines correspond to micromechanical modeling to extract the fiber-matrix interface strength from the depth-dependent fiber strain measurements. Adapted from Barabash et al. (2011) with permission.

3.2.1. In situ deformation in metal pillars

It has been widely reported that single-crystal face-centered-cubic metal pillars formed using focused ion-beam milling are stronger for smaller sizes (Uchic et al., 2004; Greer & Nix, 2006; Bei et al., 2008; Budiman et al., 2008). Budiman et al. (2008) suggested from depth-integrated microbeam Laue measurements that dislocation starvation and not strain hardening led to the apparent added strength of smaller structures. However, Maaß et al. (2009), using in situ Laue microbeam diffraction (without depth resolution) in combination with crystal plasticity simulations, concluded that plastic deformation is controlled initially by unspecified boundary constraints (e.g. preexisting defects, surface damage, misalignment), often leading to initial non-Schmid slip activation during microcompression pillar deformation. Using polychromatic and scanned monochromatic microbeams to study 2-10 µm-diameter Au pillars in situ under microcompression, it was found that deformation started with a sharp onset of diffraction peak rotation (a so-called `Laue yield') that was followed by classical crystal plasticity corresponding to activation of the highest Schmid factors. Deformation then proceeded with size-dependent strain hardening in what Maaß et al. (2009) refer to as percolative slip.

3.2.2. Indentation size effects

Although GNDs are in general related to local dislocation density tensors as discussed above (Nye, 1953; Larson et al., 2008, 2007), depth-integrated 3DXM measurements of lattice rotations under the flat faces of pyramidal Berkovich indentations have been used to provide insight into deformation as well (Feng et al., 2008; Barabash et al., 2001). For instance, Feng et al. (2008) verified experimentally that the effective strain under self-similar indenters is independent of the indentation depth. This result was further used in connection with the revised Nix-Gao model (Durst et al., 2005) to estimate the size of plastic zones under indents and estimate strain gradients and GNDs without direct depth resolution. The complexity of indent deformation (Yang, Larson, Pharr et al., 2004) suggests that direct depth-resolved measurements will provide more detailed deformation insight.

3.2.3. Deformation microstructure in bulk deformed copper

Magid et al. (2009) performed depth-integrated polychromatic X-ray microbeam diffraction investigations of large copper single crystals oriented for single slip and compressed uniaxially by approximately 10%. Local deformation patterns were mapped with 1-3 µm-diameter white X-ray beams, over large (up to 297 µm square) regions of samples cut from the interior of the original deformed ingot. As has been known from etch pit analysis, the deformation is not uniform but strongly concentrated in irregularly spaced bands parallel to {111} planes: in the case studied, 30 µm spacing. The depth-integrated polychromatic Laue diffraction patterns provided information on the magnitude of the local lattice rotations integrated over 10-20 µm along the direction of the probing microbeam (limited by photoelectric absorption) as a function of position relative to slip bands in the sample. Magid et al. further reported that the analysis of deviatoric strain from the depth-integrated Laue diffraction patterns yielded residual deviatoric stresses significantly exceeding 500 MPa and that the high-stress regions were strongly correlated with the presence of the shear bands. The magnitude of the local stress (strain) state in the material is of course not provided by etch pit analysis; this highlights the importance of microbeam diffraction analyses. However, the presence of strains corresponding to 500 MPa in 10%-compressed Cu is surprising considering that such strains are an order of magnitude larger than those found for 30%-compressed copper using SM-3DXM (Levine et al., 2006, 2011).

As mentioned in connection with the analysis of white-beam diffraction patterns for deformed silicon above and discussed in more detail in the following subsection, experience has shown that considerable caution must be exercised in quantitatively extracting deviatoric strains when even small amounts of asterism or streaking are present in Laue diffraction spots. Considering the fact that the 500 MPa stresses extracted in the 10%-deformed Cu here correlate directly with the presence of shear bands (i.e. larger streaks), it is likely that the deviatoric stress magnitudes reported are significantly inflated by diffraction spot distortions. In cases of diffraction spot streaking or distortion, it is suggested that scanned monochromatic (SM-3DXM) measurements of local dilatational strains be used to corroborate the presence of such highly stressed material.

3.2.4. Uncertainties in the determination of deviatoric strains in deformed materials

Uncertainties associated with polychromatic deviatoric strain measurements are important as a general issue and so warrant further elaboration. The source of the uncertainty is known to be the slight differences in the streaking and asterism for projections of the local lattice rotations onto different crystal planes. As shown by Larson et al. (2002, 2008), 3DXM measurements of both deviatoric and dilatational strain can be performed accurately and precisely in deformed dislocation-free brittle materials such as cylindrically bent Si; however, as pointed out by Larson et al. (2008) the same is not true for Si deformed above the brittle-ductile transformation temperature. This is because streaking and asterism cause the centroids of Laue spots to vary as a function of diffraction plane orientation, thus producing spurious deviations in Laue spot position determinations. These streaking/asterism-induced deviations, in turn, introduce spurious angular deviations into the inherently small angular displacements associated with deviatoric strains in ductile materials and hence can exaggerate deviatoric strain values severely.

The Monte Carlo uncertainty analysis of Poshadel et al. (2012) addresses such issues in a general way, demonstrating quantitatively the importance of the above-stated uncertainty considerations associated with white-beam deviatoric strain measurements in a tensile-stressed 304 stainless steel wire. Analogously to the caution expressed with respect to stress determinations in deformed Cu (Magid et al., 2009) above and in cyclically deformed nitinol (Robertson et al., 2007) below, Poshadel et al. suggest the likelihood of significant artefact-induced uncertainties in the deviatoric strain results reported for plastically deformed quartz (Chen et al., 2011) and a nickel alloy (Chao et al., 2009). Technically the centroids of distorted peaks (especially when used without depth resolution) correspond to an undefined orientation average that is not mathematically related to a direct deviatoric strain measurement.

On the other hand, scanned monochromatic measurements (SM-3DXM) determine strains for single Bragg reflections at a time and are therefore inherently much less sensitive to such effects. Accordingly, 3DXM can provide accurate dilatational strain measurements in the presence of strong diffraction spot distortions, albeit at the cost of performing energy scan measurements for each spot as discussed with respect to results in Fig. 7. Fortunately, polychromatic orientation measurements are also much less impacted by streaking because of the collective nature of Laue spot movements for rotations, in contrast to the differential nature of Laue spot angle changes associated with the extraction of deviatoric strains. In this connection we comment that the measurements of plastic deformation-induced rotations and the determinations of crystal structure phases such as martensite versus austenite discussed by Robertson et al. (2007) for cyclic deformation of nitinol shape memory alloys are likely to be accurate. However, the magnitude of the deviatoric strains determined for cyclically deformed nitinol are likely to be significantly overestimated, especially in the highly distorted regions near crack tips.

3.2.5. Microbeams in high-pressure science

A new and important application for polychromatic and scanned monochromatic microbeam diffraction to high-pressure investigations of crystal structures (Wang et al., 2010) has emerged recently. Nanoscale X-ray beams make it possible to avoid key limitations in the study of the structure of materials up to multi-megabar (>200 Mbar, with 1 bar = 10⁵ Pa) pressures, including spatially resolved measurements of multiple grains within a single sample, measurements of stress gradients, and searches for crystal domains on micrometre or submicrometre length scales within multi-micrometre grain sizes in diamond-anvil cells. Measurements on Fe, Pt and W were used as demonstrations with beam sizes of 250 and 600 nm and pressures of up to 282 GPa, showing that nanoscale X-ray beams enable single-grain X-ray diffraction studies at ultra-high pressures in polycrystalline samples of postperovskite, a high-pressure phase of MgSiO₃. A further important aspect of the use of microbeams in high-pressure studies is the ability to achieve higher pressures and more extreme conditions by allowing additional miniaturization of pressure cells.

The importance of submicrometre polychromatic and scanned monochromatic X-ray microdiffraction in high-pressure science was demonstrated through the discovery by Mao et al. (2010) that a small rhombohedral distortion from simple cubic existed in the high-pressure Ca III phase of Ca at 300 K and that a monoclinic distortion from simple cubic exists at 30 K. Powder pattern measurements showed the structure to be simple cubic, while exhaustive theoretical work showed clearly that lower-enthalpy orthorhombic and tetragonal phases should form the stable structure. Single-nanocrystal diffraction patterns resolved the (pseudo) simple cubic patterns into multiple twin spots associated with rhombohedrally (at 300 K) and monoclincally (at 30 K) distorted simple cubic phases. This result in turn resolved the unlikely implication that a simple cubic structure could form the stable high-pressure structure at low temperature.