# Challenges related to tomographic reconstruction of 3D intragranular orientation fields in the presence of orientation relationships

Z Liu, N Viganò, W Ludwig

► To cite this version:

Z Liu, N Viganò, W Ludwig. Challenges related to tomographic reconstruction of 3D intragranular orientation fields in the presence of orientation relationships. IOP Conference Series: Materials Science and Engineering, 2022, 1249 (1), pp.012030. 10.1088/1757-899X/1249/1/012030. [https://doi.org/10.5274/zenodo.575712]

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# Challenges related to tomographic reconstruction of 3D intragranular orientation fields in the presence of orientation relationships

To cite this article: Z Liu et al 2022 IOP Conf. Ser.: Mater. Sci. Eng. 1249 012030

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# Challenges related to tomographic reconstruction of 3D
intragranular orientation fields in the presence of orientation
relationships

Z Liu¹,², N Viganò² and W Ludwig¹,²

¹ MATEIS, INSA Lyon, UMR5510 CNRS, 69621 Villeurbanne, France

² ESRF, Grenoble, France

E-mail: zheheng.liu@esrf.fr

**Abstract.** X-ray diffraction contrast tomography (DCT) is a near-field diffraction imaging technique to characterize the 3D shape and crystallographic orientation of grains within polycrystalline samples. The presence of orientation relationships as such encountered between $\Sigma3^n$ annealing twins found in Cu and Ni lead to systematic diffraction spot overlap, since a significant fraction of the lattice planes is shared between the parent and twin crystal lattices. If not correctly addressed, these overlaps will lead to artifacts in the individual grain reconstruction. In this paper we introduce a strategy for joint, tomographic reconstruction of Twin Related Domains (TRD) and cost function weights into DCT to improve the grain reconstruction quality. A large-grained polycrystalline sample made from pure Ni is used for testing this approach and its ability to reveal intragranular orientation gradients related to plastic strain localization at the onset of plastic deformation in this type of materials.

## 1. Introduction

Strain localization is a plastic instability process before fracture. There are many studies on that subject, and some examples can be found in [1, 2]. During initial stages of plastic deformation in metallic alloys, strain localizes in the form of slip bands. The observation of these slip bands at the sample surface, using a combination of scanning electron microscopy techniques (SEM, EBSD, DIC) is an important method to study plasticity at the microscale. In this paper we study the feasibility to use full-field X-ray diffraction imaging techniques to characterize orientation gradients and slip bands in three-dimensions in the bulk of metallic polycrystals.

X-ray Diffraction Contrast Tomography (DCT) [3] is a non-destructive X-ray imaging technique to reconstruct the microstructure of the grains in polycrystalline sample volumes. The initial approach, hereafter termed 3D-DCT, assumes all the voxels of the grain have the same orientation and applies to well recrystallized microstructures with small intragranular orientation spread. 6D-DCT [4] on the other hand can reconstruct both the shape and intragranular orientations of the grain. It is based on a discretization of 3D orientation space, so that the 3D position space is expanded into a 6D position-orientation space. In single grain reconstructions, only a small sub-region of 3D orientation space (centered on the grain average orientation) is sampled. This concept can be extended to *cluster* reconstructions [5], i.e. sampling of multiple, disjoint regions of orientation space, enabling the joint (simultaneous) reconstruction of grains exhibiting orientation relationships. This in turn allows to correctly handle the presence of shared reflections between different domains of the cluster, which can

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now be included in the optimization process without violating the consistency of the tomographic reconstruction problem.

In this study, we apply the above outlined 6D-DCT cluster reconstruction to the case of a Twin Related Domain (TRD) in a miniature tensile sample, made from pure Ni. The reconstructed intragranular orientation gradients and signatures of strain localization, visible in the diffraction topographs are discussed.

## 2. Material and method

The miniature tensile sample made from pure Ni with a cross-section of 400×400 µm is extracted and subsequently annealed to produce a fully recrystallized grain microstructure with an average grain size of ~ 100 µm. The sample is mounted in the NANOX [6] stress rig and scanned at two different applied loads of ~ 2N and 20N, corresponding to ~ 0 and 0.2% applied strain. Each of the DCT scans comprises 7200 images with 0.05° rotation increment, using an X-ray beam energy of 43.5 keV. The detector pixel size is 1.63 µm.

The average orientations and the positions of the grains are computed in a Friedel-pair-based grain indexing step [7]. Then, the shapes and the intragranular orientations of the grains are reconstructed from its corresponding diffraction spots [4]. Usually, the reconstructions are performed grain by grain individually. However, complications may arise for materials exhibiting orientation relationships between neighboring grains. A prominent example are annealing twins in metallic materials with FCC lattice structure like Cu and Ni, for which the parent and twin orientation share subsets of common lattice directions. For example, if the grain has a Σ3 twin, half of the {2 2 0} and {3 1 1} lattice planes of the grain have common reflections with the twin. These common reflections lead to spots with overlapping intensity distributions, which would have to be excluded from the reconstruction of the parent/twin domains in order to keep consistency of the tomographic reconstruction problem. However, these materials often exhibit extended neighborhoods of TRD, containing higher order Σ3n twins, which further reduce the number of reflections which can unambiguously be assigned to an individual domain. The quality of the individual grain reconstruction can be strongly compromised, if the presence of resulting diffraction spot overlaps is not taken into account.

To solve this problem, we propose a generalization of the joint reconstruction approach outlined in [4]. Instead of reconstructing grains one by one, the entire material volume of the TRD is reconstructed at once. Firstly, the TRD recognition [8] is applied. Then, the mappings are created between the grains and the diffraction spots. Afterwards, the TRD is jointly reconstructed based on a modified version of the 6D-DCT algorithm [9].

The reconstruction problem is formulated as (1). The column vector **x** contains the unknown variables in 6D position-orientation space. **y** contains the pixel intensities of the corresponding diffraction spots. **A** is the (sparse) projection matrix, which specifies the contribution of each of the elements of **x** to each of the detector pixels **y**. λ||**H**(**x**)||₁ is the regularization term. Here, the spatial 3D intensity volume (i.e. the sum over the reconstructed orientation space contributions) is regularized by the total variation norm, promoting solutions exhibiting piecewise constant intensity (i.e. similar diffraction power for each voxel inside a grain). **W** is a diagonal matrix containing the cost function weights.

$$ \mathbf{x}^* = \arg \min_{\mathbf{x}} \frac{1}{2} \left\| \mathbf{W}^{\frac{1}{2}} (\mathbf{y} - \mathbf{A}\mathbf{x}) \right\|_2^2 + \lambda \|\mathbf{H}(\mathbf{x})\|_1, \text{ subject to } \mathbf{x} \ge 0 \quad (1) $$

The weights are inversely proportional to the noise variance. The pixels of the detector are assumed to have a Poisson noise and the variance of this Poisson noise scales with the intensity of the diffracted beams. Moreover, an estimate of σdark, the variance of the thermal noise of the detector, is calculated from the images without beam. Therefore, the weights are formulated as (2).
$$W = \sigma_{\text{dark}} [\text{diag}(\sigma_{\text{dark}} + \mathbf{y})]^{-1} = \begin{bmatrix} \frac{\sigma_{\text{dark}}}{\sigma_{\text{dark}} + \mathbf{y}_1} & & \\ & \ddots & \\ & & \frac{\sigma_{\text{dark}}}{\sigma_{\text{dark}} + \mathbf{y}_2} & \\ & & & \ddots \\ & & & & \sigma_{\text{dark}} \end{bmatrix} \quad (2)$$

### 3. Results

As a first step, 3D-DCT is used to reconstruct an approximate grain map of the sample under the load of 20N. A first refinement is obtained by switching to the 6D-DCT algorithm, yielding improved grain shape reconstructions for this material, which shows non-negligible intra-granular orientation spread, inherited from previous processing and sample handling steps. By segmenting the reconstructed grain intensity and dilating the segmented grain masks by 5 pixels, the grain map of the sample volume is created. Figure 1 shows a slice in the y-z plane of the sample. Next, using the TRD recognition procedure outlined in [8], the TRD map of the grain volume is obtained. Figure 2 shows the corresponding slice of the TRD map. Comparing these two figures, the grains A, B and C indicated in figure 1 belong to the same TRD.

<figure><img src="image_7.png" /><figcaption>Figure 1. 2D slice (y-z plane) of the grain map in reconstructed sample volume.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 2. 2D slice (y-z plane) of the TRD map in reconstructed sample volume.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 3. Diffraction spot of grain A.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 4. Overlapped diffraction spots of grain A and B.</figcaption></figure>
Figure 3 shows the (2 2 0) diffraction spot of grain A, integrated over the rotation. Its average ω and η are ~ 54° and 255°, where ω is the rotation angle of the sample and η is the azimuthal angle along the diffraction ring. Figure 4 shows an example of an overlapped diffraction spot, corresponding to the (1 1 1) reflection, common to grains A and B (their twin boundary is (1 1 1) plane). Its average ω and η are ~ 23° and 232°. These two spots have similar ω and η, so the spots show the similar structures of grain A, such as the edges and the band-like intensity contrast. In all the figures, the direction of load is upward, which is close to the crystal orientation <2 1 2> of grain A. Interestingly, the band-like image contrast does not correspond to any of the close-packed (1 1 1) planes, which are the expected slip planes in this type of material.

Figure 5 shows the slices of the individually reconstructed grains A, B and C. The inclusion of shared diffraction spots into individual grain reconstructions gives rise to ghost images, as can be observed in figure 5(a), where the increased intensity in the bottom left corner corresponds to the region occupied by the twin orientation. Automated identification and removal of shared diffraction spots can be a solution for such small clusters, containing limited number of twin related orientations. For larger clusters which may contain several tens of different twin related orientations (including higher order Σ3<sup>n</sup> twins), this approach becomes impractical, since the number of “valid” (not shared) diffraction spots decreases quickly.

<figure><img src="image_11.png" /><figcaption>**Figure 5.** 2D slices of Individual grain reconstructions of grains A, B and C.</figcaption></figure>

Figure 6(d) shows a slice of the same grain cluster, this time reconstructed using the TRD cluster reconstruction approach. Sub-figures 6(a), 6(b) and 6(c) are the slices of the grains A, B and C in the reconstructed TRD, whereas figure 6(d) is the sum of figures (a), (b) and (c), indicating that the reconstructed twin boundaries of the grains match each other.

<figure><img src="image_12.png" /><figcaption>**Figure 6.** 2D slices of the volumes of TRD joint reconstructions.</figcaption></figure>
Comparing the results of two reconstruction methods, the TRD joint reconstruction result has less artifacts. After the selection of reflections, there are 81, 58 and 58 reflections used for the individual grain reconstructions (figure 5) of the grains A, B and C respectively, while the TRD reconstruction in figure 6 uses a total of 288 reflections, making better use of the diffraction spot data.

Figures 7 and 8 show the Kernel Average Misorientations (KAM) and the Intragranular Misorientations (IGM) of the TRD at applied load of ~ 2N. There is pre-existing orientation spread, which can be caused by imperfect sample preparation. The IGM of the voxel indicates the misorientation angle between voxel orientation and the average grain orientation. The KAM indicates the average of the misorientation angles between the voxel and its 3×3×3 neighborhood.

<figure><img src="image_13.png" /><figcaption>Figure 7. 2D slice of KAM.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Figure 8. 2D slice of IGM.</figcaption></figure>

Figures 9 and 10 respectively show the slices of the KAM and the IGM of the reconstructed TRD at applied load of ~ 20N. From the images, we can see the band like structures intersecting the twin boundaries, which matches the contrast visible in the diffraction spots in figures 3 and 4. The orientation of these bands do not correspond to any of the {1 1 1} planes and we attribute them as signature of lattice misorientations created by kink bands which are known as an alternative mode of plastic strain localization in fcc materials [10].

<figure><img src="image_15.png" /><figcaption>Figure 9. 2D slice of KAM.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Figure 10. 2D slice of IGM.</figcaption></figure>
## 4. Conclusions

The presence of shared reflections, as commonly observed in twin related domains in fcc materials, can lead to reconstruction artifacts when not handled appropriately. To address this problem, we propose a joint TRD (grain cluster) reconstruction approach. After grain indexing and an initial grain reconstruction, the TRDs are recognized as a first step. Based on the TRD recognition, the domain occupied by the TRD is reconstructed in a joint (simultaneous) reconstruction, making optimal use of the available diffraction data. According to experiment results, compared to the individual reconstruction, the joint reconstruction and proposed modification of the cost function weight improve the reconstruction quality. The method has been applied to a Ni polycrystal, deformed to about 0.2% plastic strain. The grain projections, and reconstructed orientation field reveal band-like structures, aligned parallel to the (1 ¯2 1) lattice planes.

## Acknowledgments

Many thanks to Henry Proudhon and Muhammad Fakhry Hatta for preparing the samples. We thank ESRF for beam time under proposal MA4761. Agence nationale de la recherche is acknowledged for funding the PhD of ZL under the project 3DiPolyPlast.

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