# 3D microstructural and strain evolution during the early stages of tensile deformation

A. Zelenika, C. Yildirim, C. Detlefs, R. Rodriguez-Lamas, F.B. Grumsen, H.F. Poulsen, G. Winther

► To cite this version:

A. Zelenika, C. Yildirim, C. Detlefs, R. Rodriguez-Lamas, F.B. Grumsen, et al., 3D microstructural and strain evolution during the early stages of tensile deformation. Acta Materialia, 2024, 270, pp.119838. 10.1016/j.actamat.2024.119838. hal-05128598

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Full length article

3D microstructural and strain evolution during the early stages of tensile
deformation

A. Zelenika<sup>a,b,*</sup>, C. Yildirim<sup>a</sup>, C. Detlefs<sup>a</sup>, R. Rodriguez-Lamas<sup>a</sup>, F.B. Grumsen<sup>c</sup>, H.F. Poulsen<sup>b</sup>, G. Winther<sup>c</sup>

<figure><img src="image_3.png" /></figure>

ARTICLE INFO

Keywords:

Plastic deformation

X-ray imaging

Dislocation structures

Dark field X-ray microscopy

Crystallographic orientation

A B S T R A C T

Dislocation patterning and self-organization during plastic deformation are associated with work hardening, but the exact mechanisms remain elusive. This is partly because studies of the structure and local strain during the initial stages of plastic deformation are a challenge. For this reason, literature data typically cover strains from 0.05-0.1 and higher. Here we use Dark Field X-ray Microscopy to generate 3D maps of embedded 350 × 900 × 72 µm³ volumes within three pure Al single crystals. Tensile deformation was applied to true strains of 0.6%, 1.7%, and 3.5% along the [10 13 -10] direction. Orientation maps revealed the existence of two distinct types of planar dislocation boundaries both at 0.6% and 1.7% but no systematic patterning. At 3.5%, these boundaries have evolved into a well-defined checkerboard pattern, characteristic of Geometrically Necessary Boundaries, GNBs. The crystallographic alignment of the GNBs match that in polycrystals for grains of similar orientation. The GNB spacing measured perpendicular to the planes ≈ 6 µm and the misorientation ≈ 0.2°, in fair agreement with literature data for higher strains. By contrast to the sharp boundaries observed at higher strains, the boundaries are associated with a sinusoidal orientation gradient, showing that they are not yet fully formed. Maps of the elastic strain along the (111) direction exhibit strain variations of ±0.0002 with an average domain size of 3 µm.

**1. Introduction**

The plastic deformation of a crystal lattice is caused by the creation and motion of linear defects in the lattice, dislocations. These dislocations multiply and self-organize with increasing deformation into 3D structures of dislocation boundaries. In our current understanding, this is a multi-scale process leading to work hardening [1]. Well-established dislocation patterns at strains above about 5%-10% have been studied extensively, leading to classification of patterns according to crystallographic orientations [2,3]. In addition, empirical rules for the dependence on slip systems [4] have been established along with scaling laws [5,6] for the misorientation across boundaries and the spacing between boundaries. Furthermore, in this strain range the patterns exhibit self-similarity [1]. By contrast, there is a noticeable scarcity of data at lower strains where patterning is initiated. Provision of such data may shed light on the underlying mechanisms.

Transmission Electron Microscopy (TEM) is the most widely-used
technique to study dislocations, revealing both dislocation structures/
walls (patterns) and isolated dislocations thanks to the high spatial

resolution in combination with diffraction contrast [7]. One of the main
downsides of TEM is the need for thin foils with a thickness of the
order of 100 nm or less. Such foils are not necessarily representative
of the bulk, since the dislocations can interact with the free surface. In
addition, the thin foil may not contain any boundaries if the boundary
spacing is high and the boundaries are not steeply inclined to the plane
of the foil.

Other techniques such as Electron Backscatter Diffraction (EBSD) [8], Diffraction Contrast Tomography (DCT) [9], and 3D X-ray Diffrac- tion (3DXRD) [10] have all provided valuable insights, particularly into stage III deformation on polycrystals. However, these techniques either lack the angular or the direct space resolution to study the patterning in the initial stages (i.e. stage I and II) of deformation.

Recently, Dark Field X-ray Microscopy (DFXM), a diffraction-based imaging technique, has been used to image dislocations deeply embedded in mm-sized samples [11–13]. DFXM is analogous to dark-field TEM (i.e. using an objective lens), but with a superior angular resolution (~ 10⁻³°) that allows for accurate strain and orientation mapping

<footnote><sup>a</sup> European Synchrotron Radiation Facility, 71 Avenue des Martyrs, CS40220, 38043, Grenoble Cedex 9, France</footnote>

<footnote><sup>b</sup> Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark</footnote>

<footnote><sup>c</sup> Department of Civil and Mechanical Engineering, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark</footnote>

<footnote>* Corresponding author at: Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark.
E-mail address: zeleni@dtu.dk (A. Zelenika).</footnote>
<figure><img src="image_4.png" /><figcaption>Fig. 1. (a) Schematic of the DFXM setup with indication of the diffraction vector $\bar{\Omega}$, goniometer tilts $\phi$, $\chi$ and scattering angle $2\theta$. (b) $\chi$ center of mass (CoM) map of one layer in the undeformed sample. Before deformation no defects are visible.</figcaption></figure>

of dislocations, while affording a spatial resolution on the order of 100 nm [14–16]. Using an objective lens with a smaller numerical aperture than that of TEM, a single Bragg reflection is imaged at a time, and variations of components of the local strain and orientation around a given diffraction vector are mapped.

In this study, we investigate the unexplored: dislocation patterning in aluminium at strains below 4%, using DFXM. We specifically focus on well-annealed single crystals with a known orientation prior to deformation. We report on the initial patterning of dislocations and elastic strain after tensile loading.

## 2. Experimental method

Samples for tensile testing were cut out by Electrical Discharge Machining (EDM) from a 2.2 mm thick sheet of AA1090 Aluminium exhibiting columnar grains with a diameter of several centimeters. Samples were cut out from a single grain with the tensile direction being close to [10 13 -10] as determined by EBSD. The crystals are thus oriented for double slip on (111)[01-1] and (1-11)[110]. Due to the shape of the grain, the tensile length varied between 35 and 50 mm. After cutting, all samples were annealed at 540 °C for 10 h. Tensile testing was performed to true strains of 0.6, 1.7 and 3.5%. Samples of mm³ dimensions for DFXM were cut by EDM from the central part of the gauge section of these deformed specimens.

The DFXM experiments were conducted at Beamline ID06-HXM [15, 17] at the European Synchrotron Radiation Facility, ESRF. Fig. 1(a) shows a schematic of the experimental setup. We used a monochromatic beam with an energy of 17 keV. The beam was focused to a line with a FWHM of ≈ 600 nm in the z direction, illuminating a layer within the sample. The 3D information is then acquired by translating the sample in the z direction. For this study, we used the {111} Bragg reflection which is associated with a scattering angle of $2\theta = 17.98^{\circ}$. The objective was a Compound Refractive Lens (CRL), which magnified the diffracted signal by ×17.98 onto a 2D detector located about 5 m from the sample. With an additional magnification of 2 in the camera itself the effective pixel size was ≈ 209 × 679 nm and the field of view (FOV) in the sample is 350 µm × 900 µm. The details of the setup have been presented elsewhere [13].

To map the microstructure, we used two modes of DFXM: orientation and axial strain scans. Orientation scans comprise a mesh of the two orthogonal tilt angles of the sample (i.e. $\phi$ and $\chi$, see Fig. 1(a)) to measure the variations of the local orientation around the {111}

Bragg reflection. The strain scans involve a collective motion of the sample tilt $\phi$ and the $2\theta$ arm (i.e. objective and detector), mapping out the local difference in the d-spacing ($\Delta2\theta = 0.1^{\circ}$ in our scans). Both orientation and strain scans were repeated for a series of z-positions to map several layers of the xy-plane. The initial d-spacing was estimated as the mean value over the FOV assuming a macroscopically stress-free sample. Most of the maps are from the (111) reflection but data for (1-11) are also included. The crystallographic alignment of the microstructure was determined from 3D reconstructed data from the (111) reflection and further verified by means of the (1-11) data.

The orientation and strain maps that are used in this work are generated as center of mass (CoM) maps: given a series of images as function of say angle $\chi$, for each voxel one obtains a distribution of angles. The $\chi$ CoM map displays the average $\chi$ angle — as such it is reminiscent of EBSD maps. All of the CoM analysis shown in this work is based on darfix, a dedicated software package for DFXM data analysis [18]. The correlation maps were analysed using the xcorr2 function in MATLAB [19].

## 3. Results

We start by showing the microstructure of an undeformed aluminium single crystal. Fig. 1(b) shows the $\chi$ CoM map at $\phi$ and $2\theta$ fixed at the centre position of the diffraction peak. Minor fluctuations in local orientation are clearly visible, as well as a small gradient going from top left to bottom right. By inspection of both the $\phi$ and $\chi$ maps we verified that this type of inhomogeneity is different from the deformation induced structure presented below and does not influence the conclusions drawn.

### 3.1. Qualitative description of microstructure evolution

In Fig. 2(a) we show an orientation map for one layer in the 0.6% deformed sample. For all maps from this reflection, the diffraction vector [111] is inclined by 9° to the normal of the map. The horizontal and vertical axes, x and y, align with the crystallographic directions [−0.66 −0.21 0.72] and [0.60 −0.72 0.34], respectively. The features marked by dashed black lines are nearly parallel. These primary features extend through the entire sample — also in 3D. We also identify two types of more localized linear structures, annotated in green. We interpret these as two sets of emerging dislocation boundaries. The angle between the traces is about 130°.
<figure><img src="image_5.png" /><figcaption>**Fig. 2.** χ CoM map of one layer in (a) the 0.6% deformed sample and (b) the 1.7% deformed sample. Both (a) and (b) have primary bands annotated with black dotted lines, and two sets of dislocation boundaries with 130° between the traces annotated with dotted green lines. The red lines in (b) indicate dislocation boundaries with traces inclined 112° to each other. The illuminated plane is inclined 9° to the [111] diffraction vector; normal to one of the active slip planes. The x and y axes align with [−0.66 −0.21 0.72] and [0.60 −0.72 0.34]. The unannotated images are in the supplementary material. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Fig. 3.** Two χ CoM maps of two different {111} reflections of the 3.5% deformed sample: (a) (111) and (b) (1-11). The crystallographic directions of x and y for the (111) reflection are the same as in Fig. 2, i.e. x = [−0.66 −0.21 0.72] and y = [0.60 −0.72 0.34]. For the (1-11) reflection, x = [−0.72 −0.49 0.50] and y = [−0.01 −0.71−0.70]. The legend and annotations are similar to those of Fig. 2, while the rectangle in (a) indicates the FOV in Fig. 5.</figcaption></figure>

Likewise in Fig. 2(b) we show an orientation map for one layer in the 1.7% deformed sample. The same primary and secondary structures are visible. A wide band with a different structure and delineated by primary boundaries is visible in the left part of the image. In this band, the angle between the dislocation boundary traces marked by red is smaller.

In Fig. 3(a) we show corresponding orientation maps for one layer in the 3.5% deformed sample. The dislocation structure has developed as compared to Fig. 2(b) into a checkerboard pattern formed by two boundaries, GNB1 and GNB2. The angle between the boundary traces is the same as in Fig. 2(a). Alternating white and orange bands in between the boundaries marked by the green lines are now clearly visible, although the boundary between white and orange is not sharply defined. The alternating colours indicate alternating signs of the misorientation across these boundaries. At this strain, the original DFXM detector images also clearly reveal mis-orientation across the boundaries. This is in agreement with what is typically found for extended planar boundaries (termed geometrically necessary boundaries, GNBs) in electron microscopy at higher strains [20]. One of the two sets (labelled GNB1) is more pronounced than the one labelled GNB2. The spacing between the GNB1 set is higher.

Fig. 3(b) is a similar plot acquired using the (1-11) reflection of the same sample. Here the angle between the traces of the dislocation

boundaries is smaller. (The original CoM maps for both (a) and (b) with the full FOV and without annotation are available in the supplementary information.)

We emphasize that the DFXM maps are 3D. In Fig. 4 we show results for a 340 × 340 × 72 µm³ local volume of the 3.5% deformed sample. This represents the stacking of the χ COM maps for 41 layers in z with an equidistant spacing of 1.8 µm. (The movement of the sample was not exactly along the lab z axis due to the rotation of the φ motor, so the images were shifted to account for a minor component of the translation along the lab x axis.) It is clear from Fig. 4(a) that the primary bands extend through large volumes of the sample. Likewise from (b) it is evident that (at least) one set of GNBs extend over the entire z-range. Analysis of the GNB traces shows that GNB1 and GNB2 are inclined 35° to the two active slip planes. GNB1 aligns with (0.13 0.–36 0.92) and GNB2 with (0.1 0.91 0.41). The primary band aligns with (0.99 −0.03 0.02).

As mentioned DFXM also readily provides a map of the axial (elastic) strain component in the direction of the diffraction vector, here $\tilde{Q}_{111}$, by scanning $2\theta$. In Fig. 5 we compare the local variation in orientation (as determined by the CoM in the χ-direction) and this axial strain in the entire field of view of one layer of the 3.5% deformed sample. The local region of the orientation map for the same layer marked by the large rectangle is shown in Fig. 3(a). Green lines in
<figure><img src="image_7.png" /><figcaption>Fig. 4. Slices in the two directions orthogonal to Fig. 3(a) through the centre of the imaged 3D volume of the first reflection of the 3.5% deformed sample. Primary bands are annotated as a black dashed line while two sets of GNBs are annotated as a dotted green line. The angles with the z-axis are noted. The z-axis is inclined 9° to the [111] diffraction vector = [0.45 0.66 0.60], x = [-0.66 -0.21 0.72] and y = [0.60 -0.72 0.34]. The colour scale is identical to that of Fig. 3.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 5. (a) Orientation (χ CoM) map and (b) axial strain map of the entire field of view for one layer of the 3.5% deformed sample (z = 0). Units are degrees and strain, respectively. The primary bands defined by orientation changes and annotated by dashed lines in green in (a) are copied as dashed black lines in (b). The large and small rectangles mark the locations from where Figs. 3 and 6 are taken, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. Right: Comparison of maps of (a) local orientation (χ CoM map) and (b) elastic strain in direction of $\vec{Q}_{111}$ within a local region marked by the small rectangle of the layer shown in Fig. 5 for the 3.5% deformed sample. The colour scales are the same as used in Fig. 5. The circles, which are placed in identical positions in (a) and (b) at intersections between GNB1 and GNB2. Left: Statistics over microstructure for (a) and (b) respectively. (a) Histogram of χ values within the local field of view. (b) Histogram of strain distribution and the corresponding best fit to a simple Gaussian distribution (blue line) with indication of the resulting standard deviation. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

the orientation map and corresponding black lines in the strain map illustrate that the orientation change associated with the primary bands are also found in the strain map. Just like the orientations change at these bands, the sign of the strain in the immediate vicinity of them also alternates for those marked 1–3. Such a change is not clear for 4 and 5. Furthermore, for the unlabelled boundaries marked with only dashed lines in the right part of the images, the misorientation difference is clear whereas the general strain level is smaller and less correlated with the boundaries. In contrast with the GNB2s, the GNB1 directionality is also discernible in the strain map.

Next, in Fig. 6 we zoom in on a region of interest in Fig. 5 marked by the small rectangle that does not comprise the primary structure. By inspection we find that the domains in the orientation map are substantially larger than those in the strain map. Notably, there is little visual correlation between the two local maps. In particular, the intersection between the two GNBs does not stand out in the strain map.

The structure displayed in Fig. 6(a) and (b) are quantified in the left hand part of the figure. The elastic strain distribution is seen to be well described by a Gaussian with a $FWHM = 2.35\sigma \approx 1.9 \times 10^{-4}$. The histogram for the χ distribution on the other hand exhibits a range
<figure><img src="image_10.png" /><figcaption>Fig. 7. Left: autocorrelation of the orientation map displayed in Fig. 6(a). Two main directions are identified in the pattern. Right: projections of this pattern perpendicular to the two directions marked by dashed lines.</figcaption></figure>

that is 10 times larger, giving evidence of the evolution of orientation
differences.

3.2. Quantification by correlation functions

In Fig. 7 the autocorrelation of the orientation map in the xy-plane
of the (111) reflection is shown for the 3.5% deformed sample (in the
language of diffraction this is known as the point spread function).
There is clear evidence for partial ordering in a structure with two basis
vectors as marked in the figure. They exhibit an angle of 125°, in fair
agreement with the roughly 130° found between the GNB traces. Also,
the basis vectors align with the trace directions.

Next to the right in Fig. 7 this 2D distribution is projected onto
directions perpendicular to the two basis vectors. Both projections
can be seen as modulated sinusoidal functions. The full period per-
pendicular to the GNB1 system in the xy-plane is ≈29 µm, and the
corresponding domain size is therefore half of that: ≈14.5 µm. The
ordering perpendicular to the GNB2 system is less prominent, but with
a full period that within uncertainty is equal to half that of the GNB1
system. By means of the inclinations of GNB1 and GNB2 to the xy-plane,
the spacings along the directions that are perpendicular to the GNB
planes in 3D are calculated to 5.4 and 6.5 µm for GNB1 and GNB2,
respectively.

The corresponding autocorrelation function of the orientation map
in the yz-plane is presented in the Supplementary Information. The
results are consistent with extended planar boundaries that extend
through the entire 3D region of interest.

We repeated the autocorrelation analysis for the samples elongated
0.6% and 1.7% (see Fig. 8). We find no signature for ordering in the
orientation data of the kind shown in Fig. 7.

The corresponding autocorrelation function for the local strain map
for the 3.5% deformed sample, shown in Fig. 6(b), is provided in Fig. 9.
This appears less ordered except for a contribution directed along
the x-axis. Again there is clear evidence of a modulated alternating
plus/minus structure, with the same period as the GNB1 spacing. As
detailed in the supplementary information, by fitting this autocorre-
lation function the average size of the domains in the strain map is
determined to be 3 µm, a factor of 5–10 smaller than the domains in
the orientation map (Fig. 6(a)).

4. Discussion

4.1. *Instrumental*

This work demonstrates the relevance of using DFXM for studies
of plastic deformation in the early strain regime. The large volume in-
spected enables statistics and mapping on three length scales simultane-
ously: the sample scale with a large spacing between the primary bands,
the GNB structures and local variations within these. The high angular

resolution allows the identification of structures at the very early stage
and enables high accuracy quantitative measures for the local strain
to be derived. The measurements are relatively fast, e.g. the data
acquisition for the one-layer strain map shown in Fig. 5 took 4 min. The
application to in situ studies is evident and has been pursued following
this study. Correlating DFXM data directly to micro-mechanical models
is discussed in [21].

The main limitation of the technique in this study is the use of only
one reflection for the same volume. This implies that only two out of
three orientation variables and only one out of six strain components
are probed.

4.2. *Crystal subdivision*

The large FOV combined with the high angular resolution reveals
the subdivision of the crystal at two length scales. On the larger scale,
primary boundaries with a spacing of the order of 50–200 µm are
seen. The complex origin and nature of such widely spaced boundaries
is treated in depth in [22]. Notably, the present primary boundaries
aligned with [0.99 -0.03 0.02] are neither classical kink bands per-
pendicular to the slip direction nor bands of secondary slip parallel
to the slip plane. For the samples elongated by 0.6%, 1.7% and part
of one at 3.5%, these boundaries are associated with a misorientation
of less than 0.1° and the misorientation clearly alternates in sign
across neighbouring boundaries. In some of these boundaries individual
dislocations could be discerned in the detector images. In another part
of the sample at 3.5% the misorientation is larger. It cannot be ruled out
that this difference is a result of the larger strain. However, we interpret
them as boundaries formed in the initial stages of deformation. They
probably form due to mechanical constraints from the grips. These
primary boundaries delineate bands within which subdivision takes
place on a smaller length scale. As exemplified in Fig. 2b, the character
of this subdivision is not always completely identical, giving evidence
of deformation heterogeneity at the larger scale.

4.3. GNBs

The dislocation boundary structure on the smaller scale is clearly
visible at 3.5% strain with the characteristic parallel extended planar
boundaries termed geometrically necessary boundaries that are also
observed at higher strains. At 0.6% and 1.7%, the boundary structure is
much less developed. It is, however, noteworthy that the features with
the same directionality as the GNBs are visible. Still, distinct ordering
has not yet taken place as evidenced by the autocorrelation analysis.

The two GNBs deviate from the active slip planes by 35°. They
are rotated away from the these around crystallographic axes close to
[121] and [11-2], which are inclined 30° to the active slip directions.
Grains in tensile deformed aluminium polycrystals of similar orienta-
tions tensile deformed to strains of 0.1 to 0.34 have been reported to
<figure><img src="image_11.png" /><figcaption>**Fig. 8.** Left: autocorrelation of the orientation maps displayed in Fig. 2 for the samples at 0.6% strain on top, and 1.7% strain the bottom. Right: Projections of the autocorrelation maps for the X and Y axis from the centre.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 9.** Left: autocorrelation of the strain map displayed in Fig. 6(b). Right: The distribution for $y = 0$ (blue dots) along with a fit to a model of this distribution with a fixed period(red line) equal to the period of the blue curve in Fig. 7 for GNB1.</figcaption></figure>

have GNBs deviating up to 29° from the slip planes around the same axes [2,4]. It is somewhat surprising that the GNBs in a single crystal oriented for double slip on the primary and conjugate planes have a crystallographic alignment similar to those in grains with multiple slip systems. This may indicate activity also on additional slip systems. The Schmid factors for the primary and conjugate systems are 0.33 and the second highest Schmid factors are 0.28 for systems coplanar with the primary and conjugate ones.

With increasing strain, the dislocation density increases, which manifests itself in both smaller boundary spacings and higher misorientation angles. Fig. 10 shows data in the literature for GNBs not aligned with slip planes in 99.996% pure aluminium for tensile strains in the range 0.05–0.34 [1]. The data for the spacings in the figure are average spacings measured on TEM images between GNBs with an unknown inclination to the image plane. These spacings will always be higher than the perpendicular spacing in 3D. The GNB spacings in the xy-plane obtained by the autocorrelation as well as the true perpendicular spacings in 3D are added to the figure. It is seen that the spacings in the

xy-plane of 14.5 µm for GNB1 is higher and the 7.5 µm for GNB2 lower than the literature average. The average of these match the literature data well. The mean misorientation angle of 0.21° in the present data is lower than the literature data, which may be explained by the fact that the measured misorientation reflects only the spread of the probed 111 direction (and not rotations around it). (In a recent study it was shown that there is an underestimate of the misorientation measured by DFXM compared to EBSD measurements due to only a limited amount of the lattice curvature being measured in DFXM, and hence its magnitude is underestimated and so is the geometrically necessary dislocation density [23].)

The good agreement between the quantitative data for 3.5% strain and literature data for higher strains combined with the similarity in directionality of the visually observed dislocation boundaries from 0.6% to 3.5% indicate that early patterning include mechanisms also active throughout the entire strain regime. Nevertheless, it is clear from the lack of order in the autocorrelation for the 0.6 and 1.7% elongated samples that the microstructure has not yet developed into regular patterns.
<figure><img src="image_13.png" /><figcaption>**Fig. 10.** Comparison of average GNB spacings and misorientations in the present experiment with literature data for 99.996% pure tensile deformed aluminium polycrystals with GNBs of similar directionality [1]. The misorientation angles are denoted with the $\Theta$, while the boundary spacings are denoted with the D. The GNB subscript indicates the digitized data from the literature [1], the AC subscript indicates the data points obtained by the present study via autocorrelation and the subscript TRUE indicates the perpendicular spacing in 3D. The subscript M for the angle indicates that this was manually determined from inspection of the data.</figcaption></figure>

### 4.4. Elastic strain level

The maximum value of the elastic strain distribution in Fig. 6(b) is $\approx 2 \times 10^{-4}$. Using the diffraction elastic modulus of 72.8 GPa along the (111) direction [24], this corresponds to a stress of 14 MPa. This is $\approx 70\%$ of the yield stress of 20 MPa. Stresses of 25% of the flow stress have been measured for a single reflection by a microbeam scanning technique in copper single crystals after 25% elongation after unloading [25]. Another study using the same technique but measuring three reflections found smaller stresses in the range 14%–40% of the flow stress in commercially pure aluminium grains subjected to equal-channel-angular pressing (ECAP) [26]. The stress level decreased with the number of ECAP passes, i.e. plastic strain. The high stress level in the current study implies that mutual screening of the stress fields of the dislocations in the boundaries is limited.

The mean misorientation angle in Fig. 6(a) is 0.21°, which corresponds to a dislocation spacing in the boundaries of $\approx 80$ nm. The dislocations in the network are not resolved and the full strain tensor is not probed. Nevertheless, comparison to the shear stress around a simple tilt boundary of edge dislocations with 80 nm spacing reveals that the stress more than 50 nm away from the boundary is below 14 MPa. Considering the spatial resolution of the strain data, this is in good agreement with the observations.

Interestingly, the strain level obtained by Discrete Dislocation Dynamics (DDD) simulations of a copper single crystal tensile deformed to 0.85% produced strains of $\approx 1.9 \times 10^{-4}$ and misorientations of 0.01° [27]. In the DDD simulation there was clear differences between the spatial distributions of the strain and orientation data. In the present experimental data, spatial correlation is visually clear for some of the primary boundaries where the orientation and strain differences manifest themselves with a common directionality in the large FOV in Fig. 5. In the zoomed-in images in 6, this visual correlation is lost. The correlation analysis of the zoomed-in strain data found a periodicity at a length scale that roughly matches that of GNB1. The directionality is horizontal. This may indicate an effect of the almost parallel trace

of GNB2. An in-depth analysis of the strain field is beyond the scope of this paper as only one component of the strain field was probed. The two GNB planes are inclined differently to the diffraction vector, implying that they cannot be expected to show up in the strain map in the same manner. It is also noteworthy that the intersection between GNBs appears not to be associated with a strain concentration — at least not along [111]. The finding of short-range ordering of the stress with a domain size of 3 µm is evidence of the presence of additional dislocations in between the GNBs.

### 4.5. Sinusoidal shape of correlation functions

Although the GNBs are clearly visible in the CoM map of the 3.5% elongated sample, the sinusoidal shape of the autocorrelation function of this map reveals that these are not strictly two-dimensional but consist of a distribution of dislocations around a plane. This is in contrast to GNBs in aluminium at higher strains. The internal dislocation network within GNBs in 10% rolled aluminium has been observed to consist of a fairly regular grid of straight dislocation lines forming an almost perfect planar boundary [28], which further fulfils the so-called Frank equation for boundaries free of long-range stresses [29]. By contrast, boundaries with a certain width are well-known from e.g. nickel, where boundary widths in the range 0.15-0.45 µm were measured in a 10% cold rolled sample [30]. The boundary width rapidly decreased with further straining.

## 5. Conclusion

The very onset of dislocation patterning has been investigated by DFXM in a tensile deformed aluminium single crystal with the tensile axis along [10-13 10], i.e. oriented for double slip on the primary and conjugate slip planes. The results reveal that GNBs are scarce but clearly visible already at 0.6% and 1.7% strain but with no systematic ordering into patterns. A well-ordered GNB pattern consisting of two sets of intersecting boundaries with alternating misorientation comparable to that at higher strains has formed at 3.5% strain. The GNBs at 3.5% strain deviate 35° from the active slip planes around the same (211) axes that are found for grains in polycrystals of similar orientation. They have the same directionality as those at 0.6% and 1.7%. The misorientation and GNB spacing at 3.5% are in fair agreement with literature data for tensile deformed 99.996% pure aluminium. Yet, the boundaries are much more diffuse than those observed by TEM at higher deformation.

The present findings reveal that patterning starts at the early stages of plastic deformation. Although the early patterns are not well-defined, they share characteristics with those at later stages, indicating that at least some of the fundamental mechanisms underlying patterning are the same.

The axial strain along the probed crystallographic direction reveals some correlation with the primary bands and one set of GNBs in the orientation data by visual inspection of a large domain. Quantitative correlation of the periodicity with this set of GNBs was found, although the ordering was along a different direction.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

The authors acknowledge the ESRF for provision of beamtime at ID06-HXM at the European Synchrotron Radiation Facility (ESRF). AZ, FBG, GW and HFP acknowledge support from ERC Advanced Grant nr 885022, from the ESS lighthouse on hard materials in 3D, SOLID, funded by the Danish Agency for Science and Higher Education, Denmark, grant number 8144-00002B, and from DanScatt for travel cost. Szilvia Kalacska is acknowledged for EBSD.
# Appendix A. Supplementary data

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.actamat.2024.119838.

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