# Statistical properties of fractal type dislocation cell structures

Sándor Lipcsei, Szilvia Kalácska, Péter Dusán Ispánovity, János Lábár, Zoltán Dankházi, Istvan Groma

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Sándor Lipcsei, Szilvia Kalácska, Péter Dusán Ispánovity, János Lábár, Zoltán Dankházi, et al. Statistical properties of fractal type dislocation cell structures. Physical Review Materials, 2023, 27 (3), pp.033604. 10.1103/PhysRevMaterials.7.033604. emse-04059953

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Statistical properties of fractal type dislocation cell structures

Sándor Lipcsei,¹ Szilvia Kalácska,²,³ Péter Dusán Ispánovity,¹,⁴

János L. Lábár,⁵ Zoltán Dankházi,¹ and István Groma¹, *

¹Eötvös Loránd University, Department of Materials Physics,
1117 Budapest, Pázmány Péter sétány 1/a, Hungary

²Mines Saint-Étienne, Univ Lyon, CNRS, UMR 5307 LGF,
Centre SMS, 158 cours Faurel, 42023 Saint-Étienne, France

³Empa, Swiss Federal Laboratories for Materials Science and Technology,
Laboratory of Mechanics of Materials and Nanostructures,
CH-3602 Thun, Feuerwerkerstrasse 39., Switzerland

⁴Institute for Advanced Simulation: Materials Data Science and Informatics (IAS-9),
Forschungszentrum Jülich GmbH, 52425 Jülich, Germany

⁵Thin Film Physics Department, Institute of Technical Physics and
Materials Science, Centre for Energy Research, Budapest, Hungary

The dislocation microstructure developing during plastic deformation strongly influences the stress-strain properties of crystalline materials. Resent theoretical investigations based on the 2D continuum theory of straight parallel edge dislocations were able to predict a periodic dislocation microstructure. The results obtained, however, can only be considered as a very first step toward the understanding of the origin of dislocation patterning. One of the most challenging problems is the modeling of the formation of the fractal like dislocation microstructure. So, it is crucial to determine the statistical properties of such a structure developing at ideal multiple slip orientation. In the paper, by X-ray line profile analysis and the novel method of high resolution electron backscattering diffraction (HR-EBSD) a complex experimental characterization of dislocation microstructure developing in uniaxially compressed Cu single crystals is presented. With these methods, the maps of the internal stress, the Nye tensor, and the geometrically necessary dislocation (GND) density were determined at different load levels. It is found from the fractal analysis of the GND maps that the fractal dimension of the cell structure is decreasing with increasing average spatial dislocation density fluctuation. Moreover, it is shown that the evolution of different types of dislocations can be successfully monitored with the HR-EBSD based technique.

PACS numbers: 62.25.-g, 61.72.Lk, 64.70.qj, 45.70.Ht

# I. INTRODUCTION

It was first observed nearly 60 years ago that dislocations created during the plastic deformation of crystalline materials tend to form different patterns with the morphology depending on the mode, temperature and rate of deformation. There is an equally longstanding discussion regarding the physical origin of these patterns. A large variety of approaches have been proposed to model the instability leading to the spatial variation of the dislocation density, many of which are based upon analogies with pattern formation in other physical systems. It has been argued that dislocation patterns can be understood by the tendency toward the minimization of some kind of elastic energy functional (Hansen and Kuhlmann-Wilsdorf¹, Holt², Rickman and Vinas³), but the theories have never been worked out in details. Another approach proposed is to model the dislocation patterning as a reaction-diffusion phenomenon of the mobile and immobile dislocation densities (Walgraef and Aifantis⁴, Pontes et al.⁵). The fundamental problem with this approach is that it is completely phenomenological, i.e., one can not see how the different terms appearing in the evolution equations are related to the properties of individual dislocations.

In a recent series of papers⁶⁻⁸ a new theoretical approach

based on a continuum theory of dislocations, derived from the evolution of individual dislocations, was proposed for modeling the patterning process. According to the theory the main source of the instability is the nontrivial mobility of the dislocations caused by the finite flow stress, while the characteristic length scale of the pattern is selected by the “diffusion” like terms appearing in the theory due to dislocation correlation effects. Since, however, the theory is developed for a rather idealized 2D dislocation configuration, further experimental and theoretical investigations are needed to create a general comprehensive theory of dislocation patterning.

One of the most challenging issues is the characterization and modeling of the self-similar fractal-like dislocation cell structure formed in FCC crystals oriented for ideal multiple slip (for details see the pioneering works of Zaiser and Hähner⁹,¹⁰). For developing an appropriate theory of the problem one should determine experimentally the statistical properties of the fractal-like dislocation microstructures. Since copper single crystal is an easily processable model material, this paper reports about detailed experimental investigations performed on compressed Cu single crystals oriented for ideal multiple slip. In order to get a complex picture of the dislocation cell structure, as a first step, the earlier investigations performed by X-ray line profile analysis and TEM in-
vestigations on samples compressed up to different stress levels (Székely et al.¹¹) were revisited. It should be noted, that with the X-ray detectors available nowadays one can achieve much better signal to noise ratio than earlier. This improves considerably the accuracy of the parameters obtained from the X-ray profiles. Moreover, as a relatively recently developed method high-angular resolution electron backscatter diffraction (HR-EBSD) investigations were also performed on the samples. With this, the maps of the internal stress and the geometrically necessary dislocation (GND) density developing in the samples could be determined. Some of the aspects of the applied methods are developed exclusively for the specific requirements of the addressed problem, consequently in the first half of the paper the applied experimental methods are explained in detail. In the second part the obtained experimental results are discussed. The obtained statistical results (e.g. fractal dimension) have been confirmed by the simulations done by Bakó and Hoffelner¹², which predict the formation of dislocation cell structures with non-trivial fractal dimension in the absence of climb for multiple slip oriented FCC materials. Nevertheless, the earlier TEM observations by courtesy of Essmann in the work of Hähner and Zaiser¹³ on uniaxially deformed copper single crystals validates many of our results regarding the statistical parameters.

## II. EXPERIMENTAL METHODS

### A. Sample preparation

In order to study the dislocation cell formation mechanism in face-centered cubic (FCC) materials, a high purity copper single crystal was used. For the compression tests rectangular prism shaped samples with dimensions of 2.5 × 2.5 × 5 mm³ were cut with an electrical discharge machine (EDM). The orientation of each surface was of (100) type. For removing the amorphous layer created by EDM the specimens were etched in a 30% HNO₃ solution for 10 minutes. To reduce the initial dislocation density the samples were heat treated at 600 °C for 6 hours in a vacuum furnace. The average dislocation density in an undeformed sample was determined by X-ray line profile analysis (XPA) (the details of the method applied are explained below in Sec. II B). It was found that it is below the 1 × 10¹³ m⁻² lower limit that one can determine by X-ray profile analysis.

The samples were compressed from the direction of the squared shaped surfaces ensuring uniaxial deformation in the [001] direction corresponding to ideal multiple slip. The EBSD and XPA measurements were performed on the [010] surface.

Six different samples with the same size and orientation were deformed up to different strain levels. The resolved shear stress $\tau^*$ vs. strain $\varepsilon$ curve of the sample with the highest terminal deformation is shown in Fig. 1. The black dots on the curve mark the maximum stresses

and strain levels of the 6 different samples. (Up to the maximum stress levels the stress-strain curves obtained on the other samples follow the same curve as the one plotted within 5% of error.) The black line in the figure shows the hardening rate $\Theta = d\tau/d\varepsilon$ as a function of strain. As it is expected for ideal multiple slip¹⁴ the $\Theta(\varepsilon)$ curve consists of a nearly horizontal (stage II) and a decreasing (stage III) linear part. This is indicated by the dotted lines in the figure.

<figure><img src="image_2.png" /><figcaption>Figure 1. Resolved shear stress ($\tau^*$) and hardening rate ($\Theta$) vs. strain ($\varepsilon$) obtained on a compressed Cu single crystal oriented for (001) ideal multiple slip. The black dots on the curve mark the stress levels until which the 6 different samples were compressed.</figcaption></figure>

From the six specimens prepared three are close to the strain level corresponding to the transition from stage II to stage III. As it is seen below this strain region is critical for the statistical properties of the dislocation cell structure developing during the deformation.

Finally, in order to prepare the samples for TEM and HR-EBSD measurements, electropolishing was applied at 20 V, 1.2 A using Struers D2 electrolyte for 30 seconds.

### B. X-ray line profile analysis

X-ray line profile analysis is a well-established method to determine the average dislocation density, the average squared dislocation density and the dislocation polarization from the measured intensity profile. In our analysis the “restricted moments” method developed by Groma et al.¹⁵⁻¹⁷ was applied. In the evaluation of the measured data the asymptotic behavior of the different order restricted moments are analyzed. The $k^{th}$ order restricted moments are defined as

$$v_k(q) = \frac{\int_{-q}^{q} q'^k I(q') dq'}{\int_{-\infty}^{\infty} I(q') dq'} \quad (1)$$

where $q$ is the varying integration limit the restricted moment $v_k(q)$ depends on, $I(q')$ is the intensity distribution near to a Bragg peak, in which $q' = 2(\sin\theta - \sin\theta_0)/\lambda$, $\lambda$
is the wavelength of the applied X-rays, and $\theta$ and $\theta_0$ are the half of the diffraction and Bragg angles, respectively.

As it is explained in detail in¹⁵ for large enough $q$ values the asymptotic form of the 2nd order restricted moment reads as

$$v_2(q) = 2\Lambda\langle\rho\rangle \ln\left(\frac{q}{q_0}\right), \quad (2)$$

were $\langle\rho\rangle$ is the average dislocation density, $q_0$ is a parameter determined by the dislocation-dislocation correlation, and $\Lambda$ is a constant depending on the dislocation Burgers vector $\vec{b}$, the line direction $\vec{l}$, and the diffraction vector $\vec{g}$. $\Lambda$ is commonly written in the form $\Lambda = \pi|\vec{g}|^2|\vec{b}^2|C/2$ where $C$ is called the contrast factor. (For its actual value a detailed deduction and explanation can be found in¹⁵.) From the intensity profiles measured the values of $\Lambda\langle\rho\rangle$ and $q_0$ can be obtained by fitting a straight line on the asymptotic part of the $v_2(q)$ versus $\ln(q)$ plot. It should be noted that if the dislocation density is less than about $10^{13} \text{ m}^{-2}$ the instrumental broadening is in the same order as the one caused by the dislocations, so dislocation density below this limit cannot be accurately determined by the X-ray profile analysis performed in a laboratory setup applied in the investigations.

Beside the 2nd order restricted moment for our analysis the 4th order restricted moment is also important. In the asymptotic regime it is¹⁵:

$$v_4(q) = \Lambda\langle\rho\rangle q^2 + 12\Lambda^2\langle\rho^2\rangle \ln^2\left(\frac{q}{q_1}\right), \quad (3)$$

where $\langle\rho^2\rangle$ is the average dislocation density fluctuation, and $q_1$ is a parameter. For the better visualization it is useful to consider the quantity

$$\frac{v_4(q)}{q^2} = \Lambda\langle\rho\rangle + 12\Lambda^2\langle\rho^2\rangle \frac{\ln^2\left(\frac{q}{q_1}\right)}{q^2}, \quad (4)$$

which asymptotically tends to $\Lambda\langle\rho\rangle$. The actual values of the parameters $\Lambda\langle\rho\rangle$, $\Lambda\langle\rho^2\rangle$, and $q_1$ can be determined by fitting the form given by Eq. (4) to the asymptotic regime of the $v_4(q)/q^2$ versus $q$ plot.

An important statistical parameter of the dislocation microstructure developed is the relative dislocation fluctuation defined as

$$\sigma = \sqrt{\frac{\langle\rho^2\rangle - \langle\rho\rangle^2}{\langle\rho\rangle^2}} \quad (5)$$

that can be determined from the 4th order restricted moment.

It should be noted that the measured intensity $I(q')$ often contains a background which has to be subtracted before the calculation of the restricted moments. Since however, the background has different contribution to the 2nd and 4th order restricted moments, determining the average dislocation density from both moments offers

a internal checking possibility whether the background level was selected correctly.

The profile measurements have been performed with a Cu rotating anode Cu X-ray generator at 40 kV and 100 mA with wavelength $\lambda = 0.15406 \text{ nm}$. In order to reduce the instrumental broadening the symmetrical (220) reflection of a Ge monochromator was used. The $K\alpha_2$ component of the Cu radiation was eliminated by an 0.1 mm slit between the source and the Ge crystal. The profiles were registered by a linear position sensitive DECTRIS MYTHEN2 R detector with 50 $\mu\text{m}$ spatial resolution and 1280 channels. The sample-detector distance was 0.7 m resulting in an angular resolution in the order of $0.004^\circ$. During the measurements approx. $1 \times 10^2 \text{ mm}^2$ surfaces of the samples were illuminated by the x-ray, so the parameters obtained from x-ray measurements correspond to average values over areas much larger than the typical dislocation cell size (see below).

The evaluation method applied is demonstrated on the intensity distribution (Fig. 2) obtained on the sample compressed up to 43.12 MPa resolved shear stress. The corresponding restricted moments are shown in Fig. 3. As it is shown in the figure the different parameters can be determined with an accuracy of less than a few percents.

<figure><img src="image_3.png" /><figcaption>Figure 2. The X-ray line profile obtained at $\vec{g} = (020)$ on the sample compressed up to 43.12 MPa. In order to eliminate the effect of the noise the peak intensity should be at least $10^3 - 10^4$ times higher than the background, and a subsequent background subtraction should be carried out.</figcaption></figure>

### C. HR-EBSD

EBSD measurements were carried out in a FEI Quanta 3D scanning electron microscope (SEM) equipped with an Edax Hikari EBSD detector. Diffraction patterns were recorded with $1 \times 1$ binning (640 px $\times$ 480 px resolution) using an electron beam of 20 kV, 16 nA. In order to carry out statistical analysis on the collected data a 20 $\mu\text{m}$ $\times$ 20 $\mu\text{m}$ area was mapped with a step size of 100 nm on each sample. The HR-EBSD technique utilizes image cross-
<figure><img src="image_4.png" /><figcaption>Figure 3. The raw data (blue lines) and the fitted restricted moments (orange lines).</figcaption></figure>

correlation on the recorded diffraction patterns¹⁸. The local strain tensor components can be determined, and a lower bound estimate of the GND density can be given using the commercially available software. The method requires an ideally stress-free diffraction pattern as reference, that is often difficult to obtain experimentally. In the absence of such reference, it is noted that the scales should be implemented as relative and not absolute measures. Image cross-correlation based HR-EBSD calculations were performed using BLG Vantage CrossCourt v.4 software that provided the components of the elastic distortion ($\beta_{ij}^{el}$) and the stress tensor ($\sigma_{ij}$) and the also the values of the GND density ($\rho_{GND}$).

From the distortion map the Nye dislocation density tensor $\alpha_{ij}$ defined as¹⁹

$$ \alpha_{ij} = -e_{klj} \partial_k \beta_{il}^{el} \quad (6) $$

can also be determined where $e_{ijk}$ is the Levi-Civita symbol. Since, however, in the HR-EBSD measurement the distortion tensor is measured directly on the sample surface, only those components of $\alpha_{ij}$ can be calculated that are independent from the derivation in the direction perpendicular to the sample surface. So, in a coordinate system with z axis perpendicular to the sample surface only the iz components of the Nye tensor

$$ \alpha_{iz} = \partial_y \beta_{ix}^{el} - \partial_x \beta_{iy}^{el} \quad i = x, y, z \quad (7) $$

can be directly determined from a HR-EBSD measurement. With some additional assumptions one can determine further components of the $\alpha_{ij}$ tensor²⁰ but since to see the validity of the assumptions required is not straightforward only the $\alpha_{iz}$ were used in the investigations presented.

Since

$$ \alpha_{ij} = \sum_{t} b_{i}^{t} l_{j}^{t} \rho^{t}, \quad (8) $$

where the superscript t denotes a given type of dislocation present in the system with Burgers vector $\vec{b}^t$, line direction $\vec{l}^t$, and dislocation density $\rho^t$, from the measured Nye tensor components, one can make an estimate on the dislocation population in the different slip systems (for details see below).

Furthermore, to characterize the GND density the scalar quantity

$$ \rho_{\text{GND}} = \frac{1}{b} \sqrt{\alpha_{xz}^2 + \alpha_{yz}^2 + \alpha_{zz}^2} \quad (9) $$

was introduced. The GND density and the $\alpha_{iz}$ tensor components were determined using a C++ code developed by some of the Authors²¹,²².

### III. STRESS-MAP ANALYSIS WITH THE RESTRICTED MOMENT METHOD

It was already demonstrated earlier by Groma et al.²³⁻²⁵, that for a dislocation ensemble of parallel edge dislocations the asymptotic part of the probability distribution of the internal stress $p(\sigma)$ decays as

$$ p(\sigma) \approx \frac{b^2 \mu^2}{8\pi^2} C \langle \rho \rangle \frac{1}{\sigma^3} \quad (10) $$

where $\mu$ is the shear modulus and $C$ is a “geometrical” constant, similar to the contrast factor in the case of X-ray peaks, depending on the type of dislocation, the normal direction of that surface of the sample, on which the measurements are carried out and the stress component $\sigma$ considered. So, like for X-ray line broadening, in the asymptotic regime the second order restricted moment $v_2(\sigma)$ is linear in $\ln(\sigma)$. From its slope one can determine the quantity $\rho^* = C < \rho >$ often referred to as “formal dislocation density”. It should be noted, however, that the stress value obtained by HR-EBSD in a given scanning point is the average stress on the area illuminated by the incoming electron beam. As a result at large enough stress levels the probability distribution $p(\sigma)$ measured deviates from the inverse cubic decay, it turns to a much faster decaying regime²¹. Nevertheless, for most cases one can easily identify a linear regime on the second order restricted moment $v_2(\sigma)$ versus $\ln(\sigma)$ plot (see Fig. 4). From the deviation of the inverse cubic decay we can define a characteristic length scale $r_d = \mu b / \sigma_d$ where $\sigma_d$ is the stress level where the probability distribution start to deviate from the inverse cubic regime. In the investigations performed $r_d \approx 75$ nm. This means, that short-range dislocation structures (such as dipoles) narrower than $r_d$ are not “seen” by this method. So, compared to X-ray line profile analysis HR-EBSD somewhat underestimates the dislocation density (for details see below).

Since from the HR-EBSD analysis one can obtain 5 independent stress components ($\sigma_{33}$ is assumed to vanish in the HR-EBSD analysis) a “formal” dislocation density $\rho_{ij}^* = b^2 \mu^2 / (8\pi) C_{ij} \langle \rho \rangle$ can be determined from the
stress maps corresponding to different *ij* stress components, where the parameter $C_{ij}$ is the “contrast factor” of the *ij*th stress component. Unlike for the X-ray line broadening there is no existing analytical calculation to give the precise value for $C_{ij}$. In the theoretical paper on the internal stress distribution²³ $C_{ij}$ is calculated only for the shear stress generated by edge dislocations in isotropic materials in the coordinate system defined by the Burgers and line direction vectors of the dislocations. In this specific case $C_{shear} = \pi/[2(1-\nu)^2]$ where $\nu$ is the Poisson's ratio. According to our experimental results for the case studied in the paper $C_{ij}$ varies significantly (by a factor of about 5). In the results presented below we give only the average of the 5 formal dislocation densities normalized by $C_{shear}$. The issue, however, requires further investigations.

A typical stress map, stress probability distribution, and the corresponding second order restricted moment can be seen on Fig. 4.

<figure><img src="image_5.png" /><figcaption>Figure 4. Stress map, stress probability distribution, and the corresponding second order restricted moment $v_2$ versus $\ln(\sigma)$ obtained on the sample compressed up to 43.12 MPa.</figcaption></figure>

## A. TEM investigations

A TEM specimen was fabricated from the bulk copper single crystal deformed up to 43.12 MPa resolved shear stress with the aim of qualitative comparison of dislocation structures with those obtained from GND density maps. The TEM lamella preparation was carried out using a FEI Quanta 3D FEG dual-beam SEM-FIB microscope. The initial fabrication process was carried out at 30 kV acceleration voltage and ion currents of 1-30 nA. The final polishing consisted of low current (0.2 – 0.5 nA) and low voltage (2 – 5 kV) ion polishing. It is noted that to be able to investigate very large dislocation cells unusually large (20 µm × 20 µm) specimens were fabricated requiring extra care during the preparation process²⁶⁻²⁸. Bright field images of the dislocation network were recorded on a 6 × 6 cm² 4k × 4k CETA 16 CMOS camera with 14 µm pixel size, controlled by VELOX software in a Titan Themis G2 200 transmission electron microscope operated at 200 kV (see Fig.13).

## IV. FRACTAL ANALYSIS

The dislocation cell structure developing under uni-directional deformation at ideal multiple slip is known to be a so called “hole” fractal⁹ consisting of connected “walls” and “cell interiors” with a power law type size distribution. Since the GND maps obtained by HR-EBSD measurements allow us to study the dislocation microstructure on a much larger area than one can do with TEM (applied traditionally for microstructure characterization) we performed fractal dimension analysis on the GND maps at different stress levels. We have applied two different methods, the “traditional” box counting and the correlation dimension analysis.

### A. Box-counting algorithm

A common algorithm to determine the fractal dimension of a set is the well known box-counting algorithm²⁹. In the method we cover the image with an equidistant grid of lattice spacing $L$, and then count the number of boxes $N$ containing GNDs above a threshold level (for details see below). The fractal dimension $D_H$ is

$$D_H = \frac{d \ln(N)}{d \ln(L)}, \qquad (11)$$

that is obtained by fitting a straight line to the $\ln(N)$ versus $\ln(L)$ plot. ($D_H$ is often referred to as Hausdorff dimension.) The advantage of the method is that it is numerically a cheap, fast and a fairly precise procedure.
## B. Correlation dimension

One can also measure the geometrical randomness of points through the so-called correlation integral, which may be estimated for large enough systems with the correlation sum$^{30}$

$$C(\epsilon) = \frac{1}{N(N-1)} \sum_{i \neq j}^{N} H(\epsilon - |r_i - r_j|) \quad (12)$$

where $\epsilon$ is the threshold distance, $N$ is the number of non-zero points, $H$ is the Heaviside step function, $r_i$ and $r_j$ are the coordinates of the set points. The correlation integral scales with the threshold distance as$^{30}$

$$C(\epsilon) \propto \epsilon^{D_c}, \quad (13)$$

where $D_c$ is the correlation dimension. One can easily see, that for points on a circle the correlation dimension $D_c = 1$, for points on a sphere $D_c = 2$ and for points evenly distributed in a sphere $D_c = 3$. For the analysis of 2D embedded geometrical structures one may expect that $1 \le D_c \le 2$.

## C. Image filtering

The GND maps measured can not be analyzed with the method explained above in a straightforward manner. One issue is that the maps are obviously not binary ones so, one has to introduce some threshold value above which we consider the map intensity to be 1 and 0 below. The fractal dimension obtained may depend on the threshold value chosen. Another problem we face is that the GND map contains numerous random points. They may correspond to individual dislocations or narrow dislocation multipoles but certainly they should not be considered during the fractal analysis.

A simple method for global binarization is the so-called Otsu's method$^{31,32}$. (It is analogous to Fischer's Discriminant Analysis$^{33}$ method and equivalent to a globally optimized k-means clustering method$^{34,35}$.) In the simplest form it returns a binarised intensity map threshold by maximizing the inter-class variance. In order to get this threshold value first the probability distribution of the point intensity $p(I)$ is calculated numerically with some appropriate binning level chosen. After this, with a threshold level $t$ the histogram $p(I)$ is cut into two sub-histograms separated by the threshold, and the quantity

$$\sigma_w^2(t) = P_0(t)\sigma_0^2(t) + P_1(t)\sigma_1^2(t) \quad (14)$$

is calculated, where $P_0$ and $P_1$ are the probabilities of the two classes separated by $t$, while $\sigma_0^2$ and $\sigma_1^2$ are variances of the two classes. The threshold for the image binarization is selected by minimizing $\sigma_w(t)$.

Otsu's method performs exceptionally well when the histogram obtained on the image has a bimodal distribution and the background and foreground values are

separated by a deep valley. However, if the image is corrupted with additive noise or the variation of intensities between background and foreground are large compared to the mean difference, the histogram may degrade.

One may observe a fluctuating salt-and-pepper like noise on the raw Nye-tensor component maps and GND density maps (Fig. 5). This prevents the direct applicability of Otsu's method. In order to eliminate this noise, a smoothing window was applied to the measurable Nye-tensor components. The maps were convoluted with a circular averaging window of radius $r = 150$ nm. The application of a smoothing window results in a more pronounced dislocation wall structure (Fig. 5). A globally

<figure><img src="image_6.png" /><figcaption>Figure 5. Example for $\alpha_{yz}$ maps and GND density maps obtained with and without smoothing for the sample compressed up to 43.12 MPa. a) $\alpha_{yz}$ map without smoothing, b) with smoothing, c) GND map without smoothing, d) with smoothing.</figcaption></figure>

applied binarization method discussed above may ignore those dislocation walls, which may have a lower dislocation density than the thickest dislocation ensembles. In order to avoid this problem a multiscale binarization method was developed. The area map was subdivided into squared sub-areas and Otsu's method was applied separately for each sub-area (Fig. 6). By repeating this algorithm with areas with different sizes and by adding up the maps binarized with different scales we could obtain a purely bimodal histogram for the image (Fig. 6). The effect of the method for the intensity histograms is seen in Fig. 7. After the multiscale binarization method explained above the histogram is clearly bimodal allowing to define a threshold level. Those pixels were considered as dislocation walls which had a higher value than the intensity value corresponding to the minimum of the histogram valley. This method is a powerful tool to obtain not only the global, but also the globally invisible, locally present dislocation walls (see Fig. 6).
<figure><img src="image_7.png" /><figcaption>Figure 6. a) Otsu's binarization method with box size of 1 µm, b) 5 µm, c) the added binary maps at all binarization sizes, and d) the map after the final binarization.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 7. The histogram before (upper box) and after (lower box) image filtering.</figcaption></figure>

### D. Burgers vector analysis

As it was discussed above only the $\alpha_{iz}$, $i = x, y, z$; components of the Nye-tensor can be determined from a HR-EBSD measurement without any further assumption regarding the dislocation system (Fig. 13). Therefore, according to Eq. (8) the vector constructed from the available Nye-tensor components

$$ \vec{B} = (\alpha_{xz}, \alpha_{yz}, \alpha_{zz}) \qquad (15) $$

is

$$ B_i = \sum_t b_i^t \rho^t \cos(\vartheta^t) \qquad (16) $$

where $\vartheta^t$ is the angle between the line direction of the $t^{th}$ type dislocation and the surface normal vector. To characterize the type and sign of the dislocation at a given point of the scanned surface the method introduced in³⁶ is followed, that is, the quantity

$$ a_i = \cos(\varphi_i) = \frac{\vec{B} \cdot \vec{b}_i}{Bb_i} \qquad (17) $$

can be calculated where the index $i$ goes through all the 6 Burgers vectors existing in the FCC crystal³⁶. Certainly, one cannot determine the relative population of the different type of dislocations from $\vec{B}$, but according to the definition given by Eq. (17) if the $\rho^t$ density of one of the Burgers vectors is dominantly larger than the other ones, the absolute values of the corresponding $a_i$ are close to 1. Therefore, $a_i$ values can help to describe the type of dislocations at the sample surface. To visualize this the product of the $a_i$ value and the local GND density was calculated at each measurement point and plotted for the 6 possible Burgers vectors. (Typical results can be seen in Fig. 16.)

### V. RESULTS AND DISCUSSION

As a first step X-ray line profile measurements with {020} Bragg reflection were performed on the (010) surface of the 6 samples deformed up to different stress levels. According to earlier investigations on deformed Cu single crystals oriented for ideal multiple slip¹⁷ for this reflection $\Lambda = 0.783$. The intensity distributions were analysed with the restricted moment method explained earlier. Both the 2nd and the 4th order restricted moments were evaluated.

As it can be seen in Fig. 8 the Taylor linear relation between the square root of the dislocation density and the resolved shear stress is fulfilled. A relatively small deviation can be seen at lowest $\tau^* = 17.43$ MPa stress level. They are definitely larger than the accuracy of the determination of formal dislocation density. The deviations can be attributed to the fact that for small deformation level in stage II the dislocation population may differ from the one obtained in the investigations presented in¹⁷. As a consequence the contrast factor can be somewhat different from the one used. The results are in agreement with the earlier investigations of Székely et al.¹¹. However, as it mentioned earlier, the X-ray detector used in the measurements reported here has a much better signal-to-noise ratio than the one used earlier resulting in a much more improved accuracy of the current study.

In Fig. 9 the $v^4(q)/q^2$ restricted moments are plotted for the undeformed and the 6 deformed samples. It can
<figure><img src="image_9.png" /><figcaption>Figure 8. The $\sqrt{\langle\rho\rangle} - \tau^*$ relation, where $\langle\rho\rangle$ is the average dislocation density measured by X-ray and $\tau^*$ is the resolved shear stress.</figcaption></figure>

be seen even without any curve fitting that the asymptotic part of the curves tend to a constant value that increases monotonically with the applied stress. Since the asymptotic value of the $v^4(q)/q^2$ is proportional to the average dislocation density this is in accordance with the results discussed above. It is remarkable, however, that the maximum values of the curves normalized with the asymptotic value are not monotonous with the stress. It has a clear maximum at 36.11 MPa stress level. After performing the fitting of the function given by Eq. (4) the $\sigma$ value defined by Eq. (5) can be determined.

<figure><img src="image_10.png" /><figcaption>Figure 9. The $v^4(q)/q^2$ versus $q$ curves at different compression levels. The corresponding resolved shear stresses are indicated in the upper right corner.</figcaption></figure>

In agreement with the “phenomenological” feature mentioned above the $\sigma$ versus $\tau$ curve exhibits a sharp maximum at $\tau = 36.11$ MPa (see Fig. 10) corresponding to the stage II to stage III transition stress level (see Fig. 1). (For the undeformed sample it is assumed that the dislocation network is nearly homogeneous, so the fluctuation is zero.) The results obtained are in agreement with the ones reported earlier on the same material¹¹.

As it was suggested earlier by Mughrabi et al.³⁷,³⁸

<figure><img src="image_11.png" /><figcaption>Figure 10. The $\sigma(\tau^*)$ function is represented, where $\sigma$ is the average dislocation density fluctuation.</figcaption></figure>

the dislocation system can be envisaged as a composite of “hard” dislocation walls with dislocation density of $\rho_w$ and “soft” cell interiors with dislocation density $\rho_c$. Within this model

$$ \langle\rho\rangle = f\rho_w + (1-f)\rho_c \quad (18) $$

and

$$ \langle\rho^2\rangle = f\rho_w^2 + (1-f)\rho_c^2 \quad (19) $$

where $f$ is the volume fraction of the cell walls. Since according to earlier investigations³⁷,³⁸ $f$ is in the order of 0.1, and $\rho_w$ is an order of magnitude higher than $\rho_c$ the second term in $\langle\rho^2\rangle$ can be neglected so

$$ \langle\rho^2\rangle \approx f\rho_w^2 \quad (20) $$

With this the quantity $\rho_w^{\text{app}} = \langle\rho^2\rangle/\langle\rho\rangle$, that can be determined directly from the X-ray line profile, is

$$ \rho_w^{\text{app}} = \rho_w \frac{1}{1 + \frac{(1-f)\rho_c}{f\rho_w}} \quad (21) $$

If the dislocation content in the cell interiors is much smaller than in the cell walls i.e. $(1-f)\rho_c \ll f\rho_w$ then the “apparent” dislocation density $\rho_w^{\text{app}} \approx \rho_w$. According to Fig. 11 in stage II, $\rho_w^{\text{app}}$ increases monotonically and at the stage II to III transition stress level it has a maximum. In stage III at large enough stress it tends to saturate.

Based on the X-ray line profile results it can be concluded that during stage II the dislocation distribution becomes more and more inhomogeneous, dense dislocation walls are formed with an increasing $\rho_w$ dislocation density. At a given deformation level, however, the dislocation density in the walls reaches a maximum level, dislocation annihilation prevents the further increase of the dislocation density. This process is called dynamic recovery³⁷. In stage III new walls and an increase of the
<figure><img src="image_12.png" /><figcaption>Figure 11. The “apparent” dislocation density $\rho_w^{app}$ as a function of the resolved shear stress.</figcaption></figure>

dislocation density in the cell interiors is needed to accumulate more dislocations. According to Fig. 11, for large enough stress levels the term $(1-f)\rho_c/f/\rho_w$ is in the order of unity, ($\rho_{app}$ droops down to about the half of its maximum value).

As it is seen above X-ray line profile analysis is a rather powerful method to determine some average statistical properties of the dislocation microstructure, but certainly it is not able to say anything about the actual dislocation morphology.

Beside the traditionally applied TEM³⁹ the relatively recently developed HR-EBSD method offers a new perspective to directly study the dislocation microstructure. A big advantage of the HR-EBSD is that a much larger area can be studied than by TEM. Moreover, the sample preparation is much easier. Figure 12 shows the GND maps obtained on the 6 deformed samples. At each deformation level a clear cell structure can be seen with increasing volume fraction of the cell walls. In Fig. 13 the maps of the three $\alpha_{iz}$ components, the GND density, the stress component $\sigma_{yy}$, and a TEM picture obtained on the sample deformed are plotted. Similar pictures were obtained for the other stress levels studied.

According to Fig. 13, as it was assumed earlier³⁷,³⁸, long-range internal stresses develop in the cell interiors. Unlike X-ray line profile analysis, HR-EBSD is a direct method to determine the local stress state of the sample, so the result obtained is a direct evidence of the presence of long-range internal stresses.

The dislocation density was also determined from the stress maps by the restricted moment analysis of the internal stress distribution. In order to reduce the error, the average of the $\rho_{ij}^*$ values were calculated for the 5 independent components of the stress tensor. The results obtained are plotted in Fig. 14. As it is seen there is correlation between the $\rho_\sigma$ average dislocation density obtained from the stress maps and the $\langle\rho\rangle$ density found by the X-ray line profile analysis, but the relation is clearly not linear. One can also note, that in some points (for example in the fourth), higher difference present. This

<figure><img src="image_13.png" /><figcaption>Figure 12. The GND density maps obtained on samples deformed up to (a) 17.43, (b) 26.5, (c) 36.11, (d) 43.12, (e) 55.04, and (f) 67.22 MPa.</figcaption></figure>

is probably due to the local nature of the HR-EBSD, but also due to the fact that the measurements were carried out on different samples, not on the same surface and place. Moreover, the $C_{ij}$ geometrical factor may vary with stress and due to the finite volume illuminated by the electrons we cannot detect the small dislocation dipoles (see above). So, the HR-EBSD internal stress analysis is a possible method for the determination of the dislocation density, but the issue requires further investigations to be able to produce dislocation density values with high precision.

The average GND density $\rho_{GND}$ defined by Eq. (9) was also determined from the Nye's tensor maps. According to Fig. 14, as a general trend $\rho_{GND}$ increases with increasing deformation but due to the large dislocation density fluctuation, in order to get more precise GND density values, one should perform EBSD measurements on a very large area that was not possible with the setup used.

After the image binarization with the method explained above the fractal dimensions of the $\rho_{GND}$ maps plotted in Fig. 12 were also determined by both the box counting ($D_H$) and the correlation dimension ($D_c$) analysis. The results obtained are plotted in Fig. 15. It is found that $D_c \approx 0.95D_H$ so, the two methods give the same fractal dimension within experimental error. This consistent correlation confirms the formation of a special dislocation structure with non-integer (fractal) di-
<figure><img src="image_14.png" /><figcaption>Figure 13. The maps of the (a) $\alpha_{13}$, (b) $\alpha_{23}$, and (c) $\alpha_{33}$ components, (d) the GND, the (e) $\sigma_{22}$, and (f) a TEM picture obtained on the sample deformed up to 43.12 MPa. Notice that the scale and the observation site on the TEM picture is different than on the other ones.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Figure 14. The dislocation densities obtained from the stress probability distribution (black curve) and from the Nye's tensor components (green curve) versus the dislocation density obtained by X-ray line profile analysis.</figcaption></figure>

It is, however, a nontrivial and somewhat unexpected result that the fractal dimension is decreasing with increasing relative dislocation density fluctuation $\sigma$. With other words in the stage II. deformation regime the fractal dimension decreases with increasing stress level but it starts to increase in stage III. So like the relative dislocation density fluctuation the fractal dimension also has an extremum value at the stage II-to-III. transition. This means that the fractal dimension is not controlled directly by the average dislocation density but the rela-

<figure><img src="image_16.png" /><figcaption>Figure 15. In the upper image, the correlation dimension $D_c$ is presented versus the Hausdorff-dimension $D_H$ obtained by box counting. In the down figure, the average dislocation density fluctuation is shown in function of the Hausdorff-dimension.</figcaption></figure>

tive dislocation density fluctuation. This behavior has a rather important implication for the general form of the theory of dislocation patterning. A proper theory should predict the dislocation density fluctuation, not only the average dislocation density.

In Sec. IV D, a method was outlined to analyze Burgers vectors based on the projection of vector $\vec{B}$ to the different possible Burgers vectors$^{36}$. For the sample deformed up to 43.12 MPa the different $\rho_{GND} \cdot a_i$ maps obtained are plotted in Fig. 16. Similar behavior is found for the other samples. As it is seen, some of the walls have a positive (red) or negative (blue) net Burgers vector while the other ones are more dipole like with a positive net Burgers vector on one side and a negative one on the other side.

This picture somewhat refines the composite model proposed by Mughrabi et al.$^{37,38}$. In the original form of the model the sources of the long-range stress are the two dislocation walls allocated on the two sides of a cell wall. The dislocation walls are formed by the reaction of dislocations in two slip systems resulting a Burgers vector parallel to the surface of the cell wall, i.e., the GND
<figure><img src="image_17.png" /><figcaption>Figure 16. The $ρ_{GND} · a_i$, $i = 1..6$ maps obtained on the sample compressed up to 43.12 MPa.</figcaption></figure>

structure imagined is dipole like. However, an elongated region with finite length *d* having a net Burgers vector can also generate long-range internal stress within the connected *d* × *d* sized area. So, the sources of the long-range stresses are not necessary dipole-like walls as assumed earlier.

## VI. SUMMARY AND CONCLUSIONS

Copper single crystals oriented for (100) ideal multiple slip were compressed uniaxially up to different stress levels. The dislocation microstructures developing in the samples were studied by X-ray line profile analysis and HR-EBSD.

The main conclusions are:

* It is shown that HR-EBSD offers a new efficient method to study dislocation microstructure with

much less sample preparation effort than TEM conventionally applied;

* The presence of the long-range internal stress developing in the cell interiors is directly seen by HR-EBSD measurements. Moreover, the stress maps can be directly measured.

* Some of the walls have a positive or negative net Burgers vector while the other ones are more dipole-like with positive net Burgers vector on one side and negative one the other side;

* According to X-ray line profile investigations on compressed Cu single oriented for ideal multiple slip the relative dislocation density fluctuation exhibits a sharp maximum at stage II to III transition stress level;

* The most important finding of the investigations is that the dislocation cell structure is well described by a hole fractal with fractal dimension decreasing monotonically with the relative dislocation density fluctuation. So, it is directly controlled by the level of fluctuations developing in the system.

The results obtained can be directly compared to the prediction of the theoretical models, so they can help to inspire and validate them. Finally, it is important to emphasize that there are still a lot of issues that should be addressed. One very important question is related to size effect, especially the formation of cells in micropillars, that may lead to new interesting results. Moreover, it would be important to study in-situ the cell formation process.

## ACKNOWLEDGMENTS

The authors are grateful to Prof. Angus Wilkinson and Prof. Géza Györgyi for their valuable comments. This work has been supported by the National Research, Development and Innovation Office of Hungary (PDI and IG, project No. NKFIH-FK-138975). JLL acknowledges the support by VEKOP-2.3.3-15-2016-00002 of the European Structural and Investment Funds.

<footnote>* Corresponding author:groma@metal.elte.hu</footnote>

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