Dislocation densities in cold worked copper by electron and X-ray diffraction methods

Sample preparation and experimental methods

Technical purity as-received (cold drawn) Cu samples were processed by ECAP for one pass at room temperature. The specimens were 10 mm in diameter and ∼60 mm in length. Processing by ECAP was carried out using a 90° die. The yield strengths of the ECA-pressed specimens were determined by Vickers hardness measurements. Vickers microindentation hardness tests were carried out on the polished surfaces of the samples by a Buehler 1105 micro-hardness tester. The applied load was 0.981 N (100 g). Ten indentations were performed on each sample. Further details of the preparation of samples and determination of the yield strength are given later.

The X-ray diffraction experiments were carried out in a special high-resolution diffractometer (Rigaku, RA-MultiMax9) dedicated to line profile analysis with a plane Ge (220) primary monochromator operated at the Cu Kα fine focus rotating copper anode at 40 kV and 100 mA. The distance between the source and the monochromator was 300 mm and a slit of about 200 µm was put before the monochromator, at a distance of 260 mm from the X-ray source. At this distance, the Kα1 and Kα2 components were detected by the Ge (220) crystal at a large enough separation allowing for cutting off the Kα2 component by the 200 µm wide slit. The Cu Kα1 beam had the size of about 0.15 × 2.0 mm on the specimen surface.

The scattered radiation was detected by three imaging plate (IP) detectors with the linear spatial resolution of 50 µm. The IPs were placed at a distance of 300 mm from the specimen covering the angular range between 2Θ = 30 and 150°. The diffraction geometry is of parallel-beam type; therefore, the specimen does not have to be moved while the angular resolution is sufficiently good over the entire angular range of measurement.

The diffraction patterns were obtained by integrating the intensity distributions along the corresponding Debye-Scherrer arcs on the IPs. The X-ray beam was positioned on the specimen surface by using a low depth of field microscope coupled to a screen. The measurements were performed at about the centre region of the cross-section surface of the specimens. For the X-ray measurements, the samples were chemically etched in order to remove any surface layer damaged by the sample preparation procedure. The instrumental effect was considered by measuring a standard NIST SRM-660a LaB₆ sample under the same conditions that were applied for the copper specimens.

For EBSD measurements, the samples were ground and polished by traditional metallographic methods. Finally, they were electro-polished in the solution of 15% phosphoric acid (H₃PO₄), 42.5% methanol (CH₃OH) and 42.5% water (H₂O) at 15 V for ∼30 s. The EBSD analysis was performed by a Philips XL30 scanning electron microscope equipped with a TSL type EBSD system and an OIM Analysis 6.2 software configuration. On the surfaces of the samples 300 × 200 µm areas were scanned at 25 kV acceleration voltage and 250× magnification with different step sizes (0.7, 1.0, 1.4 and 2 µm). As it can be seen later these step sizes are much smaller than grain sizes.

Methodology of the GND density determination from EBSD misorientations

An original software was written to calculate GND density from misorientations measured between the adjacent points of EBSD images. In EBSD measurements, these points were positioned on lattice points of a square grid fitting on the planar surface of the sample. Each EBSD measurement results an output file including the Euler angles (as orientation) and the Cartesian coordinates of the points in the reference system fixed to the sample. From these output data a dislocation density tensor, the so-called Nye tensor is calculated by the above-mentioned software written in programming language C#, using mostly the mathematical algorithm suggested in references [1,2,10].

The main steps of calculation of dislocation density from misorientation relations can be written as follows [3]. First, orientation tensors are determined for each point from the Euler angles. Then, misorientation matrix is calculated from orientation tensors of two adjacent points considering symmetries of the given crystal lattice with symmetry matrices. From all possible misorientation matrices, only one is chosen for which the misorientation angle is minimal. It is easy to calculate misorientation angles from the quaternionic form of matrices. In mathematics, the quaternions are number systems which consist of a scalar and a three-dimensional vector. Then, this result is applied to determine Nye's dislocations tensor: (1) Since orientations of single points can be measured only in a plane but not perpendicular to it, so some elements of the dislocations tensor cannot be determined. Therefore, Nye's tensor is an incomplete matrix and only a local apparent dislocation density corresponding to one pixel can be calculated from the extant tensor elements as: (2) where b is the absolute value of Burgers vector.

X-ray line profile analysis

XLPA gives quantitative details about average subgrain size (what means the average size of coherently scattering domains, which in case of metals correlate well with the size of dislocation cells [11]), TD densities, type and arrangement of dislocations [12–14]. Originated from physical microstructure models, theoretical diffraction profile functions are determined to describe the influence of coherently scattering domain size and dislocation structures on the form of the diffraction profiles [11–14]. XLPA is an appropriate method to determine TD density in severe plastic deformed metals [15–20].

The lower detection limit of XLPA for TD density depends on the instrumental broadening of the applied diffraction system. It is ∼10¹³ 1/m² in the case of the present diffraction configuration [21,22]. However, in annealed metals, the values of TD densities vary from 2 × 10¹¹ to 3 × 10¹² 1/m² [23], which values are lower than the above-mentioned detection limit. That was the reason why cold worked (drawn) copper sample was applied for this study.

TD density was determined by using the convolutional multiple whole profile (CMWP) fitting procedure [24,25]. The parameters of the microstructure obtained by this procedure are listed in Table 1.

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Table 1

The numerical results of the CMWP procedure.

Sample	q	ρ (×1014 m−2)	M	<x>area (nm)	m (nm)	σ
Cu as-received	2.0 (±0.1)	4.9 (±0.5)	2.1 (±0.2)	130 (±10)	100	0.33
Cu ECAP-1p	2.2 (±0.1)	9.1 (±0.9)	4.4 (±0.4)	110 (±10)	76	0.27

The parameter q describing the edge/screw character of dislocations. It depends on the elastic constants of the crystal and the edge or screw character of the dislocations. The parameter ρ is the dislocation density, M is the dislocation arrangement parameter. <x>_area is the area-weighted mean crystallite size, m and σ are the median and the variance of coherently scattering domain size.

The area-weighted mean crystallite size is about 170 nm for the as-received sample, and 110 nm for the ECA-pressed sample. It has been reduced by ∼35% already after a single ECAP pass. However, they are much less than the average grain size determined by EBSD. XLPA is sensitive to the coherently scattering domain size, so not only the high-angle grain boundaries are detected, but the low-angle boundaries also [24]. Subgrains with low-angle domain boundaries, as well as dislocation walls are the structures the XLPA is sensitive to, in contrast with the microscopy techniques (SEM, EBSD) which cannot resolve these low-angle grain boundaries. This is why the subgrain size determined by diffraction methods like XLPA is always smaller than the grain size determined by EBSD.

According to the XLPA measurements, dislocation density increases from 4.9 × 10¹⁴ m⁻² in the as-received sample to 9.1 × 10¹⁴ m⁻² in the ECA-pressed sample. As it will be seen later (Figure 4), they are much larger than the values obtained from EBSD. It should be noted that the latter procedure gives only the density of the geometrically necessary dislocations, while XLPA provides the TD density (including both geometrically necessary and statistically stored dislocations). It is also remarkable that the dislocation density and the area-weighted mean crystallite size obtained for ECA-pressed copper after a single pass by XLPA is in good agreement with the value determined by the other author (71 nm and 12 × 10¹⁴ m⁻² [19]).

The experimental value of parameter q is 2.0 and 2.2 ± 0.1 for the as-received and the ECA-pressed sample, respectively. The theoretical values calculated for pure edge and screw dislocations are 1.6 and 2.3 [11]. The experimental value of q for the as-received sample agrees well with the arithmetic average (1.95) of the values calculated for pure edge and screw dislocations, which means that the character of dislocations is half edge–half screw. The experimental value of q for ECA-pressed Cu reveals that it contains more screw dislocations.

The M parameter increases with the ECA pressing. This indicates that with increasing deformation the dipole character of the dislocation structure becomes much weaker. XLPA also gives a parameter describing the edge/screw character of dislocations.

Automated EBSD measurements

In order to calculate GND densities through EBSD measurements, at first a misorientation range should be determined. This range with a higher and a lower limit contains the misorientations considered for calculations. A higher misorientation limit should be avoided for taking into account grain boundaries (as high-angle misorientations) and misidentified points (tiny grains or single pixels with high-angle boundaries inside a larger grain mostly caused by structural faults).

Figures 1 and 2 show misorientation distributions for the as-received and ECA-pressed samples obtained by EBSD with 1.0 µm step size. It reveals that the majority of misorientations which can be taken into account to calculate GND densities are smaller than ∼5° in the as-received sample and ∼10° in the ECA-pressed sample. Therefore, the higher limit of misorientation range should be 10°. These charts also imply that the ECA-pressed sample contains more of those kinds of misorientations what can be considered for calculating of GND density.

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Figure 1

Misorientation distributions for the as-received sample obtained by EBSD with 1.4 µm step size.

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Figure 2

Misorientation distributions for the ECA-pressed sample obtained by EBSD with 1.4 µm step size.

However, there is an uncertainty in the determination of the Euler angles from the Kikuchi patterns obtained by EBSD which can result an error in the calculation of the dislocation density from misorientations. In order to estimate the error in the misorientation angles, the same measurements were carried out on an undeformed single crystalline Si wafer. In this single crystal orientation does not change, therefore differences between the crystallographic orientations of the neighbouring EBSD image pixels should be practically zero. Figure 3 shows the misorientation distribution for this Si single crystal obtained with an applied step size of 2.0 µm. This distribution reveals that certain misorientations can be observed under 1.5° even if the material is practically defect-free. Therefore, misorientations higher than 1.5° can only be used in the calculation of the dislocation density. The other thing why a lower limit should be applied is the high fractions of small misorientations under 0.1–0.2° (see Figures 1 and 2) what would increase unreasonably the calculated dislocation densities.

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Figure 3

Misorientation distribution for Si single crystal obtained by EBSD with of 2.0 µm step size.

Nevertheless, if both misorientations between a pixel and its two neighbours (at the right side and below the studied pixel) are smaller than 1.5°, the dislocation density is taken as zero in that pixel. If any of the two misorientations is higher than 10°, the pixel is ignored from the dislocation density calculation. For all other pixels, the dislocation density was calculated from the misorientations, as described in the section ‘Methodology of the GND density determination from EBSD misorientations’.

Figures 4 and 5 show inverse pole figures (IPF) with grain boundaries between 1.5 and 10° for the as-received and ECA-pressed samples obtained by EBSD with 1.4 µm step size. These maps obviously reveal that ECA-pressed sample contains more misorientations which can be taken into account for GND calculations. They also indicate that these misorientations are within single grains surrounded by high-angle grain borders.

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Figure 4

Inverse pole figure (IPF) with grain boundaries between 1.5 and 10° for the as-received sample obtained by EBSD with 1.4 µm step size.

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Figure 5

Inverse pole figure (IPF) with grain boundaries between 1.5 and 10° for the ECA-pressed sample obtained by EBSD with 1.4 µm step size.

Finally, Figure 6 shows the densities of GND and TDs as a function of applied EBSD step size determined by EBSD and XLPA, respectively. This figure indicates correctly that GND densities are lower than TD densities. Figure 6 also reveals that the smaller the step size, the higher the calculated GND density. This is because the larger the step size, the more misorientations are omitted for calculating GND densities during scanning.

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Figure 6

Densities of GND and TDs as a function of applied EBSD step size.

Fitted curves show that values of GND densities would be higher than TD densities at step sizes smaller than 0.1–0.2 µm which is the size range of the area-weighted mean crystallite size. It may indicate that the applied step sizes (0.7–2 µm) would be too large. However, with regard to the high uncertainty (1.5°) in the determination of the Euler angles from the Kikuchi patterns, in case of lower uncertainty (with another EBSD configuration) higher GND densities would be calculated than TD densities at larger step sizes than 0.1–0.2 µm because more misorientation would be considered for calculating GND density. Therefore, using smaller step sizes than 0.1–0.2 µm seem to be unnecessary in case of the applied EBSD configuration.

Correlation between yield strengths and dislocation densities

As EBSD IPF maps show (Figures 4 and 5), microstructure of the samples simultaneously contains both low-angle cell boundaries and high-angle grain boundaries. Under these conditions, two strengthening contributions should be considered: dislocation strengthening due to the presence of low-angle boundaries (inside grains) and grain boundary strengthening due to the presence of medium- to high-angle boundaries.

Generally, the yield stress of a polycrystalline material, σy, is described by the following equation [26]: (3) where σ₀ is the friction stress for pure copper (20 MPa [20,27]), σ_p, the precipitation hardening, σ_s, the solid solution hardening, σ_d, the hardening of dislocations, and σ_gb, is the grain boundary strengthening. In our case, the precipitation and solid solution hardening can be ignored because samples do not contain any alloying elements or precipitations.

The dislocation strengthening can be written by the well-known Taylor equation: (4) where α is a constant (it is taken as 0.33 [20]),G is the shear modulus (G = 42 GPa [20]), b is the length of the Burgers vector (b = 0.256 nm [20]) and M is the Taylor factor (M = 3 for untextured polycrystalline materials [20]). The grain boundary strengthening can be written by the Hall–Petch equation: (5) where k_HP is the Hall–Petch slope (k_HP = 0.14 MPa m^1/2 [27]), and d is the effective grain size or the spacing of high-angle boundaries, since they are expected to act as barriers of moving dislocations during deformation.

Average grain sizes were calculated by the OIM software. During this calculation, those grains were considered as individual grains which were surrounded at least 15° grain boundaries. According to this method, average grain sizes were given 34 µm for the as-received and 32 µm for the ECA-pressed sample. Therefore, the applied step sizes seem to be appropriate because all step sizes are much smaller than these values.

Finally, the yield stress can be written as follows: (6) Measured yield stress was determined by the empirical relationships between the hardness HV and yield strength, which approximately follow the relationship for cold rolled Cu [28]: (7) and for ECA-pressed 1p Cu [28]: (8) Results of these calculations can be seen in Table 2.

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Table 2

Results of calculations for yield strengths.

	Cu as-received	Cu ECAP-1p
Average grain size (µm)	∼34	∼32
Hardness (HV)	94	125
Dislocation strengthening (σd, MPa)	238	324
Grain boundary strengthening (σgb, MPa)	24	25
Yield stress – measured from hardness (MPa)	254.1	361.3
Yield stress – calculated from Taylor equation (MPa)	282.1	369.0
Dislocation strengthening / calculated yield stress	0.84	0.88

Calculations indicate that dislocation strengthening is much more dominant than grain boundary strengthening in the samples. That means the majority of the strength of the studied samples originates from dislocations and not from grain boundaries: it is ∼84 and ∼88% of the total calculated strength for the as-received and the ECA-pressed samples, respectively. These proportions also show the effect of severe plastic deformation which increased the dislocation density in the ECA-pressed sample.