Supplement to ‘Dislocation Structure from Conventional EBSD Maps’
Ulrich Faul
1 Dislocation Energies
Note: ‘Sup. Figure’ refers to figures in this Supplement, ‘Figure’ to figures in the main text; dito for tables.
Pantleon (2008) discusses the problem that cubic crystals have many more slip systems than there are components of the curvature tensor available from 2-D EBSD maps (e.g.
48 slip systems for a bcc lattice vs. 6 components of the curvature tensor). This leaves inversions for dislocation density severely underdetermined. Minimization of dislocation energy is one way to provide additional constraints. Wallis et al. (2016) point out that with six slip systems for olivine the inversion has the same number of equations as there are unknowns. Consequently the six slip systems are uniquely determined from the accessible components of the curvature tensor.
The influence of dislocation energy on the inversion in MTEX was tested by assigning different energies to the six slip systems used here. The test area for calculation of the dislocation densities was the top left quarter of the map of 7506-2 in Figure 7 for three different energy sets (Sup. Table 1): the energy for all dislocations was arbitrarily set to
1; 2.) the energy for screw and edge dislocations was set according to Hull and Bacon (2011) (Escrew = (1 - ν) * Eedge) with Poissons ratio ν = 0.27 for olivine; 3.) the energies were scaled according to Heinisch et al. (1975), their Table 2. The slip system with the maximum energy in the set used here ([001](100)) was set to 1, the other five slip systems were then scaled according to their energy in Heinisch et al. (1975), Table 2 to fractions of this maximum energy.
Table 1: Scaled Dislocation Energies
disloc. type [100] [001] [100](010) [100](001) [001](100) [001](010) sum energies
uniform 1 1 1 1 1 1 6
HB 0.73 0.73 1 1 1 1 5.46
He75 0.46 0.65 0.73 0.74 1.00 0.88 4.47
HB:Hull and Bacon (2011), He75:Heinisch et al. (1975)
Sup. Table 2 shows that the calculated total dislocation densities for the three cases vary by about 20% relative to those calculated with Hull and Bacon (2011). The total densities scale with the ratios of the sums of the dislocation energies in Sup. Table 1.
Importantly, the dislocation structures in maps of the individual slip systems for the different energies are indistinguishable, showing that the calculated dislocation structures do not depend on the energies assigned to the slip systems (Sup. Figure 1).
2 Denoising
In comparison to the dislocation energies, the smoothing parameter αhas a more signifi- cant effect on calculated densities, both on their values as well as structures. A too low value ofα(Test 4) results in apparently higher dislocation densities, which in fact are due to underdamped noise. This is obvious particularly for already noisy [001](010). More
Table 2: Calculated Dislocation Densities for Energy and Filter Tests
test energy α thresh. total dens. [100](010) [100](001) [001](100) [001](010) [100] [001]
deg. *E12/m2 *E11/m2 *E11/m2 *E11/m2 *E11/m2 *E11/m2 *E11/m2
1 HB 5 5 2.41 9.95 1.15 0.71 7.33 2.88 2.1
2 1 5 5 2.6 9.96 1.20 0.99 7.34 3.84 2.79
3 He75 5 5 1.89 7.29 0.87 0.47 6.53 1.88 1.86
4 He75 1 5 3.8 8.17 2.38 0.98 17.90 4.00 4.50
5 He75 10 5 1.94 7.45 0.73 0.47 7.08 1.87 1.81
6 He75 5 2 1.89 7.29 0.87 0.47 6.53 1.88 1.86
α: smoothing parameter in hQ filter, threshold angle is the value used with the hQ filter
problematic, the increased noise level obscures coherent structures. This is evident for example for the two screw dislocation types.
A too high value of α creates artificially lumpy structures (Test 5) with a slightly higher value for the total dislocation density. The value of α used throughout (α = 5) results in a (near) minimum value of the dislocation densities. No effort was made to find the absolute minimum, since varying α by at least ±1 only produces marginally different results, and the minimal value is likely different for different maps.
3 Angular Threshold for Denoising
Changing the threshold value of the angle above which no smoothing takes place has no discernible effect on densities or structures (Test 6). Figures 5b, 9a and 12a (right axis) show that in the first instance the orientation gradients both for the sub-boundaries in the naturally deformed San Carlos olivine and the sub-boundaries and gradients in the experimentally deformed samples are much smaller than the threshold value and are therefore subject to the filter. As Figures 5, 9, 12 and 13 show, coherent, sharp (step- function-like) sub-boundaries with orientation changes to > 0.01−0.02◦ are recovered
Figure 1: Dislocation densities calculated with the parameters shown in Sup. Table 2, with test numbers in the bottom left of each suite of dislocation types.
after denoising. Hielscher et al. (2019) noted that the total variational (“global”) filter recovers low and high angle boundaries in their tests.
4 Improved EBSD Acquisition
Along with the amount of data the substantially improved quality of indexing also affects resolution of misorientation axes for small angular change since the work by Prior (1999).
Earlier phosphor screens were smaller, capturing a smaller solid angle of the diffraction pattern. The CCD cameras were of lower resolution and sensitivity. The average number of bands for indexing of olivine in early maps was six, with a mean MAD of ≥0.7◦. The present maps for the undeformed single crystal were acquired with an average of 9.99 bands and mean MAD 0.18◦, while for the deformed single crystals the average number of bands was 9.8 with mean MAD of 0.25◦.
5 Additional Material for [110]c Sample 7506-2
Sup. Figure 2 shows the dislocation density along Line 2 in Figure 7b (see also Figure 12).
The prominent sub-boundary with a large step in angular orientation at the beginning of the line is formed by all dislocation types, except that of the main slip system, [100](010), causing its low correlation coefficient along the line. [001] screw dislocations are well correlated with angular orientation change along this line, including the right arm of the inverted V shapes rays.
Sup. Figure 3 shows a map at 0.19 µm step size with details of the prominent sub- boundary in Figure 7. At this resolution the sub-boundary shows faceting with segments of [100](001) edge dislocations, and segments dominated by both types of screw dislocations.
The main dislocation type, [100](010) is nearly absent.
Figure 2: Dislocation densities along Line 2 in Figure 7b.
Figure 3: Dislocation densities of the six dislocation types of 7506-2 for the higher resolu- tion map of this sample. The black arrow points to surface imperfections, the red arrow indicates a crack. The imperfections in the carbon coated surface are visible for all slip systems, while the crack only appears in [100](001) edges and [100] screws. This suggests that the former is a surface artifact, while the latter is sample intrinsic.
6 [011]c Sample 7406-2
Sup. Figure 4 shows a map covering a 720 x 460µm area of 7406-2. The map shows both vertical and oblique horizontal structures. In comparison to 7506-2 orientation changes are smaller and appear more localized. The KAM map shows incipient sub-boundaries predominantly perpendicular to the crystal [001] direction.
Figure 4: (a) EBSD map with a stepsize of of 1.5 µm of deformed single crystal 7406-2 color coded by misorientation from the mean orientation. (b)Kernel average misorienta- tion map. Colorscale represents misorientation in degrees capped at 0.2◦. (c) The same map color coded to show orientation changes. The mean orientation of the crystallo- graphic axes are shown, as well as lines further discussed in the text.
6.1 Dislocation Density Maps
Dislocation densities shown in Figure 5 indicate bands of alternating high and low dislo- cation density for [001](010) edge dislocations intended to have the highest resolved shear stress in the [011]c geometry. These bands are near perpendicular to the [001] crystal- lographic direction. Misorientation axes in Figure 6 from a line parallel to [001] (Figure 4b) are clustered near [100], consistent with activity of the [001](010) glide system. Fur- ther, Figure 7 shows that orientation gradients are best correlated with [001](010), and
uncorrelated with b = [100] edge dislocations. For a line parallel to the [100] crystal- lographic axis with small angular change, orientation changes are best correlated with [100](010). At the stepsize of 0.2 µm of the second map all slip systems show some correlation with orientation changes, but [001](010) is again best correlated (main text, Table 2). Screw dislocations of both Burgers vectors are moderately correlated with the orientation changes for the [011]c sample.
Figure 5: Dislocation densities by dislocation type for the [011]c sample 7406-2. Arrow in the top left panel indicates a polishing a scratch. The [001](010) glide system shows somewhat discontinuous bands of high dislocation density. [100](010) is dominated by horizontal structures corresponding to the orientation changes seen in Figure 4.
6.2 Fine Structure of 7406-2
The high resolution map reveals further features from the top of the deformed crystal.
Scratches due to polishing are visible in the band contrast image (Figure 8a) and can be matched with features in the dislocation density map (Figure 8b). Close inspection shows
Figure 6: (a) Angular change in orientation along Line 2 in Figure 4b relative to the first point of the line (left axis), and corresponding gradient between neighboring pixels (right axis). (b)Misorientation axes in crystal coordianates for the change in orientation between neighboring pixels.
that scratches produce straight, groove-like structures, with high dislocation density at the edges of the scratch and low densities in the center. By contrast, deformation induced structures show only a single, high density band of dislocations, which is also not straight.
They can therefore clearly be distinguished from scratches. Particularly noticeable in the dislocation density map are lenticular structures of high dislocation density.
A chip observable in the band contrast image (indicated by the blue arrow) as well as other cracks are surrounded by high dislocation density of [001](010) edge dislocations.
The cracks in the band contrast image are visible in reflected light and extend throughout the section, i.e. can not be removed by polishing. Overall this may suggest that these cracks, with lenticular chips surrounded by high dislocation density represent cracks at the top of the sample, close to the platen faces.
Figure 7: Correlation of dislocation density for the six dislocation types with angular change in orientation along Line 2 in Figure 4b. The correlation coefficient shown in the top right of each panel is calculated after smoothing (solid line) of the raw data (dashed).
[001](010) is best correlated with the orientation gradient; both types of screw dislocations are also involved in the orientation changes.
6.3 Summary for 7406-2
For the [011]c orientation bands of alternating high and low dislocation density of the [001](010) slip system perpendicular to [001] are much narrower, approximately 10 µm wide (SI Figures 2 and 3). However, the mapped section was cut perpendicular to the compression direction, adjacent to the platens. The structures at the top of the map from the plane strain section of the [110]c sample indicate some friction between platen and crystal, resulting in heterogeneous stress. Overall for the [011]c sample dominance of [001](010) slip systems is again consistent wit the inference of Durham and Goetze (1977) for this orientation.
Figure 8: (a)Band contrast image of higher resolution map for 7406-2. (b)Corresponding dislocation density map of [001](010). Black arrows indicate scratches, red arrows indicate dislocation structures, continuous as well as lenticular. Blue arrow in both a and b points to a chip that is surrounded by high dislocation density.
7 Noise and Scratches
In the main text the noisiness of [001](010) in sample 7506 was described. Another example of the orientation dependence of noise is given by comparison of [001](100) edge dislocation of 7506-2 (Figure 8) and the as received San Carlos single crystal (Figure 6).
The map of this slip system of the former, experimentally deformed samples is less noisy, with lower average dislocation density than the latter naturally deformed crystal with overall lower dislocation density.
Scratches from polishing are evident in maps of the [100](001) slip system of 7506 (Figure 8) as well as in maps from 7406 (SI Figure 5). Additionally cracks at the platen interface result in dislocation structures (SI Figure 5). Room temperature indentation experiments have been used to investigate dislocation structures by TEM (e.g. Gabori-
aud et al. (1981)). Inspection of images (optical, secondary electron and band contrast) allows distinction of these features from deformation structures due to high temperature deformation.
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