O. Muránsky, L. Balogh,

M. Tran, C.J. Hamelin,

J.-S. Park, M. R. Daymond

ondrej.muransky@ansto.gov.au

Only thanks to the support of many people this paper has happened. Special thank you to T. Palmer (ANSTO), T. Nicholls (ANSTO), Z. Zhang (ANSTO), L. Edwards (ANSTO), and M.R. Hill (UC). I hope you find the paper useful. /OM

On the Measurement of Dislocationsand Dislocation Structures using EBSD and HRSD Techniques

To maintain compatible deformation across variously oriented grains in a polycrystalline aggregate, the voids and overlaps between the individual grains, which would otherwise appear due to the crystallites (grains) anisotropy are corrected by the storing a portion of dislocations in the form of geometrically-necessary dislocations (GNDs). Plastically deformed material also stores so-called statistically-stored dislocations (SSDs), which are stored by mutual random trapping. Both GNDs and SSDs arrange themselves into energetically favourable configurations, forming geometrically-necessary boundaries (GNBs) and incidental dislocation boundaries (IDBs), respectively.

01|

Dislocations & Sub-Grain Structure

02|

Experiment

Ni C Si P Fe Mn Cr Mo Cu V Nb Ti Al

bal. <0.01 0.07 <0.01 0.03 <0.01 <0.01 <0.01 0.01 <0.01 <0.01 0.07 <0.01

Ni-201

EBSD orientation map showing the overall equiaxed grain structure of our solution-annealed Ni201 before testing.

Interrupted tensile tests were performed to varying levels of imparted plastic strain. Samples were extracted from the gauge length for EBSD and HRSD measurement.

EBSD Measurements

03|

04|

Electron Back-Scatter Diffraction (EBSD)

EBSD, is a scanning electron microscope (SEM) based technique that gives crystallographic information about the microstructure of a sample. The data collected with EBSD is spatially distributed

and is visualised in so-called EBSD orientation maps.

GNDs have a geometrical consequence giving rise to a curvature of the crystal lattice, which can be measured by EBSD technique. The crystal orientation (1,,2) changes only when the electron beam crosses an array of GNDs that has a net non-zero Burger’s vector.

05|

EBSD & Dislocations

A schematic representation of lattice curvature components calculation between two neighbouring crystals misoriented (∆θ) by a rotation around the common crystallographic axis [100]c ([uvw]c) and separated by pixel separation distance (Δx2). Note, that in this example: κ12≈Δθ1/Δx2, and κ22,κ32 = 0.

06|

Lattice Curvature

Lattice Curvature Tensor

pixel separation distance, x

𝜅𝑖𝑗 =𝜕𝜃𝑖𝜕𝑥𝑗

≈∆𝜃𝑖Δ𝑥𝑗

𝜅𝑖𝑗 =

𝜅11 𝜅12 𝜅13𝜅21 𝜅22 𝜅23𝜅31 𝜅32 𝜅33

Lattice Curvature Tensor Components

𝜅11 ≈Δ𝜃1Δ𝑥1

; 𝜅21 ≈Δ𝜃2Δ𝑥1

; 𝜅31 ≈Δ𝜃3Δ𝑥1

𝜅12 ≈Δ𝜃1Δ𝑥2

; 𝜅22 ≈Δ𝜃2Δ𝑥2

; 𝜅32 ≈Δ𝜃3Δ𝑥2

Misorientation Angle

W. Pantleon, Scripta Materialia, 58, 2008, pp. 994-997.

07|W. Pantleon, Scripta Materialia, 58, 2008, pp. 994-997.

Lattice Curvature & GND Density

𝜅𝑖𝑗 =

𝑡=1

𝑁

𝑏𝑗𝑡𝑙𝑖𝑡 −

1

2𝛿𝑖𝑗𝑏𝑚

𝑡 𝑙𝑚𝑡 𝜌𝐺

𝑡

N = 1 .. 36 – number of possible dislocation types

36

G

2

G

1

G

36

3

36

2

2

3

2

2

1

3

1

2

36

2

36

2

2

2

2

2

1

2

1

2

36

1

36

2

2

1

2

2

1

1

1

2

36

3

36

1

2

3

2

1

1

3

1

1

36

2

36

1

2

2

2

1

1

2

1

1

36

1

36

1

2

1

2

1

1

1

1

1

32

22

12

31

21

11

2

1

2

1

2

1

2

1

2

1

2

1

lblblb

lblblb

lblblb

lblblb

lblblb

lblblb

6 known lattice curvatures (measured)

36 possible dislocation types

Burgers Vector

GND Density

Edge Dislocations: 12Screw Dislocations: 6

Dislocation Types: 2 x 18 = 36dislocations of opposite

sign needs to be distinguished

1,947,792 possibilities!

Line Vector

Not Accessible

11

21

31

12

22

32

13

23

33

Not Accessible

Not Accessible

Lattice Curvature Tensor (ij)

08|A. Arsenlis, Acta Matterialia, 47, 1999, pp. 1597-1611.

Dislocation Types (fcc)

6Burger’s Vectors

Number of Dislocation Types

b t

1

2

3

4

5

6

b 011 a /2

b 101 a /2

b 011 a /2

b 101 a /2

b 110 a /2

b 010 a /2

b t

1 1 1

2 1 2

3 2 3

4 2 4

5 3 5

6 3 6

7 4 7

8 4 8

9 5 9

10 5 10

11 6 11

12 6 12

t b n

t b n

t b n

t b n

t b n

t b n

t b n

t b n

t b n

t b n

t b n

t b n

13 1

14 2

15 3

16 4

17 5

18 6

t b

t b

t b

t b

t b

t b

12Line Vectors

Edge = 12Screw = 6

6Line Vectors

18 2 = 36

dislocations of oppositesign needs to be distinguished

111 011

111 011

111 101

111 101

111 011

111 011

111 101

111 101

111 110

111 110

111 010

111 010

12

Def

orm

atio

n M

od

es

09|W. Pantleon, Scripta Materialia, 58, 2008, pp. 994-997.

Lower-Bound GND Density

𝜅𝑖𝑗 =

𝑡=1

𝑁

𝑏𝑗𝑡𝑙𝑖𝑡 −

1

2𝛿𝑖𝑗𝑏𝑚

𝑡 𝑙𝑚𝑡 𝜌𝐺

𝑡

N = 1 .. 36 – number of possible dislocation types

36

G

2

G

1

G

36

3

36

2

2

3

2

2

1

3

1

2

36

2

36

2

2

2

2

2

1

2

1

2

36

1

36

2

2

1

2

2

1

1

1

2

36

3

36

1

2

3

2

1

1

3

1

1

36

2

36

1

2

2

2

1

1

2

1

1

36

1

36

1

2

1

2

1

1

1

1

1

32

22

12

31

21

11

2

1

2

1

2

1

2

1

2

1

2

1

lblblb

lblblb

lblblb

lblblb

lblblb

lblblb

6 known lattice curvatures (measured)

36 possible dislocation types

𝑡=1

6

𝑤𝑡𝜌𝐺𝑡 = 𝑚𝑖𝑛

Burgers Vector

GND Density

Simplex Optimisation Algorithm► With a set of 6 linear

equations and 36

unknowns, a large

number of possible

solutions exists

whereby a unique

solution cannot be

obtained. It is therefore

necessary to constrain

the solution using

physically-based

constraints.

𝜌𝐺 =

𝑡=1

36

𝜌𝐺𝑡 ≈ min

𝑡=1

6

𝜌𝐺𝑡

𝐸𝑠𝑐𝑟𝑒𝑤 = 1 − 𝜐 𝐸𝑒𝑑𝑔𝑒

Not all dislocation types are equally energetically

favourable.

Lower-bound GND Density

𝑤𝑡 = Ԧ𝑏𝑡 Ԧ𝑙𝑡Edge Dislocations: 12Screw Dislocations: 6

Dislocation Types: 2 x 18 = 36dislocations of opposite

sign needs to be distinguished

1,947,792 possibilities!

Line Vector

Density of geometrically-necessary dislocations (GND, ρG) maps calculated from the EBSD-measured Euler Angles (1,,2) for specimens with 0% (as-received), 7.8% and 13.9% of imparted plastic strain.

GNDs arrange themselves into energetically favourable configurations forming geometrically-necessary boundaries (GNBs) subdividing grains into the sub-grains.

Discrete measurements provide information on spatial distribution of GND across the microstructure.

Step size (h) = 200 nm Magnification = 153x

10|

GND Density

dG = Τ1 ρG

GND DensityDislocation

spacing

Non-uniform distribution of GNDs in the microstructure as GNDsarrange themselves into energetically favourable configurations subdividing grains into the sub-grains.

Spacing between geometrically-necessary dislocations (GND, dG) recalculated from the GND density (ρG) for specimens with 0% (as-received), 7.8% and 13.9% of imparted plastic strain.

11|

GND Spacing

GND Density - High Resolution

Density of geometrically-necessary dislocations (GND, ρG) calculated from the EBSD-measured Euler Angles (1,,2).

Spacing between geometrically-necessary dislocations (GND, dG) recalculated from the GNDdensity (ρG).

Step size (h) = 20 nm Magnification = 1000x

12|

GND Density

Density of geometrically-necessary dislocations (GND, ρG) maps calculated from the EBSD-measured Euler Angles (1,,2).

13|

Distribution (histogram) of discrete GND density (ρG) measurements for specimen with 0% (as-received), 7.8% and 13.9% of imparted plastic strain (εp).

𝑓 𝜌𝐺|𝜇, 𝜎 =1

𝑥𝜎 2𝜋𝑒𝑥𝑝

− ln 𝜌𝐺 − 𝜇 2

2𝜎2

𝑚(𝜌𝐺) = 𝑒𝑥𝑝 𝜇 +𝜎2

2

Log-Normal Distribution

Mean & Variance

VARIANCE of the log-normal distribution

The variance of the GND density distribution describes the heterogeneity of the GND distribution across variously oriented grains within the microstructure, the mean can be then taken as the microstructure-averaged (bulk) GND density.

𝑣(𝜌𝐺) = 𝑒𝑥𝑝 2𝜇 + 𝜎2 (𝑒𝑥𝑝 𝜎2

14|

Microstructure-Averaged GND Density

MEAN of the log-normal distribution

𝑓 𝜌𝐺|𝜇, 𝜎 =1

𝑥𝜎 2𝜋𝑒𝑥𝑝

− ln 𝜌𝐺 − 𝜇 2

2𝜎2

𝑚(𝜌𝐺) = 𝑒𝑥𝑝 𝜇 +𝜎2

2

Log-Normal Distribution

Mean & Variance

VARIANCE of the log-normal distribution

MEAN of the log-normal distribution

Due to the increase in heterogeneity of GND distribution with imparted plastic strain, a larger number of grains is required to reach solution convergence.

𝑣(𝜌𝐺) = 𝑒𝑥𝑝 2𝜇 + 𝜎2 (𝑒𝑥𝑝 𝜎2

The development of the mean GND density as a function of number of analysed grains in GND density maps for specimen with 0% (as-received), 7.8% and 13.9% of imparted plastic strain (εp).

15|

Microstructure-Averaged GND Density

The development of the mean GND density distribution as a function of imparted plastic strain (εp) for all tested specimens.

The development of the variance of GND density distribution as a function of imparted plastic strain (εp) for all tested specimens.

16|

Microstructure-Averaged GND Density

Map showing the ratio of screw dislocations to the total number of dislocations in the solution (6) for the specimen with 13.9% imparted plastic strain.

Screw dislocation ratio as a function of imparted plastic strain for all tested specimens.

The uniqueness of the solution is not guaranteed. Only pure edge and pure screw dislocations have been considered in the calculation.

17|

GND Types in Solution

HRSD Measurements

18|

- 200 200 μm- = 0.018972 nm- 200 0.3-s intervals

- Detector: 2048 2048 pixels- Pixel Size: 200 200 μm- Distance: 1873 mm

19|

HRSD Set-Up

High-resolution synchrotron diffraction (HRSD) set-up at 1-ID high-energy beamline at the Advanced Photon Source (APS), Argonne National Laboratory (ANL).

𝐼𝑇𝑂𝑇𝐴𝐿 = 𝐼𝑆𝐼𝑍𝐸 ∗ 𝐼𝑆𝑇𝑅𝐴𝐼𝑁 = ℱ−1 𝐴𝑆𝑖𝑧𝑒𝐴𝑆𝑡𝑟𝑎𝑖𝑛

𝐴𝑆𝑖𝑧𝑒 = ℱ 𝐼𝑆𝐼𝑍𝐸 𝐴𝑆𝑡𝑟𝑎𝑖𝑛 = ℱ 𝐼𝑆𝑇𝑅𝐴𝐼𝑁

The total diffraction peak shape (which includes peak broadening) ITOTAL of a is the convolution of the shape contribution caused by the size of coherently scattering domains (sub-grains) ISIZEand the contribution caused by strain fields of present dislocations ISTRAIN.

Size (sub-grain) contribution to the

peak shape.

Convolution is defined as the invers Fourier transform of the product of the individual Fourier transform of the components.

Strain (dislocation) contribution to the

peak shape.

20|

Diffraction Peak Broadening

T. Ungar et al., J. Appl. Cryst. 34, 2001, pp. 298-310.

The broadening due to the size of the coherently diffracting domains (sub-grains) is the same for all hkldiffraction peaks, while the broadening component due to the strain field of present dislocations varies between diffraction peaks. This variation in the strain (dislocation) broadening is not monotonous due to the anisotropic behaviour described by the dislocation contrast factors.

- XA = 60 nm- ρT = 21E+14 m-2

- q = 2- M = 1.6

Williamson-Hall Plot

T. Ungar et al., J. Appl. Cryst. 34, 2001, pp. 298-310.21|

Diffraction Peak Broadening

Comparison of full diffraction patterns for specimens with 0% (as-received) and 13.9% of imparted plastic strain. The different behavior of size (sub-grain) and strain (dislocation) peak broadening can be resolved if many peaks are available.

The diffraction peak broadening was analysed using the eCMWP (extended Convolutional Multiple Whole Profile) LPA software

2D diffraction pattern (Debye-Scherrer rings) of specimen with 13.9% of imparted plastic strain.

22|

Diffraction Peak Broadening

Total dislocation density (ρT) and size of the coherently scattering domains (SCDs) obtained by line profile analysis (LPA) of HRSD patterns as a function of imparted plastic strain (εp) - open symbols represents individual measurements along the sample loading axis, and solid symbol represents the mean values.

23|

Total Dislocation Density & Sub-Grain Size

EBSD + HRSDMeasurements

24|

Comparison of the HRND-measured total dislocation density (ρT) and the EBSD-measured density of GNDs (ρG), together with expected dislocation densities calculated using the modified Taylor’s model, and single-slip Ashby’s model.

25|

EBSD- & HRSD- Measured Dislocation Density

Both GNDs and SSDs contribute to the work-hardening of the material. SSDs represent more than 80% of all the present dislocations.

Comparison of the HRSD-measured size of CSDs (red circles) with EBSD-measured spacing of GNDs (dG) (blue squares), and the estimated minimum size of CSDs (green triangles) from EBSD-measured density of GNDs (ρG).

Size of SSDs from G

G from XA

GNDsdensity

EBSDstep size

𝑋 𝐴 ≥ 𝑋 𝐴,𝑚𝑖𝑛 =2

3

1

∆𝑥𝜌𝐺=

2

3

𝑑𝐺2

∆𝑥

GNDs spacing

GND Density & Size of CSDs

𝜌𝐺 ≥2

3

1

∆𝑥 𝑋 𝐴

EBSDstep size

Size of SSDs

This defines the connection between EBSD-measured G and HRSD-measured XA one can then estimate G from XA.

26|

EBSD measures the lower-bound 𝜌𝐺, while HRSD measures 𝜌𝑇.

The minimum detected 𝜌𝑇 measured by HRSD is about 1E13 m-2, while the minimum 𝜌𝐺 measured by EBSD is about 2E12 m-2.

EBSD is more sensitivity to the small amount of plastic deformation in the material, while HRSD gets more accurate with higher amount of plastic deformation.

There is a connection between EBSD-measured 𝜌𝐺 and HRSD-measured size of CSDs (XA).

EBSD = Density of GNDs (𝜌𝐺), + estimate the minimum Size of CSDs

HRSD = Total Dislocation Density (𝜌𝑇), size of CSDs (XA), + estimate of minimum density of GNDs (𝝆𝑮)

27|

Conclusions

𝑋 𝐴 ≥ 𝑋 𝐴,𝑚𝑖𝑛 =2

3

1

∆𝑥𝜌𝐺=

2

3

𝑑𝐺2

∆𝑥

GNDsdensityEBSD

step size

GNDs spacing

Thank you for your time and interest in this work. We hope you

will find it useful.

Developed Matlab code for calculation of GNDs is available as a supplementary material with out Acta Materialia paper.