on the measurement of dislocations and dislocation ... acta... · edge dislocations: 12 screw...
TRANSCRIPT
O. Muránsky, L. Balogh,
M. Tran, C.J. Hamelin,
J.-S. Park, M. R. Daymond
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.