The Orientation Distribution

1
The Orientation Distribution

• A. D. Rollett
• Advanced Characterization Microstructural
  Analysis
• Lecture 2 part B

2
Impact of Anisotropic Grain Boundary Properties

• Aluminum foil annealed at 550C to obtain
  microstructures from which g.b. properties were
  measured (a story for a later time).
• Snapshots obtained of the grain structure at two
  different times.
• Structure meshed (finite element) and grain
  growth simulated with and without the measured
  properties.

3
Computational verification of Grain Growth
Annealing
EBSD Microstructure
EBSD (OIM)
Microstructure
Mesh Generation LAGRIT
?
Grain Simulation Grain3d
Melik Demirel, MRS Proceedings, Fall 2000
4
Computational Model

• Microstructure kinetics is modeled in 3-D using
  gradient-weighted moving finite elements.

• Boundary conditions quasi-periodic

A. Kuprat, SIAM J. ON SCI. COMP., Vol. 22, 535,
2000
5
EBSD Boundary Detection
Philips-XL 40 FEGSEM
EBSD patterns
Boundary and Triple Junction Information (Gabor
Wavelet Filter) D. Casasent, ECE, CMU
Filtered Image
150 m
6
Sample Preparation Limitations

• Al-foil (99.98 pure) 120 micron thickness
  (ALCOA)
• Annealed 9 hours at 550C to get columnar
  structure (N2 environment)
• Electropolished (60 seconds, 10V)
• Microhardness indents (locate the scanning area)

7
Experiment Annealing of Al-foil
BEFORE
AFTER
150 m
150 m
Columnar Aluminum foil Annealed at 550 C for
20 minutes
8
Low Angle Grain Boundary Mobility
001
Low
117
105
113
205
215
335
203
8411
111
High
(Yang, Mullins, Rollett, 2000)
101
323
727
9
Experimental microstructures?Simulation inputs
Mesh Generation
Initial Configuration
Top view
3-D view
D. George, LAGRIT http//www.t12.lanl.gov/lagrit

10
Simulation Result (Isotropic)
Simulation
Final Configuration
Initial Configuration
Experiment
11
Simulation Result (Anisotropic)
Simulation
Final Configuration
Initial Configuration
Experiment
12
Lecture Objectives

• Outline conversions from Miller indices to
  matrices to Euler angles.
• Outline the representation of discrete ODs
• Outline the conversion from volumes to OD and
  vice versa.

13
Notation

• Vector-Matrix v is a vector, A is a matrix
• Index notation explicit indexes (Einstein
  convention)vi is a vector, Ajk is a matrix
  (maybe tensor)
• Scalar (dot) product c ab aibi
• Vector (cross) product c ci a x b a ? b
  eijk ajbk

14
Miller indices to vectors

• Need the direction cosines for all 3 crystal
  axes.
• Construct a transformation matrix from the
  vectors.

15
Rotation of axes in the planex, y old axes
x,y new axes
y
y
v
x
q
x
N.B. Passive Rotation/ Transformation of Axes
16
Euler Angles, Animated
e3ZsampleND
e3
001
010
e3
zcrystale3
f1
ycrystale2
e2
f2
e2
e2YsampleTD
xcrystale1
100
F
e1
e1
e1XsampleRD
17
Definition of an Axis Transformatione old
axes e new axes
Sample to Crystal (primed)


e3
e3

e2

e2


e1
e1
18
Form matrix from Miller Indices
19
Geometry of hklltuvwgt
Sample to Crystal (primed)


e3
e3 (hkl)
Miller indexnotation oftexture
componentspecifies directioncosines of
xtaldirections tosample axes!
001
010

e2


e2 t

e1 uvw

e1
100
20
Euler angles to Matrix
Rotation 1 (f1) rotate axes (anticlockwise)
about the (sample) 3 ND axis Z1. Rotation 2
(F) rotate axes (anticlockwise) about the
(rotated) 1 axis 100 axis X. Rotation 3 (f2)
rotate axes (anticlockwise) about the (crystal)
3 001 axis Z2.
21
Euler angles to Matrix, contd.
AZ2XZ1
22
Matrix with Bunge Angles
A Z2XZ1
uvw
(hkl)
23
Matrix with Kocks Angles
a(Y,Q,f)
(hkl)
uvw
Note obtain transpose by exchanging f and Y.
24
Euler Angle Conventions
Bunge and Canova are inverse to one anotherKocks
and Roe differ by sign of third angleBunge
rotates about x, Kocks about y (2nd angle)
25
Matrix with Roe angles
(hkl)
uvw
a(y,q,f)
26
Conversions
27
Compare Matrices
uvw
(hkl)
uvw
(hkl)
28
Miller indices from Euler angle matrix
Compare the indices matrix with the Euler angle
matrix.
n, n factors to make integers
29
Euler angles from Miller indices
Inversion ofthe previousrelations
30
Discrete OD

• Normalization also required for discrete OD
• Sum the intensities over all the cells.
• 0?f1 ?2p, 0?F ?p, 0?f2 ?2p0?f1 ?90, 0?F ?90,
  0?f2 ?90

31
Volume fraction calculations

• Choice of cell size determines size of the volume
  increment, which depends on the value of the
  second angle (F or Q).
• Some grids start at the specified value.
• More typical for the specified value to be in the
  center of the cell.
• popLA grids are cell-centered.

32
Discrete ODs
dAsinFdFdf1?A?(cosF)?f1
Each layer ?VS?A?f290()
f1
Total8100()2
0
20
10
90
80
0
f(10,0,30)
10
F
? F10
f(10,10,30)
20
Section at f2 30?f210
80
f(10,80,30)
90
? f1 10
33
Centered Cells
dAsinFdFdf1?A?(cosF)?f1
f1
Different treatment of end cells
90
0
20
10
0
? F5
f(10,0,30)
F
? F10
.
10
f(10,10,30)
20
80
90
f(10,90,30)
? f1 10
? f1 5
34
Volume fraction calculation

• In its simplest form sum up the intensities
  multiplied by the value of the volume increment
  (invariant measure) for each cell.

35
Discrete orientation information
WorkDirectory /usr/OIM/rollett
OIMDirectory /usr/OIM ... 4.724
0.234 4.904 0.500 0.866 1.0
1.000 0 0 4.491 0.024 5.132
7.500 0.866 1.0 1.000 0 0
4.932 0.040 4.698 19.500 0.866
1.0 1.000 0 0 4.491 0.024 5.132
20.500 0.866 1.0 1.000 0 0
4.491 0.024 5.132 21.500 0.866
1.0 1.000 0 0 4.932 0.040 4.698
22.500 0.866 1.0 1.000 0 0
4.932 0.040 4.698 23.500 0.866
1.0 1.000 0 0 4.932 0.040 4.698
24.500 0.866 1.0 1.000 0 0
f1
F
f2
x
y
(radians)
36
Binning individual orientations in a discrete OD
f1
0
20
10
90
80
0
10
F
? F10
20
Section at f2 30
individualorientation
80
90
? f1 10
37
OD from discrete points

• Bin orientations in cells in OD, e.g. Euler space
• Sum number in each cell
• Divide by total number of grains for Vf
• Convert from Vf to f(g) (90x90x90
  space) f(g) 8100 Vf/?(cosF)?f1?f2

cell volume
38
Discrete OD from points

• The same Vf near F0 will have much larger f(g)
  than cells near F 90.
• Unless large number (gt104, texture dependent) of
  grains are measured, the resulting OD will be
  noisy, i.e. large variations in intensity between
  cells.
• Typically, smoothing is used to facilitate
  presentation of results always do this last and
  as a visual aid only!