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## Grain Boundary character: 5parameter descriptions

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Title: Grain Boundary character: 5parameter descriptions

1
Grain Boundary character 5-parameter
descriptions
Analysis
• 27-750, A.D. Rollett, Spring 2003

2
Tilt versus Twist
• Definitions a tilt boundary is one in which the
rotation axis lies in the boundary plane.
• A twist boundary is one in which the rotation
axis is perpendicular to the boundary plane.
• Example a coherent twin boundary (in fcc) is a
pure twist boundary, 60 lt111gt.

3
Properties
• Relevance? For low angle boundaries, twist
boundaries have screw dislocation structures
tilts have edge dislocations.
• The properties of twist and tilt boundaries are
sometimes different evidence in fcc metals
suggests that only 40 lt111gt tilt boundaries are
highly mobile, but not the twist 40 lt111gt type
(in recrystallization).

4
Notation
• How do we write down a mathematical description
of a 5-parameter grain boundary? Answer there
are 3 useful methods.
• Disorientation plane
• Plane of 1st grain plane of 2nd grain twist
angle
• Matrix (4 x 4)

5
Fundamental Zone
• What is the smallest possible range of parameters
needed to describe a 5-parameter grain boundary?
• Disorientation based use the same FZ for
disorientation hemisphere for the plane
• Boundary plane based use SST for the first plane
double-SST for the second plane (0,2p for
the twist angle
• Matrix no FZ established.

6
Disorientation based RF-space
Disorientations can be described in the
Rodrigues-Frank spacewhose FZ for cubic
materials is a truncated pyramid. The boundary
plane requires a full hemisphere in order to be
described.
Misorientation axis, e.g. 111
7
Disorientation based boundary plane based
2nd plane normal requires double SST
1st plane normal requires single SST

f 0
f 2p

8
Matrix description
• Construct the matrix from the 3x3 orthogonal
rotation matrix that describes the misorientation
and add a columnrow to describe the plane
normals of each side of the boundary Morawiec,
A. (1998). Symmetries of the grain boundary
distributions. Int. Conf. on Grain Growth,
Pittsburgh, PA, TMS. The determinant of the
resulting matrix, B, is 1, as it is for the
symmetry matrices can be constructed from the
conventional 3x3 matrices.

9
The Number of Grain Boundary Types
10
Measurement
• We can measure the orientation of the grains on
either side of a boundary as gA and gB, as well
as the boundary normal, n.
• Compare the misorientation axis in specimen
axes with the boundary normal by forming the
scalar product.
• If b0, we have a tilt boundary if b/-1, then
it is a pure twist boundary.

11
Tilt-twist character
• If cos-1(b)0, boundary is pure twist if
cos-1(b)90, boundary is pure tilt.

n
b
12
Ambiguities
• Caution! Although the misorientation axis is
unambiguous for low angle boundaries, it is not
necessarily so for high angle boundaries thanks
to the crystal symmetry.
• Finding the disorientation and locating the
misorientation axis in the SST, makes the choice
unambiguous, but other selections may be
physically reasonable.

13
Grain Boundary Plane
• The planes that make up a boundary are readily
identified.
• Transform the boundary normal to crystal axes in
either crystal.
• We can specify the boundary completely by
identifying indices for both sides of the
boundary and a twist angle between the lattices.

14
Example of G.B. plane
• If we measure the boundary normal as the vector
(1,0,0), and the orientation of the grain on the
A side is given by

ns
B
A
x1
Then the boundary normal in crystal(A)
coordinates is gAnS 1/v3(1,1,1)reasonable
gA112lt111gt!
x2
15
GB planes, disorientation
• Calculation of the disorientation with particular
choices of symmetry operators affects the
location of the rotation axis.

16
GB planes, disorientation, contd.
• Must re-calculate the planes and tilt/twist
characterbut, we can calculate the
tilt/twist character in the sample axes, if we
choose.

If cos-1(b)0, boundary is pure twist if
cos-1(b)90, boundary is pure tilt.
17
(hkl)1(hkl)2Twist
• A standard representation of a grain boundary
that is particularly useful for symmetric tilt
boundaries is the (hkl)1(hkl)2Twist definition.
This specifies the two planes the comprise the
boundary, and the twist angle between the two
lattices. The difference between (hkl)1 and
(hkl)2 defines the tilt angle between the
lattices.

18
Example(s) of GB plane
• Reminder full description of g.b. type requires
plane as well as misorientation.
• Example strong lt111gt fiber leads to boundaries
that are pure tilt boundaries with lt111gt
misorientation axes.
• Similarly, strong lt100gt fiber leads to pure lt100gt
tilt boundaries.
• Thus, in a drawn wire (e.g. fcc metal) with a
mixture of lt111gt and lt100gt fiber components, the
grain boundaries within each component will be
predominantly lt111gt and lt100gt tilt boundaries,
respectively.

19
Microstructure
• Euler angles1. (10,55,45)2. (87,55,45)3.
(32,55,45)4. (54,55,45)...

q1213q2355 q3422 etc.
2
3
1
4
2
3
cube- on- corner
1
20
Grain Boundary descriptions
2
Specimen axes- misorientation axes (0,0,1)-
g.b. normals (x,y,0)/v(x2y2)Crystal Axes -
misorientation axes 1/v3(1,1,1)- g.b. normals ?
1/v3(1,1,1), i.e., in the zone
110-112Boundary Planes are limited tothe
zone of (111).Misorientations include
S3,7,13b,19b.
3
1
2
3
_
_
1
4
21
lt111gt Fiber RF-space
Misorientations lie on R1R2R3 line
Boundary planes lie on 111 zoneand are pure tilt
lt111gt boundaries
Misorientation axis
22
Viewing Five Parameter Grain Boundary
Distributions
(MRD)
Planes for all boundaries, 40 rotation around
lt111gt
View point by point
(MRD)
Planes for all boundaries
Average over Dg
Average over n (conventional MDF)
or
23
Definitions
(hkl)1 (hkl)2
Twist angle
24
Experimental Information
• In a typical experiment, we measure (a) the
orientations of the two grains adjacent to a
boundary, gA and gB. In addition, we measure the
boundary normal, ns, in specimen axes (outward
pointing with respect to, e.g., grain A).
• Objective is to provide a unique, 5-parameter
description, based on the experimental
information.

25
Boundary Normal
• Define the normal to a boundary plane as the
outward pointing normal based on the grain to
which the normal is referred.

ns(A)
B
Example n(A) (1,0,0) n(B) (-1,0,0)
A
ns(B)
x1
x2
26
Locate Plane Normal in SST
• The interface-plane method seeks to emphasize the
crystallographic surfaces that are joined at the
boundary. The obvious choice of fundamental zone
would appear to be the 001-101-111 unit triangle
for each plane as will become apparent, however,
a combination is needed of a single unit triangle
for one plane and a double unit triangle,
001-101-111-011 for the plane on the other side
of the boundary, combined with the fifth (twist)
angle.

27
Equivalent Descriptions of a 5-parameter
Boundaryswitching symmetries
28
Equivalencies
• Geometry the misorientation carries the boundary
normal from one crystal into the other ?gAB
gBgAT nB ?gAB nA.
• Rule 1 if you apply symmetry to one orientation
(crystal), then you must also apply it to the
boundary plane. Thus if we have some property,
f, of a grain boundary, f(?g,nA) f(?gOAT,
OAnA) f(OB?g, nA).
• Rule 2 centrosymmetry has this effect f(?g,nA)
f(?g,-nA).
• Rule 3 switching symmetry applies f(?g,nA)
f(?gT,-?gTnA).

29
Locate Normals in SST
• Apply symmetry operators to locate the first
boundary normal in the SST, possibly (switching
symmetry) relabeling A as B and vice versa.

Then repeat the process for the second plane
normal, except this time, the result falls in a
double triangle
30
Unit triangles for plane normals
1st triangle
2nd triangle
31
Coincidence of triangles?
Look down the misorientationaxis at the grain
boundary planeof a twist boundary
Case A coincidence not possible by rotation
(hkl)
Case B coincidence is possible by rotation
Conclusion it is possible to have the same
rotation axis in the two crystals and yet for
there to be no twist angle that yields zero
misorientation.
32
Demonstration
• To demonstrate the difference between (hkl)(hkl)
and (hkl)(khl), we can draw the crystal axes.
Clearly in case 1, the axes can be made to
coincide at some twist angle (? no boundary). In
case 2, however, this cannot be accomplished at
any twist angle.

nA hkl
nA hkl
- - -
nB hkl
nB hkl
2 (hkl)(khl)
1 (hkl)(hkl)
33
Examples planes inside the SST
34
Example dissimilar planes inside the SST
35
Rodrigues space
(123)(213)
(123)(123)
The twist angle is notated at each plotted point.
Note how the misorientation axis varies between
the two sequences of grain boundaries.
36
Special Cases
• The exception to the above analysis occurs when
one of the planes lies on a mirror plane, i.e. in
the zone of either lt100gt or lt110gt. That is to
say, there is always a zero-misorientation twist
angle when you combine (hhl) or (h00) with
itself.
• Why? The mirror plane means that (hkl) and (khl)
are equivalent.
• A more subtle point is that (hhl)(mno) is
equivalent to (hhl)(nmo) except for a reversal
in the sense of twist. Specifically,
(hhl)(mno)(f) ? (hhl)(nmo)(-f).
• How to demonstrate this? Calculate the full
disorientationplane.

37
Special Cases
• No boundary (hkl)1 -1(hkl)2 twist0.
• Pure Twist (hkl)1 -1(hkl)2 twist ? 0.
• Symmetrical Tilt (h,k,0)1 -1(-h,k,0)2
twist0.
• Asymmetric tilt (hkl)1 ? -1(hkl)2 twist0.
• Note only proper rotations O(432) permitted as
symmetry operators to permute the indices of the
planes.

38
More Special Cases
• If (hkl)1 (hkl)2 twist0, then we have a
twin relationship that is equivalent to (hkl)1
-1(hkl)2 twist180. Note that matching (hkl)
does not necessarily allow a zero-boundary
condition to be found.

39
Conversions
• It is useful to be able to convert from one type
of description to another.
• First we describe conversion from ?gn to
(hkl)A(hkl)BTwist.
• Then we describe the inverse process to convert
from (hkl)A(hkl)BTwist to ?gn.
• See Takashima, M., A. D. Rollett and P.
Wynblatt (2000). "A representation method for
grain boundary character." Philosophical Magazine
A 80 2457-2465..

40
Calculation of Tilt, Twist parameters, given the
grain orientations and boundary plane
• Given the orientations of the two grains we can
compute the misorientation and decompose it into
a pair of tilt and a twist rotations

The tilt angle, y, is obtained as (the twist
rotation does not alter the position of the plane
normal)
41
Tilt-Twist Decomposition
Note it is not possible to obtain the twist
angle from this relationship based on a knowledge
of the misorientation and tilt angles because the
inverse trigonometric functions are limited to a
range of 0xp.
RTWIST n
f
q
2(1 cosq) (1 cosf)(1 cosy)
RTILT
y
42
Tilt Rotation
• The tilt axis is obtained from the cross product
of the normals
• Rotation matrix for the (active) tilt rotation,
R(rtilt,y)A-gtB

43
Twist Angle
• Form the tilt rotation matrix, and calculate the
twist rotation matrix from

Then obtain the twist angle thus
Note x and nB may be parallel or anti-parallel,
which is importantbecause the range of the twist
angle is 0f2p.
44
5-parameter conversions nAnBtwist??gn
• Given (hkl)A(hkl)Btwist, with the normals
defined as outward-pointing normals
• Obtain the tilt angle from rotation required to
bring the two normals (with inversion of nB) into
coincidence
• Vector product gives the rotation axis

45
nAnBtwist??gn, contd.
• Rotation matrix for the (active) tilt rotation,
R(rtilt,y)A-gtB
• Twist rotation axis nB angle twist angle, f,
giving R(rtwist,f)A-gtB

46
nAnBtwist??gn, no. 3
• Misorientation (axis transformation) product of
tilt, twist rotations not a disorientation! ?
gTAB R(rtwist,f) R(rtilt,y)
• Apply symmetry to identify the disorientation
(i.e. minimum angle, axis in the SST) ?g(gBOc)
-1(gAOc) Oc?gABOc-1

47
nAnBtwist??gn, no. 4
• Choose whether to use nA or nB to characterize
the plane.
• Apply the (same) symmetry operator as used to
identify the disorientation to the boundary
plane
• nA,nB,twist lt-gt ?gAB,nA
• Other conversions given elsewhere.

48
Examples
• In this next section, we give some examples of
graphical representations of various sets of
grain boundaries.
• One example is a series of symmetrical tilt
boundaries.
• Another example is a set of experimentally
measured grain boundaries in Al.
• A third example is a set of boundaries measured
in an Al foil.

49
Symmetric Tilt Boundaries lt110gt
Symmetric tilt boundaries arebased on the
concept of a tiltaxis lying in the plane of
theboundary and with boundaryplanes
(symmetry)plane. Example lt110gt tilt
boundariesare based on rotations abouta 110
axis. It is convenient andaccurate to think
about rotatingeach crystal by equal and
oppositeamounts (through half the specified
misorientation angle).
nA
nB
50
Symmetric Tilt Boundaries lt100gt
Symmetric tilt boundaries arebased on the
concept of a tiltaxis lying in the plane of
theboundary and with boundaryplanes
(symmetry)plane. Example lt100gt tilt
boundariesare based on rotations abouta 100
axis. It is convenient andaccurate to think
about rotatingeach crystal by equal and
oppositeamounts (through half the specified
misorientation angle).
nA
nB
51
lt110gt symmetric tilts disorientations
Plot of the misorientations for a series of
symmetric tilt grain boundaries based on
rotations about lt110gt. For tilt angles between 0
and 60, and between 120 and 180, the
misorientations lie on the lt110gt line in the
first section (R30). For tilt angles between
60 and 120, the misorientations follow a
complex path through Rodrigues-Frank space that
includes the S3 and S17 positions.
52
lt110gt symmetric tilts (hkl)twist
nA (111)
Boundary plane normals for a series of lt110gt
symmetric tilt boundaries. The normal for the B
crystal is plotted in a quadrant which is a
double unit triangle. The range of normals for
the A crystal is delineated by a box in each
triangle.
nA (011)
nA (001)
53
lt110gt symmetric tilts ?gplane
near twin (S3)
Boundary plane normals for the same set of
symmetric tilt boundaries as shown in the
previous figure. Poles on the upper hemisphere
are plotted as solid points and poles on the
lower hemisphere as gray points. The
misorientation axis, indicated by a circle on the
projected sphere, has been placed in a single
unit triangle, and is also plotted in a Rodrigues
space triangle, as in fig. 2.
lt110gt axes
low angle boundaries
54
Example of Annealed Al bulk spec.interface
plane description
55
Example of Annealed Al (bulk)disorientation
plane description
lt110gt axes
low angle boundaries
56
Example of Annealed Al (foil)disorientation
plane description
Strong 001lt100gtcube texture presentin the
foil.
57
Boundary Plane Analysis from Single Plane Section
• At first sight, it is not possible to measure the
full 5-parameter nature of a boundary from a
single section plane. Serial sectioning is
required in order to accomplish the
characterization.
• Statistical stereology can mitigate this problem
we perform measurement of boundary tangents,
coupled with analysis in the crystal boundary
plane space.

58
Boundary Tangents
• Measure the (local) boundary tangent the normal
must lie in its zone.

gB
ns(A)
B
ts(A)
A
x1
gA
x2
59
G.B. tangent disorientation
• Select the pair of symmetry operators that
identifies the disorientation, i.e. minimum angle
and the axis in the SST.

60
Tangent ? Boundary space
• Next we apply the same symmetry operator to the
tangent so that we can plot it on the same axes
as the disorientation axis.
• We transform the zone of the tangent into a great
circle.

Boundary planes lie on zone of the boundary
tangent in this examplethe tangent happens to
be coincident withthe disorientation axis.
Disorientation axis
61
Tangent Zone
• The tangent transforms thus tA OAgAtS(A)
• This puts the tangent into the boundary plane (A)
space.
• To be able to plot the great circle that
represent its great circle, consider spherical
angles for the tangent, ct,ft, and for the zone
(on which the normal must lie), cn,fn.

62
Spherical angles
chi declination phi azimuth
f
c
63
Tangent Zone, parameterized
• The scalar product of the (unit) vectors
representing the tangent and its zone must be
zero

64
Integration
• In order to obtain a distribution of the grain
boundary character, an integration must be
performed in the space of the grain boundary
plane.
• Each boundary (segment) contributes uniform
intensity over the length of its associated
tangent zone.
• Wherever many of the tangent zones intersect,
there must be a high frequency of that boundary
type.

65
Example coherent twins
• If there is a high density of coherent twin
boundaries, these correspond to the pure twist
boundary on 60lt111gt.
• Such a condition will be evident in many tangent
zones for 60lt111gtdisorientation types
crossing at the location of the disorientation
axis.

66
Another Illustration
The probability that the correct plane is in the
zone is 1. The probability that all planes are
sampled is lt 1.
Poles of possible planes
The grain boundary surface trace is the zone axis
of the possible boundary planes.
Trace pole
67
Recovering the Distribution from Planar Sections
The correct planes are observed more frequently
than the incorrect planes
3.0
2.0
1.0
Subtract background
68
Illustration of Stereology Approach
Randomly sample two types of planes. Movie
shows evolution of the distribution as
observations are accumulated.
69
Summary
• The Coincident Site Lattice is a useful concept
for identifying boundaries with low misfit (thus,
low energy).
• In general, five parameters needed to describe
crystallographic grain boundary character (the
macroscopic degrees of freedom).
• Statistical stereology offers some methods for
avoiding serial sectioning.