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Grain Boundary character 5-parameter

descriptions

- Advanced Characterization Microstructural

Analysis - 27-750, A.D. Rollett, Spring 2003

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.

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).

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)

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.

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

Disorientation based boundary plane based

2nd plane normal requires double SST

1st plane normal requires single SST

f 0

f 2p

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

misorientation matrix, ?g. Additionally, 4x4

symmetry matrices can be constructed from the

conventional 3x3 matrices.

The Number of Grain Boundary Types

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.

Tilt-twist character

- If cos-1(b)0, boundary is pure twist if

cos-1(b)90, boundary is pure tilt.

n

b

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.

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.

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

GB planes, disorientation

- Calculation of the disorientation with particular

choices of symmetry operators affects the

location of the rotation axis.

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.

(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.

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.

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

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

lt111gt Fiber RF-space

Misorientations lie on R1R2R3 line

Boundary planes lie on 111 zoneand are pure tilt

lt111gt boundaries

Misorientation axis

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

Definitions

(hkl)1 (hkl)2

Twist angle

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.

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

graduate

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.

graduate

Equivalent Descriptions of a 5-parameter

Boundaryswitching symmetries

graduate

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).

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

graduate

Unit triangles for plane normals

1st triangle

2nd triangle

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.

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)

Examples planes inside the SST

Example dissimilar planes inside the SST

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.

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.

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.

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.

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..

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)

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

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

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.

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

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

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

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.

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.

Symmetric Tilt Boundaries lt110gt

Symmetric tilt boundaries arebased on the

concept of a tiltaxis lying in the plane of

theboundary and with boundaryplanes

symmetrically disposedabout a low index

(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

Symmetric Tilt Boundaries lt100gt

Symmetric tilt boundaries arebased on the

concept of a tiltaxis lying in the plane of

theboundary and with boundaryplanes

symmetrically disposedabout a low index

(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

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.

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)

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

Example of Annealed Al bulk spec.interface

plane description

Example of Annealed Al (bulk)disorientation

plane description

lt110gt axes

low angle boundaries

Example of Annealed Al (foil)disorientation

plane description

Strong 001lt100gtcube texture presentin the

foil.

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.

Boundary Tangents

- Measure the (local) boundary tangent the normal

must lie in its zone.

gB

ns(A)

B

ts(A)

A

x1

gA

x2

G.B. tangent disorientation

- Select the pair of symmetry operators that

identifies the disorientation, i.e. minimum angle

and the axis in the SST.

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

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.

Spherical angles

chi declination phi azimuth

f

c

Tangent Zone, parameterized

- The scalar product of the (unit) vectors

representing the tangent and its zone must be

zero

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.

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.

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

Recovering the Distribution from Planar Sections

The correct planes are observed more frequently

than the incorrect planes

3.0

add 2

add 1

2.0

1.0

Subtract background

add 3

Illustration of Stereology Approach

Randomly sample two types of planes. Movie

shows evolution of the distribution as

observations are accumulated.

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.