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Grain Boundaries

- 27-765, Advanced Characterization
- Spring 2001
- A.D. Rollett

Objectives

- Introduce the grain boundary as a defect of

particular interest. - Outline the basic properties of a grain boundary

- energy mobility. - Describe the crystallography of grain boundaries,

especially the Rodrigues vector. - Special boundaries such as CSL structures.

References

- A. Sutton and R. Balluffi, Interfaces in

Crystalline Materials, Oxford, 1996. - V. Randle O. Engler (2000). Texture Analysis

Macrotexture, Microtexture Orientation Mapping.

Amsterdam, Holland, Gordon Breach. - Frank, F. (1988). Orientation mapping.

Metallurgical Transactions 19A 403-408.

What is a Grain Boundary?

- Boundary between two grains.
- Regular atomic packing disrupted at the boundary.
- In most crystalline solids, g.b. is very thin

(one/two atoms). - Disorder (broken bonds) unavoidable for

geometrical reasons therefore large excess free

energy.

Thermodynamics

- Large excess free energy means that it always

costs energy to create a boundary. - There is never an equilibrium population of g.b.s

even at high temperatures contrast with

vacancies. - Polycrystal always tends towards a single

crystal. - Commercial materials always have 2nd phase -

therefore coarsening prevented.

Grain Boundary Structure

- High angle boundaries can be thought of as two

crystallographic planes joined together (with or

w/o a twist of the lattices). - Low angle boundaries can be thought of as built

up of dislocations, especially for pure tilt

boundaries to be explained. - Transition in the range 10-15, the dislocation

structure changes to a high angle boundary

structure.

Pure Tilt Boundaries Dislocations

- Low angle boundaries can be made up of arrays of

parallel edge dislocations if the rotation

between the lattices is small and the rotation

axis lies in the boundary plane.

Differences in Orientation

- Preparation for the math of misorientations the

difference in orientation between two grains is a

rotation just as is the rotation that describes a

texture component. - Convention we use different methods to describe

g.b. misorientation than for texture (but we

could use Euler angles for everything, for

example).

Simple Boundary Types

- Tilt boundary is a rotation about an axis in the

boundary plane. - Twist boundary is a rotation about an axis

perpendicular to the plane.

Rotations at a Grain Boundary

z

gB

In terms of orientations rotate back from

position Ato the reference position.Then rotate

to position B.Compound (compose)the two

rotations to arriveat the net rotation

betweenthe two grains.

y

gA-1

referenceposition(001)100

x

Net rotation gBgA-1

Alternate Diagram

TJACB

gB

gBgA-1

gD

gC

gA

TJABC

Representations

- What is different from Texture Components?
- Miller indices not useful (except for axis).
- Euler angles can be used but untypical.
- Reference frame is usually the crystal lattice,

not the sample frame. - Application of symmetry is different (no sample

symmetry!)

Grain Boundaries vs. Texture

- Why use the crystal lattice as a frame? Grain

boundary structure is closely related to the

rotation axis. - The crystal symmetry applies to both sides of the

grain boundary however, only one set of 24

symmetry operators are needed to find the minimum

rotation angle.

Disorientation

- Thanks to the crystal symmetry, no two cubic

lattices can be different by more than 62.8. - Combining two orientations can lead to a rotation

angle as high as 180 applying crystal symmetry

operators modifies the required rotation angle. - Disorientation minimum rotation angle between

two lattices (and axis in the SST).

Grain Boundary Representation

- Axis-angle representation axis is the common

crystal axis (but could also describe the axis in

the sample frame) angle is the rotation angle. - 3x3 Rotation matrix, ?ggBgA-1.
- Rodrigues vector 3 component vector whose

direction is the axis direction and whose length

tan(angle/2).

Rotation Axis, Angle

gB

?ggBgA-1? gAgB-1

gD

gC

gA

Switching symmetryA to B is indistinguishable

from B to A

rotation axis, common to both crystals

Example Twin Boundary

lt111gt rotation axis, common to both crystals

q60

- Porter Easterling fig. 3.12/p123

Crystal vs Sample Frame

Components ofthe rotation axisare always

(1/v3,1/v3,1/v3) inthe crystal framein the

sample framethe componentsdepend on

theorientations ofthe grains.

z

gB

y

gA-1

q60

referenceposition(001)100

x

Rodrigues vectors

- Rodrigues vectors, as popularized by Frank

Frank, F. (1988). Orientation mapping.

Metallurgical Transactions 19A 403-408., hence

the term Rodrigues-Frank space for the set of

vectors. - Useful for representation of misorientations.
- Fibers based on a fixed axis are always straight

lines in RF space (unlike Euler space).

Rodrigues vector, contd.

- Many of the boundary types that correspond to a

high fraction of coincident lattice sites (i.e.

low sigma values in the CSL model) occur on the

edges of the Rodrigues space. - CSL boundaries have simple values, i.e.

components are reciprocals of integers e.g. twin

in fcc (1/3,1/3,1/3) ? 60 lt111gt. - Also useful for texture representation.

Rodrigues vector, contd.

- We write the axis-angle representation as (

,q) - From this, the Rodrigues vector is r

tan(q/2)

Conversions matrix?RF vector

- Conversion from rotation matrix, ?ggBgA-1

Conversion from Bunge Euler Angles

- tan(q/2) v(1/cos(F/2) cos(f1 f2)/22 1

- r1 tan(F/2) sin(f1 - f2)/2/cos(f1

f2)/2 - r2 tan(F/2) cos(f1 - f2)/2/cos(f1

f2)/2 - r3 tan(f1 f2)/2

P. Neumann (1991). Representation of

orientations of symmetrical objects by Rodrigues

vectors. Textures and Microstructures 14-18

53-58.

Conversion from Roe Euler Angles

- tan(q/2) v(1/cosQ/2 cos(Y F)/22 1
- r1 -tanQ/2 sin(Y - F)/2/cos(Y F)/2
- r2 tanQ/2 cos(Y - F)/2/cos(Y F)/2
- r3 tan(Y F)/2

Combining Rotations as RF vectors

- Two Rodrigues vectors combine to form a third,

rC, as follows,where rB follows after rA. rC

(rA, rB) rA rB - rA x rB/1 - rArB

vector product

scalar product

Combining Rotations as RF vectors component form

Range of Values of RF vector components

- Q. If we use Rodrigues vectors, what range of

values do we need? - A. Since we are working with a rotation axis

that is based on a crystal direction then it is

logical to confine the axis to the standard

stereographic triangle (SST).

Shape of RF Space

z, r3

y, r2

origin

distance (radius) from origin represents the

misorientation angle

xyz, r1r2r3

xy, r1r2

x, r1

Limits on RF vector components

- r1 corresponds to the component //100 r2

corresponds to the component //010 r3

corresponds to the component //001 - r1 gt r2 gt r3 gt 0
- 0 r1 (v2-1)
- r2 r1
- r3 r2
- r1 r2 r3 1

45 rotation about lt100gt

Alternate Notation (R1 R2 R3)

- R1 corresponds to the component //100 R2

corresponds to the component //010 R3

corresponds to the component //001 - R1 gt R2 gt R3 gt 0
- 0 R1 (v2-1)
- R2 R1
- R3 R2
- R1 R2 R3 1

Fundamental Zone

- By setting limits on all the components (and

confining the RF vector to the SST) we have

implicitly defined a Fundamental Zone. - The Fundamental Zone is simply the set of

orientations for which there is one unique

representation for any possible rotation. - Note the standard 90x90x90 region in Euler space

contains 3 copies of the FZ for

cubic-orthorhombic symmetry

Size, Shape of the Fundamental Zone

- We can use some basic information about crystal

symmetry to set limits on the size of the FZ. - Clearly we cannot rotate by more than 45 about a

lt100gt axis before we encounter equivalent

rotations by going in the opposite direction

this sets the limit of R1tan(22.5)v2-1. This

defines a plane perpendicular to the R1 axis.

Size, Shape of the Fundamental Zone

- Similarly, we cannot rotate by more than 60

about lt111gt, which sets a limit of (1/3,1/3,1/3)

along the lt111gt axis, or vR12R22R32tan(30)1

/v3. - Other symmetries set the limits shown above.
- FZ has the shape of a truncated pyramid.

RF-space

lt111gt

r1 r2 r3 1

lt110gt

lt111gt

lt100gt, r1

lt110gt

lt100gt, r1

Misorientation Distributions

- The concept of a Misorientation Distribution

(MODF) is analogous to an Orientation

Distribution (OD or ODF). - Probability distribution in the space used to

parameterize misorientation, e.g. 3 Euler angles

f(f1,F,f2), or 3 components of Rodrigues vector,

f(R1,R2,R3). - Probability of finding a given misorientation

(specified by all 3 parameters) is given by f.

Area Fractions

- Grain Boundaries are planar defects therefore we

should look for a distribution of area (or area

per unit volume, SV). - Fraction of area within a certain region of

misorientation space, ?W, is given by the MODF,

f, where W0 is the complete space

Normalization of MODF

- If boundaries are randomly distributed then MODF

has the same value everywhere, i.e. 1 (since a

normalization is required). - Normalize by integrating over the space of the 3

parameters (as for ODF). - If Euler angles used, same equation applies

(adjust constant for size of space)

Rodrigues vector normalization

- The volume element, or Haar measure, in Rodrigues

space is given by the following formula r

tan(q/2) - Can also write in terms of an azimuth and

declination angle

r vR12 R22 R32 tanq/2 c cos-1R3 z

tan-1R2/R1

Density of points in RF space

- The variation in the volume element with

magnitude of the RF vector (i.e.with

misorientation angle) is such that the density of

points (for a random distribution) increases

rapidly with distance from the origin. - Mackenzie, J. K. (1958). Second paper on

statistics associated with the random orientation

of cubes. Biometrica 45 229-240.

Mackenzie Distribution

- Frequency distribution with respect to

disorientation angle for randomly distributed

grain boundaries.

Density in the SST

- Density or in area

Experimental Example

- Note the bias to certain misorientation axes with

the SST, i.e. lt101gt and lt111gt.

Experimental Distributions by Angle

Random

Choices for MODF Plots

- Euler angles use subset of 90x90x90 region,

starting at F72. - Axis-angle plots, using SST (or 001-100-010

quadrant) and sections at constant angle. - Rodrigues vectors, using either square sections,

or triangular sections through the fundamental

zone.

MODF for Annealed Copper

2 peaks 60lt111gt, and 38lt110gt

FZ in RF space the truncated pyramid

fundamental zone truncated pyramid

Sections through RF-space

- The R-F space is sectioned parallel to the

100-110 plane - Each triangular section has R3constant.
- Special CSL relationships on 100, 110, 111

base of pyramid

Symmetry Operations

- Note that the result of applying any available

operator is physically indistinguishable from the

starting configuration (not mathematically equal

to!). - Two crystal symmetry operators

Various Symmetry Combinations

- Fundamental zones in Rodrigues space (a) no

sample symmetry with cubic crystal symmetry (b)

orthorhombic sample symmetry (c) cubic-cubic

symmetry for disorientations. after Neumann,

1990

Symmetry planes in RF space

- The effect of any symmetry operator in Rodrigues

space is to insert a dividing plane in the space. - This arises from the geometrical properties of

the space (extra credit prove this property of

the Rodrigues-Frank vector).

Delimiting planes

- For the combination of O(222) for sample symmetry

and O(432) for crystal symmetry, the limits on

the Rodrigues parameters are given by the planes

that delimit the fundamental zone. These include

six octagonal facets orthogonal to the lt100gt

directions, at a distance of tan(p/8) (v2-1)

from the origin, and eight triangular facets

orthogonal to the lt111gt directions at a distance

of tan(p/6) (v3-1) from the origin.

Symmetry planes in RF space

4-fold axis on lt100gt

3-fold axis on lt111gt

Maximum rotation

- The vertices of the triangular facets have

coordinates (v2-1, v2-1, 3-2v2) (and their

permutations), which lie at a distance (23-16v2)

from the origin. This is equivalent to a

rotation angle of 62.7994, which represents the

greatest possible rotation angle, either for a

grain rotated from the reference configuration,

or between two grains.

Quaternions

- A close cousin to the Rodrigues vector is the

quaternion. - It is defined as a four component vector in

relation to the axis-angle representation as

follows, where uvw are the components of the

unit vector representing the rotation axis, and q

is the rotation angle.

Graduate material

Quaternion definition

- q q(q1,q2,q3,q4) q( u.sinq/2, v.sinq/2,

w.sinq/2, cosq/2) - Alternative puts cosine term in 1st positionq

( cosq/2, u.sinq/2, v.sinq/2, w.sinq/2).

Historical Note

- This set of components was obtained by Rodrigues

prior to Hamiltons invention of quaternions and

their algebra. Some authors refer to the

Euler-Rodrigues parameters for rotations in the

notation (l,L) where l is equivalent to q4 and L

is equivalent to the vector (q1,q2,q3).

Rotations represented by Quaternions

- The particular form of the quaternion that we are

interested in has a unit norm (vq12q22q32

q421) but quaternions in general may have

arbitrary length. - Thus for representing rotations, orientations and

misorientations, only quaternions of unit length

are considered.

Why Use Quaternions?

- Among many other attractive properties, they

offer the most efficient way known for performing

computations on combining rotations. This is

because of the small number of floating point

operations required to compute the product of two

rotations.

Conversions matrix?quaternion

Conversions quaternion ?matrix

- The conversion of a quaternion to a rotation

matrix is given byaij (q42-q12-q22-q32)dij

2qiqj 2q4Sk1,3eijkqk - eijk is the permutation tensor, dij the

Kronecker delta

Roe angles ? quaternion

- q1, q2, q3, q4 -sinQ/2 sin(Y - F)/2 ,

sinQ/2 cos(Y - F)/2, cosQ/2 sinY F)/2,

cosQ/2 cos(Y F)/2

Bunge angles ? quaternion

- q1, q2, q3, q4 sinF/2 cos(f1 - f2)/2 ,

sinF/2 sin(f1 - f2)/2, cosF/2 sinf1

f2)/2, cosF/2 cos(f1 f2)/2

Note the occurrence of sums and differences of

the 1st and 3rd Euler angles!

Combining quaternions

- The algebraic form for combination of quaternions

is as follows, where qB follows qA qC qA

qBqC1 qA1 qB4 qA4 qB1 - qA2 qB3 qA3

qB2qC2 qA2 qB4 qA4 qB2 - qA3 qB1 qA1

qB3qC3 qA3 qB4 qA4 qB3 - qA1 qB2 qA2

qB1qC4 qA4 qB4 - qA1 qB1 - qA2 qB2 - qA3 qB3

Positive vs Negative Rotations

- One curious feature of quaternions that is not

obvious from the definition is that they allow

positive and negative rotations to be

distinguished. This is more commonly described

in terms of requiring a rotation of 4p to

retrieve the same quaternion as you started out

with but for visualization, it is more helpful to

think in terms of a difference in the sign of

rotation.

Positive vs Negative Rotations

- Lets start with considering a rotation of q

about an arbitrary axis, r. From the point of

view of the result one obtains the same thing if

one rotates backwards by the complementary angle,

q-2p (also about r). Expressed in terms of

quaternions, however, the representation is

different! Setting ru,v,w again, - q(r,q) q(u.sinq/2,v.sinq/2,w.sinq/2,cosq/2)

Positive vs Negative Rotations

- q(r,q-2p) q(u.sin(q-2p)/2,v.sin(q-2p)/2, w.sin

(q-2p)/2,cos(q-2p)/2) q(-u.sinq/2,-v.sinq/2,

-w.sinq/2,-cosq/2) -q(r,q)

Positive vs Negative Rotations

- The result, then is that the quaternion

representing the negative rotation is the

negative of the original (positive) rotation.

This has some significance for treating dynamic

problems and rotation angular momentum, for

example, depends on the sense of rotation. For

static rotations, however, the positive and

negative quaternions are equivalent or, more to

the point, physically indistinguishable, q ? -q.

Quaternion acting on a vector

- The active rotation of a vector from X to x is

given byxi (q42-q12-q22-q32)Xi 2qiSjqjXj

2q4SjXjSkeijkqk - eijk is the permutation tensor, dij the

Kronecker delta

Computation combining rotations

- The number of operations required to form the

product of two rotations represented by

quaternions is 16 multiplies and 12 additions,

with no divisions or transcendental functions. - Matrix multiplication requires 3 multiplications

and 2 additions for each of nine components, for

a total of 27 multiplies and 18 additions. - Rodrigues vector, the product of two rotations

requires 3 additions, 6 multiplies 3 additions

(cross product), 3 multiplies 3 additions, and

one division, for a total of 10 multiplies and 9

additions. - The product of two rotations (or composing two

rotations) requires the least work with Rodrigues

vectors.

Symmetry operators

Symmetrically Equivalent Quaternions for Cubic

Symmetry

Negative of a Quaternion

- The negative (inverse) of a quaternion is given

by negating the fourth component, q-1

(q1,q2,q3,-q4) this relationship describes the

switching symmetry at grain boundaries.

Finding the Disorientation Angle with Quaternions

- The objective is to find the quaternion that

places the axis in a specified unit triangle

(e.g. 0ltultvltw) with the minimum rotation angle

(maximum fourth component). If one considers the

action of the diads on lt100gt, the result is

obtained that (q1,q2,q3,q4)? (q4,q3,-q2,-q1) ?

(-q3,q4,q1,-q2) ? (q2,-q1,q4,-q3) .

- This means that one can place the fourth

component in any other position in the

quaternion. Since the first three components

correspond to the rotation axis, uvw, we know

that we can interchange any of the components

q1,q2 and q3 and we can change the sign of any of

the components. These rules taken together allow

us to interchange the order and the sign of all

four components of the quaternion.

- If this is done so as to have q4gt q3gt q2gt q1gt0,

i.e. all four components positive and arranged in

increasing order, then the only three variants

that need be considered are as follows, because

we are seeking the minimum value of the rotation

angle (i.e. the maximum value of the fourth

component, q4)(q1,q2,q3,q4)(q1-q2, q1q2,

q3-q4, q3q4)/v2(q1-q2q3-q4, q1q2-q3-q4,

-q1q2q3-q4, q1q2q3q4)/2

- Therefore with operations involving only changes

of sign, a sort into ascending order, four

additions and a comparison, the disorientation

can be identified. The contrast is with the use

of matrices where each symmetrically equivalent

variant must be calculated through a matrix

multiplication and then the trace of the matrix

calculated each step requires an appreciable

number of floating point operations, as discussed

above