Grain Boundaries

1
Grain Boundaries

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

2
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.

3
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.

4
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.

5
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.

6
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.

7
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.

8
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).

9
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.

10
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
11
Alternate Diagram
TJACB
gB
gBgA-1
gD
gC
gA
TJABC
12
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!)

13
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.

14
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).

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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).

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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
17
Example Twin Boundary
lt111gt rotation axis, common to both crystals
q60

• Porter Easterling fig. 3.12/p123

18
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
19
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).

20
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.

21
Rodrigues vector, contd.

• We write the axis-angle representation as (
  ,q)
• From this, the Rodrigues vector is r
  tan(q/2)

22
Conversions matrix?RF vector

• Conversion from rotation matrix, ?ggBgA-1

23
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.
24
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

25
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
26
Combining Rotations as RF vectors component form
27
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).

28
Shape of RF Space
z, r3
y, r2
origin
distance (radius) from origin represents the
misorientation angle
xyz, r1r2r3
xy, r1r2
x, r1
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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
30
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

31
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

32
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.

33
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.

34
RF-space
lt111gt
r1 r2 r3 1
lt110gt
lt111gt
lt100gt, r1
lt110gt
lt100gt, r1
35
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.

36
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

37
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)

38
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
39
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.

40
Mackenzie Distribution

• Frequency distribution with respect to
  disorientation angle for randomly distributed
  grain boundaries.

41
Density in the SST

• Density or in area

42
Experimental Example

• Note the bias to certain misorientation axes with
  the SST, i.e. lt101gt and lt111gt.

43
Experimental Distributions by Angle
Random
44
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.

45
MODF for Annealed Copper
2 peaks 60lt111gt, and 38lt110gt
46
FZ in RF space the truncated pyramid
fundamental zone truncated pyramid
47
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
48
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

49
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

50
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).

51
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.

52
Symmetry planes in RF space
4-fold axis on lt100gt
3-fold axis on lt111gt
53
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.

54
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
55
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).

56
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).

57
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.

58
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.

59
Conversions matrix?quaternion
60
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

61
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

62
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!
63
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

64
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.

65
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)

66
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)

67
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.

68
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

69
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.

70
Symmetry operators
71
Symmetrically Equivalent Quaternions for Cubic
Symmetry
72
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.

73
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) .

74

• 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.

75

• 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

76

• 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