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Grain Boundaries

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Title: 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).

15
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).

16
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
29
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
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