Grain Boundary and Interface lectures and it's notes

1
MT 60006
Grain Boundaries and Interfaces
3-0-0

2
Text Books
 Louisette Priester, Grain Boundaries, From Theory to
Engineering, Springer 2013
 G.A.Chadwick and D.A.Smith, Grain Boundary Structure and
Properties” Academic Press, London, 1976
 Introduction to Texture Analysis, O. Engler and V. Randle, CRC
Press, Taylor & Francis Group, NW, 2010
Reference Books
 V. Randle, The role of coincidence site lattice in grain
boundary engineering, The university press, Cambridge, 1996
 V. Randle, The measurements of grain boundary geometry,
Institute of Physics publishing, London, 1993

3
What is Grain Boundary
Interface between two grains,
or crystallites
 Essentially it’s a defect
 Orientation difference exists
across the boundaries

4
Orientation difference
O. Engler, V. Randle, Introduction to Texture Analysis, CRC Press, Second Edition, p. 40

Grain boundary is one of the important constituent of the
microstructure and therefore it is not surprising that physical,
chemical, and mechanical property of materials are controlled
by the number, character and structure of the boundaries.
Significant improvement of material properties as a
ramification of optimum grain boundary structure has opened
up new vistas in the realm of materials design for both in
structural as well as functional application.
This course would aim to cover the aspects of structure and
thermodynamics of grain boundary, strategy to measure grain
boundary character, approach and methodology to optimize
grain boundary character distribution in order to enhance
materials performances.
Importance of grain boundaries and interfaces

Detailed Syllabus and lecture-wise breakup
 Description, structure and thermodynamics of grain boundary
and interface - Degrees of freedom, low and high angle
boundaries – dislocation model – tilt and twist boundaries –
stacking fault and twin boundaries (4 to 5 lectures)
 Interphase boundaries – coherent, semi-coherent and
incoherent interphase – Antiphase boundaries (2 lectures)
 Description of orientation – Ideal orientation – Euler rotations
– Rodrigues vector and Rodrigues space (3 lectures)
 Interface networks, dihedral angles - Interfacial energy and
its anisotropy – Determination of interfacial energies (3 lectures)
 Introduction to ‘coincidence site lattice (CSL)’ theory – concept of
‘special boundary’ (3 lectures)
 Grain boundary character distribution (or Interface character
distributions in the case of multi-phase materials) –Interface
texture – misorientation (3-parameter) vs. boundary normal (5-
parameter) (2 lectures)

Detailed Syllabus and lecture-wise breakup....continued
Strategy to measure ‘five parameter’ grain boundary character
distribution – Five parameter stereological analysis – serial
sectioning and 3D EBSD – pseudo 3D EBSD (5 to 6 lectures)
 Role of interfacial phenomena in deformation and failure of
materials (viz, creep, grain boundary sliding, grain boundary
migration, grain boundary embrittlement etc) (5 to 6 lectures)
 Interfacial phenomena in thin films and composite materials, bulk
magnetic materials, solar cells (4 to 5 lectures)
 Introduction to grain boundary engineering (GBE) – Mechanisms of
GBE - Possible routes for GBE - Applications of GBE to improve
material properties viz corrosion, segregation, fracture etc (5 to 6
lectures)

8
Hall-Petch relation
i = Friction stress
k = Locking parameter
George E Dieter, Mechanical Metallurgy, McGraw Hill book company, SI Metric Edition, p. 189

9
Dislocation pile-up at grain boundaries
Number of dislocation piled-up
When the source is located at center
George E Dieter, Mechanical Metallurgy, McGraw Hill book company, SI Metric Edition, pp. 181-182

10
Hall-Petch relation…..continues
Critical stress to slip past the barrier
Rearranging
George E Dieter, Mechanical Metallurgy, McGraw Hill book company, SI Metric Edition, pp. 189-190

11
Is Twin a grain boundary?
Classical picture of twinning

12
Difference between Crystal & Lattice
Slide Courtesy: Rajesh Prasad, Department of Applied Mechanics, IITD
(https://www.slideshare.net/MukhlisAdam/basic-crystallography)

13
Difference between Crystal & Lattice

14
Unit Cell

15
Unit Cell

16
Crystal system and Bravais Lattice
O. Engler, V. Randle, Introduction to Texture Analysis, CRC Press, Second Edition, pp. 17

17
Cubic Bravais Lattice

18
Why Base Centered Cubic is not in the Bravais list?
Both posses similar symmetry…..but simple
tetragonal is primitive

19
Why can not FCC be considered as BCT?
FCC possesses higher symmetry
Slide courtesy: Rajesh Prasad, Department of Applied Mechanics, IITD

20

21
What is symmetry?
If any object is brought into self-
coincidence after any particular
operation, it said to possess
symmetry with respect to that
operation

22
Rotational symmetry
(http:// fab.cba.mit.edu/

23
Rotation Axis

25
Symmetry of Crystal Systems

26
Symmetry in tetragonal and cubic system

27

28
George E Dieter, Mechanical Metallurgy, McGraw Hill book company, SI Metric edition, p. 106

29
For cubic crystal
George E Dieter, Mechanical Metallurgy, McGraw Hill book company, SI Metric edition, p. 106

30
Description of Orientation
Orientation Matrix
Ideal Orientation (Miller or Miller-Bravais Indices)
Euler Angles
Angle/Axis Rotation
Rodrigues Vector
The main mathematical parameters that are used
to describe an orientation are as follows:

31
Ideal Orientation (Miller or Miller-Bravais Indices)
Schematic illustration of the relationship between the crystal and
specimen axes for the (110)[001] Goss orientation, that is, the normal
to (110) is parallel to the specimen ND, or Z axis and [001] is parallel to
the specimen RD, or X axis.

32
Preferred orientation (texture)
Na S. M. & Flatau A. B. , Surface-energy-induced selective growth of (001) grains in magnetostrictive ternary
FeGa-based alloys, Smart Materials and Structures, 21(5)-2012.

33
Acknowledgement: Slide taken from CMU

34
Description of Orientation by Orientation Matrix
Relationship between the specimen coordinate system XYZ (or RD,
TD, ND for a rolled product) and the crystal coordinate system
100,010,001 where the (cubic) unit cell of one crystal in the specimen
is depicted

35
Description of Orientation by Orientation Matrix
g = Rotation or Orientation Matrix
CC and CS are the crystal and specimen
coordinate systems, respectively.

36
Orientation Matrix – some features!
A typical orientation matrix (g)
 Orientation matrix is ortho-normal and the inverse of the
matrix is equal to its transpose
 It contains non-independent elements. cross product of
any two rows (or columns) gives the third and for any
row or column the sum of the squares of the three
elements is equal to unity

37
Can we calculate (hkl)[uvw]
from the g matrix??
YES……..see the next slides

38
Relation between Orientation Matrix
and Miller Indices
Rotation matrix g and Miller indices (hkl)[uvw] are
related through
Example
In practice, the direction cosines from the orientation matrix are “idealized”
to the nearest low-index Miller indices.

40
Calculate the (hkl)[uvw] from the following g matrix
Answer: (1 2 3)[6 3 -4]
O. Engler, V. Randle, Introduction to Texture Analysis, CRC Press, Second Edition, pp. 24 & 28

41
 The Euler angles refer to three rotations that, when
performed in the correct sequence, transform the
specimen coordinate system onto the crystal
coordinate system — in other words, specify the
orientation g.
 There are several different conventions for expressing
the Euler angles. The most commonly used are those
formulated by Bunge.
The Euler Angles

42
1. φ1 about the normal direction ND, transforming the
transverse direction TD into TD′ and the rolling direction
RD into RD′
2. Φ about the axis RD′ (in its new orientation)
3. φ2 about ND″ (in its new orientation)
The Euler Angles (Bunge definition)

43
The Euler Angles (Bunge definition)
https://www.youtube.com/watch?v=tmtGEHTBSdQ
You may watch the video through the following link:

44
The Angle/Axis of Rotation
 Euler angles showed how an orientation can be described
by the concept of three rotations that transform the
coordinate system of the crystal onto the specimen.
 The same final transformation can be achieved if the
crystal coordinate system is rotated through a single
angle, provided that the rotation is performed about a
specific axis.
 This angle and axis are known as the angle of rotation
and axis of rotation, or more briefly the angle/axis pair.

45
The Angle/Axis of Rotation
Diagram showing the angle/axis of rotation between two cubes

46
The Angle/Axis of Rotation from g matrix
Direction of the rotation axis [uvw] is:
(g32 – g23), (g13 – g31), (g21 – g12)
Rotation angle:
The angle/axis of rotation is extracted from the
orientation matrix (g) as follows:

47
Angle/Axis Description of Misorientation
A misorientation is calculated from the orientations of
grains 1 and 2 by
where M12 is the matrix that embodies the misorientation
between g2 and g1, where g1 is arbitrarily chosen to be the
reference orientation.

48
Crystallographically Related Solutions
 There are 24 solutions for a orientation matrix (as well as
misorientation matrix) of a material having cubic symmetry.
 Symmetry operations —
 Two rotations of 120° about each of the four 〈111〉,
 Three rotations of 90° about each of the three 〈100〉,
 One rotation of 180° about each of the six 〈110〉,
 Plus the identity matrix.
We have stated that the crystal coordinate system and
specimen coordinate system are related by the orientation
(rotation) matrix.
 However, specification of both these coordinate systems
is not usually unique, and a number of solutions can exist
depending on the symmetry of both the crystal and the
specimen.

49
Twenty four equivalent description for (123)[63-4]

50
Twenty four equivalent description for (123)[63-4]….continued

51
Crystallographically Related Solutions
 The crystallographically related solutions are generated
by pre-multiplying the misorientation matrix M by a
symmetry operator Ti:
M′ = TiM
 The 24 matrices Ti for cubic
symmetry are:

52
Distribution of misorientation angles for a randomly texture polycrystal in
cubic system
Mackenzie distribution
Out of 24 possible
misorientation angles
in cubic system, the
lowest misorientation
angle is often termed
as ‘disorientation’
angle
Mackenzie JK. Second paper on statistics associated with the random disorientation of cubes. Biometrika
1958;45:229

53
10 20 30 40 50 60
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Number
Fraction
Misorientation Angle (Degree)
Data
In typical deformed specimens
Distribution of misorientation angles for a deformed polycrystal in cubic
system
Indicates low angle boundaries

54
10 20 30 40 50 60
0.0
0.1
0.2
0.3
0.4
0.5
Number
Fraction
Misorientation Angle (Degree)
Data
In a typical low SFE annealed specimens
Distribution of misorientation angles for an annealed polycrystal in cubic
system
Indicates twin

55
Visualization of Grain Boundary in 3D

56
Variables that define a grain boundary. xA; yA; zA and xB, yB,
zB are the axes of the coordinates parallel to crystallographic
directions in grains A and B, respectively. O is the rotation
axis and is the rotation (misorientation) angle necessary to
transfer both grains to an identical position. n determines the
orientation of the grain boundary plane
Five Degrees of Freedom of Grain Boundary
Pavel Lejcek, Grain Boundary Segregation in Metals, Springer, First edition. p. 6

57
 The relationship between the rotation axis (o) and the grain
boundary plane normal (n) leads to definition of the tilt
grain boundaries and the twist grain boundaries
 For tilt boundaries: the rotation axis (o) is perpendicular to
the grain boundary plane normal (n)
For twist boundaries: the rotation axis (o) is parallel to the
grain boundary plane normal (n)
Tilt and Twist grain boundaries

58
Twist grain boundaries
No twist Twist ‘’
V. Randle The Measurement of Grain Boundary Geometry, CRC press, First edition, p. 10

59
GB plane
Twist grain boundaries......GB plane
https://en.wikipedia.org/wiki/Grain_boundary

60
Symmetrical tilt grain boundaries
When the boundary plane represents the plane of the mirror
symmetry of the crystal lattices of two grains, it is described
by the same Miller indices from the point of view of both
adjoining grains. This boundary is called symmetrical tilt
boundaries.

61
Symmetrical tilt grain boundaries.....GB Plane
GB plane
https://en.wikipedia.org/wiki/Grain_boundary

62
When the boundary plane does not represents the plane of
the mirror symmetry of the crystal lattices of two grains, the
boundary is called asymmetrical tilt boundaries.
Asymmetrical tilt grain boundaries
Symmetry plane
Misorientation axis
https://slideplayer.com/slide/17071731/

63
More representation of Asymmetrical and
Symmetrical tilt grain boundaries

64
The categorisation of the grain boundaries can be represented
by so-called interface-plane scheme based on relationship of
the Miller indices of individual contacting planes 1 and 2 in a
bicrystal and the twist angle ‘' of both planes
Tilt and Twist boundary categorisation

65
Tilt and Twist component of a mixed grain boundary
Question
Show that the misorientation θ/<UVW> of a mixed
grain boundary can be decomposed into
combination of tilt and twist component
Answer
See the next slide

The total misorientation /UVW can be decomposed into two sequential
operations: a tilt rotation followed by a twist rotation. The tilt angle component
of the total misorientation is about an axis (nT) (i.e. perpendicular to both N1 and
N2) and subsequent twist operation is about ntwist (parallel to N1 or N2) (Fig. a). A
tilt rotation of  about an axis nT, aligns the two normals. nT and  are given by:
nT = (N1 × N2)/│( N1 × N2)│
sin = │( N1 × N2)│
Fig. b shows the positions of the coordinate axes of both grains reoriented after
the tilt so that N1 and N2 are now parallel. Finally in Fig. c a twist rotation of  is
performed about ntwist (parallel to N1 or N2). Hence the total misorientation may
be written as:
M(, UVW) = M(ntwist, ) M(nT, )

67
 Based on misorientation angle ()
 Low angle GB (≤15°)
 High angle GB (>15°)
 Coincidence Site Lattice (CSL) boundary
Grain Boundary

68
Low Angle GBs

69
Primary intrinsic dislocations in Low Angle Boundaries
 A low angle boundary or sub-boundary can be represented
by an array of dislocations
 The simplest such boundary is the symmetrical tilt boundary
 The boundary consists of a wall of parallel edge dislocation
aligned perpendicular to the slip plane.
symmetrical tilt boundary
If the spacing of the
dislocations of Burgers
vector b in the boundary is
d, then the crystals on
either side of the boundary
are misoriented by a small
angle () where
  b/d
F.J. Humphreys et al., Recrystallization and Related Annealing Phenomena, Elsevier, 3rd edition p 114

70
 CSL is basically the sites at which the lattice of the two
crystals forming a boundary would coincide if they are
extended into one another
 ‘’ is reciprocal density of coinciding sites
CSL Boundary
5 Coincidence Site Lattice

 Lincoln - seven characters
 Became president in 1860
 Assassinated on Friday
 Accompanied by wife
 Inside a theater named Ford
 Next president was Andrew
Johnson
 He was born in 1808
 Lincoln was assassinated by
John Wilkes Booth - 15
characters
 John Wilkes Booth was born
in 1839
72
 Kennedy - seven characters
 Became president in 1960
 Assassinated on Friday
 Accompanied by wife
 Inside a car made by Ford
 Next president was Lyndon
Johnson
 He was born on 1908
 Kennedy was assassinated
by Lee Harvey Oswald - 15
characters
 Lee Harvey Oswald was
born in 1939
Coincidence !!

73
 CSL is basically the sites at which the lattice of the two
crystals forming a boundary would coincide if they are
extended into one another
 ‘’ is reciprocal density of coinciding sites
CSL Boundary
5 Coincidence Site Lattice

74
CSL Boundary
 In practice, only CSLs having a relatively short periodicity,
i.e. low  (up to  29), are of interest.
Misorientation angle/axis for CSLs with -values upto 35

75
Ranganathan Law
 A generating function can be used to obtain values of  and
the accompanying values of misorientation angle () and the
misorientation axis (UVW).
 If UVW is chosen, then  and  are given by:
θ = (2 tan-1[(y/x)(N1/2)]
 = x2 + y2 N
N = U2 + V2 + W2
S. Ranganathan, Acta Crystallographica, 21 (1966) 197

76
θ = (2 tan-1[(y/x)(N1/2)]
 = x2 + y2 N
N = U2 + V2 + W2
Ranganathan’s Generating function for UVW = 123

77
Calculate it now!!!
UVW 
100 13a
111 13b
111 3
110 3
110 9
110 27a
θ = (2 tan-1[(y/x)(N1/2)]
 = x2 + y2 N
N = U2 + V2 + W2
x,y 
5,1 22.6
7,1 27.8
3,1 60
2,1 70.5
4,1 38.9
5,1 31.6
Note: You should, in general, try with higher value of x and lower of y in
order to get the disorientation value.

78
CSL rule at Triple Junction
There are geometrical rules governing the relationship between
three CSLs which meet at a triple junction
 They share a common misorientation axis
 The sum of two of the misorientation angles gives the third
The product or quotient of two of the -values gives the third
A x B = (A x B) …………(i)
A x B = (A / B) …………(ii)
The second relationship applies only if A/B is an integer
and A>B

79
CSL rule at Triple Junction….some example
60/111 – 21.79/111 = 38.21/111
i.e. 3 - 21a = 7
70.53/110 + 70.53/110 = 141.06/110
i.e. 3 + 3 = 9

80
A 3 boundary interacted with a 15 boundary to form a triple junction consisting
of three CSL boundaries. If they share a common <210> misorientation axis, then
what will be the possible third CSL boundary in the junction. Calculate its
misorientation angle applying Ranganathan relationship. You need to show all the
steps in your calculation.
Exercise….do it now!
131.81/210 + 48.19/210 = 180/210
i.e. 3 + 15 = 5
131.81/210 - 48.19/210 = 83.62/210
i.e. 3 - 15 = 45b
Note: You should, in general, try to calculate all the possible misorientation
values of the sigma for a given axis to satisfy this relationship.

81
A 5 boundary interacted with another 5 boundary to form a triple junction
consisting of three CSL boundaries. If they share a common <100> misorientation
axis, then what will be the possible third CSL boundary in the junction. Calculate
its misorientation angle applying Ranganathan relationship. You need to show all
the steps in your calculation.
36.9/100 + 36.9/100 = 73.8/100
i.e. 5 + 5 = 25

82
A 3 boundary interacted with another 9 boundary to form a triple junction
consisting of three CSL boundaries. If they share a common <110> misorientation
axis, then what will be the possible third CSL boundary in the junction. Calculate
its misorientation angle applying Ranganathan relationship. You need to show all
the steps in your calculation.

83
Advantage and limitation of CSL concepts
Advantage
 Simple Geometrical concept – Easy to understand & implement
 Quantitative estimation of CSL boundary fraction in poly-
crystalline materials can easily be made
 Linkage between CSL fraction and materials property can be
made
Limitations
 Purely geometrical – does not consider inter-atomic bond
strength
 CSL model is only misorientation based (consider 3 DOFs) – it
does not consider the boundary plane

Grain Boundary and Interface lectures and it's notes

Recommended

Recommended

More Related Content

Similar to Grain Boundary and Interface lectures and it's notes

Similar to Grain Boundary and Interface lectures and it's notes (20)

Recently uploaded

Recently uploaded (20)

Grain Boundary and Interface lectures and it's notes