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Crystallographic Textures – Basics

Dr. V. Subramanya Sarma

Department of Metallurgical and Materials Engieneering

Indian Institute of Technology Madras

E-mail: [email protected]

OUTLINE

• Introduction

• Why texture is important

• Classification of textures

• How textures develop

• Representation of textures

• Measurement of textures

INTRODUCTION

• Majority of engineering materials are polycrystalline

• In latin, textor means weaver• In materials science, texture way in which a material is woven

• Each grain is a single crystal whose orientation differs from that of the neighbour

• Many possibilities of arranging the individual crystallites

A fully textured sheet (Strong texture)

A texture-less sheet (Random)

INTRODUCTION

Random Uni-axial or fibre Bi-axial

Texture influences the following properties:

• Elastic modulus

• Yield strength

• Tensile ductility and strength

• Formability

• Fatigue strength

• Fracture toughness

• Stress corrosion cracking

• Electric and magnetic properties

..........and many other properties.

Why textures ?

Why textures ?

• Properties depend on the texturepresent in the material

• Tailoring texture to achieve desiredproperties

Textures can be classified as:

• Macrotexture

• Microtexture

• Mesotexture

CLASSIFICATION

Macrotexture (Global / Bulk texture from volume)

Can be correlated to the average properties of the material

Determined by X-Ray, Neutron Diffraction

Microtexture (spatially resolved texture)

Determined by SEM/EBSD, TEM/SAD

Mesotexture (relates to misorientaion / axis of boundaries)

CLASSIFICATION

Grains marked with * have same orientation a) In a cluster b) Located randomly c) Have a different size compared to rest of the grainsd) Are located in as special region i.e., surface

(i) Crystallisation / solidification (from a non-crystalline/liquid state)

(ii) Plastic deformation (by glide or slip and twinning)

(iii) Annealing (recrystallisation / grain growth )

(iv) Phase transformation (due to orientation relationship)

(v) Thin film growth (substrate orientation and strain energy)

How textures develop


Deformation textures

• Constraint imposed during deformation causes lattice rotation

• Under tension increases and decreases

How textures developDeformation textures

• Under compression decreases and increases

• Rotation of the planes depends on stress state

How textures developAnnealing textures

• Stored energy driving force for recrystallisation• Minimisation of interfacial energy driving force for grain growth


Annealing textures

• Grain boundary mobility related to the boundary character and precipitates / solute

• Dissolution of precipitates could lead to abnormal grain growth

How textures developTransformation textures

• Texture in austenite will control the transformation texture in steel

Bain Orientation relationship

x y

• In order to specify an orientation, it is necessary to set up terms of reference, each of which is known as a coordinate system

• Specimen coordinate system: Coordinate system chosen as the geometry of the sample

• Crystal coordinate system: Coordinate system based on crystal orientation. In general [100], [010], [001] are adopted

Textures representation

There are two coordinate systems:•Sample (specimen) coordinate system•Crystal coordinate system


Main mathematical parameters that are used to describe an orientation are:

• Orientation matrix • Ideal orientation (Miller or Miller–Bravais indices)• Euler angles • Angle/axis of rotation • Rodrigues vector

All these descriptors are employed to process and represent different aspects of macrotexture and microtexture measurements

• Orientation is defined as 'the position of the crystal coordinate system with respect to the specimen coordinate system',

• where CC and CS are the crystal and specimen coordinate systems respectively and g is the orientation matrix

• The fundamental means for expressing g is the rotation or orientation matrix

Textures representationOrientation Matrix

• The first row of the matrix is given by the cosines of the angles between the first crystal axis, [l00], and each of the three specimen axes, X, Y, Z, in turn

Textures representationIdeal Orientation

A practical way to denote an orientation is via the Miller indices written as (hkl)[uvw] or {hkl}⟨uvw⟩

Thus, rotation matrix g and Miller indices (hkl)[uvw] are related through

In practice, the direction cosines from the orientation matrix are “idealized” to the nearest low-index Miller indices, for e.g. eg:

Textures representationStereographic projection

Stereographic projection is a graphical technique for representing the angular relationships between planes and directions in crystals in 2D

– Can be used to calculate angles between planes etc.– Is used in the representation of orientation of crystals

• We can represent the orientation of a plane using the normal to that plane

• If we inscribe a sphere around the crystal of interest, the point(s) where the normal(s) intersect the sphere are the poles of the planes

{100} poles of a cubic crystal

The projection of a plane (trace) passing through the origin of the crystal onto the surface of the sphere is a great circle

• The projection of a plane that does not pass through the origin is a small circle

• We can in principle measure the angle between two plane normals on the surface of the sphere to find the angle between two planes

– We make this measurement along a great circle (MLK in figure)


Great circles for the two marked planes


Making measurements on the surface of a sphere is tricky• Project everything from the spherical surface onto a plane– Pick a diameter of the sphere, put plane perpendicular to diameter and in contact with one end (or through the middle of the sphere), project from other end of diameter through entity to be projected onto the plane

• As drawn, entities in hemisphere near B will end up outside the basic circle. Points on hemisphere including A will end up inside.

– To avoid this problem, change projection point to the other end of diameter and distinguish points in the two hemispheres by marking them with different symbols (usually open versus filled in)


• Problems involving the stereographic projection are often handled using a Wulff net

• Imagine a globe with lines of latitude and longitude marked on the surface.

• Orient the globe so that the NS axis is parallel to the projection plane and project all the lines onto the plane

• The longitude lines end up as great circles in the projection and the latitude lines as small circles

• The lines in the projection can be used to read off angular coordinates

• Just like using latitude and longitude to specify geographical location

Wulff net


http://www.jcrystal.com/products/winwulff/

• Projection from 3D to 2D

Stereographic projection



A standard projection shows the angular relationships

between different poles for a given crystal orientation

– Useful for identifying crystal orientations

Standard projection

Note all reflections on a common great circle belong to the same zone. The zone axis lies at 90° to the zone

Standard projections for cubic crystals


Standard projections for hexagonal crystal for a given c/a

Textures representationPole Figure


{001} <100>

Plot (100) (111) (110) pole figures


{100} <011>



{111} <01-1>


111

-111

-1-11

1-11



u + v + w = 3 cos 118

u - v + w = 3 cos 90

-u - v + w = 3 cos 19.5

h + k + l = 3 cos 28

h - k + l = 3 cos 68

-h - k + l = 3 cos 83

RD [uvw] = [-0.816, -0.407, 0.440] ~ [- 2-11]

ND (hkl) = (0.219, 0.440, 0.870) ~ (124)

Textures representationConsidering the crystal symmetry, there are 24 different ways in which a crystal with cubic symmetry can be arranged

These matrices describe the symmetry operations—

2 rotations of 120° about each of the 4 -⟨111⟩ - 8

3 rotations of 90° about each of the 3 ⟨100⟩ - 9

1 rotation of 180° about each of the 6 ⟨110⟩ - 6

Plus the identity matrix - 1

Textures representationEuler angles / Euler space

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:

1. φ1 about the normal direction ND, transforming the transversedirection 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)


Diagram showing how rotation through the Euler angles φ1, Φ, φ2, in order 1, 2, 3 as shown describes the rotation between the specimen and crystal axes

Although an orientation can be uniquely defined by a single point in Euler space, 3D graphs are hard to interpret Therefore ODF is a 2D representation of Euler Space

Textures representationEuler Space

Textures representationEuler Space

Textures representationAngle-Axis representation

Consider the orientation of crystal lattices, X1Y1Z1 is replaced by the specimen coordinate system as a reference, for e.g. RD, TD, ND, and X2Y2Z2 is replaced by the crystal axes 100,010,001

Three independent variables that embody an orientation comprise one variable for therotation angle θ and two variables for the rotation axis r, since r is normalized

The transformation from the angle/axis pair θ/r to the orientation matrix g is given by


The angle/axis of rotation is extracted from the orientation matrix as follows:


If θ = 0° or 180°, sinθ becomes 0. In that case, r can be derived from

Angle/Axis Description of Misorientation

Instead of choosing the specimen axes for the reference orientation, it could be the axes (orientation) ofanother grain in the specimen, usually the neighboring grain.

The orientation of grain 2—a grain having crystal axes parallelto X2Y2Z2—is then expressed relative to the orientation of grain 1, which is similarly a grain having crystal axesparallel to X1Y1Z1

The essential difference between an angle/axis pair description of the orientation of a single grain and anangle/axis pair description of the misorientation between the crystal lattices of two grains is simply thechoice of reference axes

Rodrigues Vector and Rodrigues Space

The Rodrigues vector R combines the angle and axis of rotation into one mathematical entity

A population of R vectors, each of which represents anorientation or misorientation, can be represented in a 3-D space known as Rodrigues–Frank (RF) space or Rodrigues space.


The most powerful aspect of Rodrigues parameterization for the representation of orientations and

misorientations is that only rectilinear, that is, straight-line geometry is involved, which renders the space

easy to handle and visualize.


• The axis of rotation gives the direction of the R vector. Hence rotations about the same axis of rotationlie on a straight line that passes through the origin.

• The angle of rotation gives the length of the R vector. Hence small-angle boundaries cluster close tothe origin. These first two points are especially relevant to the analysis of misorientations

• Orientations that include a common direction, for example, a fiber texture, lie on a straight line that ingeneral does not pass through the origin.

• The edges of zones in Rodrigues space are straight lines, and the faces of zones are planar.