Introduction to Crystallographic Texture - IIT Kanpur Gurao/HRXRED.pdf · Introduction to...

Introduction to Crystallographic Texture

Nilesh Prakash GuraoAssistant Professor

Materials Science and EngineeringIndian Institute of Technology Kanpur, Kanpur India

3/3/2016 HRXRED-2016 IIT Kanpur 2

D Raabe 2011

Microstructure

Cheong et al. Acta Mater. 55 (2007) 1757

Herbig et al Acta Mater. 59 (2011) 590

Structure at micron scale Distribution of phases in 3D Size, shape, orientation


10%

25%

Initial

5%

35%50%

N. P. Gurao, S. Suwas, Materials Letters 2013

Rolling of Nickel at IISc


0

1000

2000

3000

4000

5000

0 2 4 6 8 10

Displacement (mm)

Load

(N)

A

B

C

E

D

A

B C D

E

200 µmS. Sinha, J. A. Szpunar, NAP Kiran Kumar, N. P. Gurao, MSEA 2015

Twinning in stainless steel at USASK


Annealing 500 C 1 hour

Reload Tension 1000 N

Compression 990 N

Tension 500 N Tension 1000 NTension 750 N

Twinning in titanium at IITK

S. Sinha, N. P. Gurao, manuscript under preparation

Texture is derived from “textor” in Latin

The way things are woven

Fabrics, rocks and materials

Morphological texture

Crystallographic texture

Preferred orientation of crystallites

Definition

3/3/2016 6HRXRED-2016 IIT Kanpur

Kanjeevaram Saree Rock

Morphological texture


Crystallographic texture

Single orientation

Random orientation

Preferred orientation



Electron Back Scatter Diffraction

Utility

Texture in polycrystalline materials is ubiquitous

Goss oriented Silicon steel

Earing of aluminum cans

Substrate for semiconductor tapes

Fatigue properties of aluminum and titanium alloys

Quality of epitaxial films: YBa2Cu3O7-δ Pmmm space group c┴epitaxy

Recoverable strain in SMA


desirable cup showing earingwww.alumatter.org.uk

Gall et al. Acta Mater. 47 (1999) 1203.

Seabaugh et al. J. Intel. Mater. Struct. 15 (2004) 209.

PMN-PT templated with SrTiO3


Orientation and Texture

Orientation is relative

Need of reference frame

Orientation is relationship between the object under consideration and the reference

For materials• Sample constitutes the reference frame• Crystal constitutes the object

Texture is relationship between sample and crystal frame of reference



Representing texture

Texture Relative orientation

Crystal and Sample frame of reference

Both must be orthogonal

Ortho-normalization of unit cell if α or β or γ ≠ 90◦

Science of texture evolved for rolling

Three important directions

Rolling Direction (RD), Transverse Direction (TD), Normal Direction (ND)

Crystal directions [100], [010] and [001]

RD

ND

TD

100

010

001



Description of orientation

{hkl}<uvw>

Euler Angles

Axis angle pair Rodrigues-Frank vector

Quaternions

Simplest description of texture

Relationship between crystallographic planes and direction and sample planes and direction

(hkl) ┴ ND and <uvw> ║ RD

For HCP, plane normal and corresponding direction are not necessarily parallel

Ortho-normalization ensures that complete orientation is described without any ambiguity

Texture description



Cc = g Csg orientation matrix

Texture description: Mathematical

RD

ND

TD

100010

001

Euler angles

Rotate φ1 about ND so that RD is perpendicular to plane containing ND and [001]

RD RD’ and TD TD’ Rotate Φ about RD’ so that ND and [001] coincide

ND [001] and TD’ TD’’ Rotate φ2 about ND so that crystal and sample frame coincide

RD’ [010] and TD’’ [100]


Additional parameters

Axis-angle pair (Rodrigues-Frank vector)A unique axis can be found out about which if rotated by a unique angle, the sample and crystal frame of reference coincide

QuaternionsMathematically elegant description of encoding axis-angle informationRotation of θ about [uvw] axis is represented as

θθθ

21sin)(

21cos),,,(

)(21

3210 kwjviueqqqqqkwjviu

+++==++


rR ⋅

=

2tan θ


Representing texture

Pole Figure: distribution of crystallographic direction/plane w.r.t. sample frame of reference

Orientation Distribution function: distribution of orientation in terms of Euler angles

Inverse Pole figure: distribution of sample direction w.r.t. crystal frame of reference


Pole figures


Stereographic projection

RD

ND

TD

100

010001

Cube {100}<001> orientation

011


011 011 011

{100}<001> in different pole figures

RD

ND

TD

010

001100

100 pole figure 101 pole figure 111 pole figure



Type of projection

Angle true

Area true


Random texture

For a polycrystal

Single orientation Multiple spots

Deviation from exact orientation Scatter


Contour levels

Density of poles

Compared with that of random sample

Smoothened contours

Multiples of random distribution (mrd)


Inverse pole figure

FCC tension


FCC rolling

ND IPF RD IPF TD IPF

Easiest description of texture

Most commonly used for processes where only one direction is uniquely defined

Uniaxial deformation , thin film deposition or solidification

Multiple IPF needed to describe complicated textures like rolling.


Euler angles and Orientation space

Roe Bunge Williamsφ ϕ2+π/2 αθ Φ ρψ ϕ1-π/2 β=ψ +tan-1(tanαcosρ)


Every (hkl)[uvw] orientation has a unique (φ1, Φ, φ2) location in Euler space


Volume of Orientation space

∫ ∫ ∫= = =

=π

φ

π

φ

π

φ

φφφφ2

0 0

2

021

1 2

sin dddVODF


Volume element dV = dAdΦ 2 = sin Φ dΦ dΦ 1 dΦ 2

V = 8π2

Volume of Orientation space


sc gOOg =′

Crystal and Sample Symmetry

Cubic 24 rotation matrices (432) Hexagonal 12 rotation matrices (62)

Tension/Compression (Axial symmetry) Rolling (222)


Cubic-Orthotropic

a) b) c)

f)e)d)

111 pole figure of 90% rolled a) Ni b) Ni-10%Co c) Ni-20%Co d) Ni-30%Co e) Ni-40% Co and f) Ni-60%Co


Cubic-axial

Initialtexture

Simulatedtexture

Experimentaltexture

Compression direction inverse pole figure of different FCC materials


100 pole figure of FCC material showing a) 110 and b) 111 fibre

a) b)

φ1 = 0-2π

Φ = 0-πΦ2 = 0-2π

For rolling

Actually plotted volume

For axial symmetry only one section required as all φ2 sections are similar

124*248 22 ππ

==V

4

2π=V

Reduced volume of orientation space


Pole Figure to Orientation Distribution Function

Pole figures 2D information

Texture 3D Orientation Distribution Function (ODF)

Developed for cubic crystals with orthotropic symmetry Rolling

Extended to all crystal structures and all processes

An ODF describes the frequency of occurrence of particular crystal orientations in an imaginary three-dimensional orientation space defined by Euler angles


Two methods to determine ODF Spherical Harmonics Discrete Methods

Incomplete PF data is input

Spherical Harmonics C co-efficient (co-efficient of spherical harmonic functions) Even and odd ODF Problems with truncation and ghost error


Direct methods

WIMV method (Williams-1968, Imhof-1977, Mathies and Vinel-1982) The Vector method (Ruer and Baro-1977; Vadon and Heizmann-1991) The component method (Helming and Eschner-1990) The ADC (Arbitrary Defined Cells) method (Pawlik-1986)



Experimental ODF


Texture quantification

Maximum intensity in a pole figures/ ODF

strength of texture

Texture index quantitative parameter to compare similar texture

Normalized sum of squared density values Δgi = ODF cell volume

Applicable for similar texture

ii

ggfF ∆= ∑ 22

2 )]([8

1π


Better quantitative estimate by plotting fibres and volume fraction of texture components

Fibres are obtained by plotting f(g) as a function of Euler angle

Volume fraction of a component {hkl}<uvw> is obtained by determining fraction of orientations in the viscinity of given orientation


Volume fraction of different texture components in differently rolled

Copper

Beta fibre plot for Cu and Cu-Zn alloys


Evolution of texture during processing

Solidification only one direction important Fibre texture

Deformation two directions important {hkl}<uvw>Complete Euler space required for complex deformation processes like

Equal Channel Angular Pressing, cross rolling

Recrystallization two direction important {hkl}<uvw>Retain similar symmetry as corresponding deformation texture

Thin film deposition growth direction Fibre texture

Phase transformation two directions {hkl}<uvw>

Measuring texture

Macro-texture Bulk sample

Micro-texture Microstructure + Texture

X-ray and neutron diffraction

Electron diffraction

Synchrotron can offer both


Diffraction and Texture


X-ray diffraction is routinely used to study the crystalline structure of materials

Information about texture is embedded in this information

Even optical microscopy shows some orientation dependent contrast

Additional diffraction techniques like neutron, synchrotron and electron diffraction offer better statistics and spatial resolution respectively


Polarized Light Micrograph

Nano-cellulose Magnesium


Micro-texture from Electron Back Scatter Diffraction

0.0

0.1

0.2

0.3

0.4

10 20 30 40 50 60

Num

ber F

ract

ion

Misorientation Angle [degrees]

Misorientation Angle

0

10

20

30

40

50

60

70

0 10 20 30

Mis

orie

ntat

ion

[deg

rees

]

Distance [microns]

Misorientation Profile50% rolled austenite sample

Gurao and Suwas, unpublished work


Riot of colours in α+γ steels

Gurao and Suwas, unpublished work


Braggs Law

nλ = 2dsinθ

d d sinθ d sinθ

X-ray Diffraction for bulk texture


Name and formulaReference code: 00-001-1260

PDF index name: Nickel Empirical formula:NiChemical formula: Ni

Crystallographic parametersCrystal system: Cubic

Space group: Fm-3m Space group number: 225

a (Å): 3.5175 Alpha (°): 90.0000

Measured density (g/cm^3): 8.90 Volume of cell (10^6 pm^3): 43.52

Z: 4.00 Status, subfiles and quality

Status: Marked as deleted by ICDDSubfiles: Inorganic Quality: Blank (B)

ReferencesPrimary reference: Hanawalt et al., Anal. Chem., 10, 475, (1938)Optical data: Data on Chem. for Cer. Use, Natl. Res. Council Bull. 107

Unit cell: The Structure of Crystals, 1st Ed.

XRD Database


http://ww1.iucr.org/cww-top/crystal.index.html

Powder diffraction


XRD of bulk electrodeposited nano-Ni

Lattice parameter, phase diagramsTexture, Strain (micro and residual)Size, microstructure (twins anddislocations)


Estimating Texture

Intensity ratio

XRD on different faces

Extended concept for plotting inverse pole figure

200

111

II

powder

powder

sample

sample

IIvs

II

200

111

200

111


Diffraction in TEM

Diffraction in polycrystal in Bragg-Brentano geometry

Source Detector


Diffraction in polycrystal in Schulz reflection geometry

Source Detector

xy oscillation

β

α


RD

TD

Schulz reflection geometry


Fix Bragg-Brentano condition for one peak

Parallel beam optics

Polycapillary or Gobel mirror

Sample oscillation to improve statistics

Rotation about axis normal to plane of sample (β)

Tilt along axis in the plane of the sample (α)


Pole figure measurement

Continuous and discrete scans


Limitation of Schulz method

De-focusing error at higher tilt

Incomplete pole figures α = 70-75◦

Pole figures not measured at θ < 10◦


α path length of X-rays Irradiated volume

Absorption is balanced for thicker sample

Absorption correction for thinner samples


Background correction I corrected = I measured (α,β) - IBG(α)

Defocusing correctionI corrected = (I measured (α, β) - IBG(α))/D(α)

NormalizationInormalised (α, β) = Icorrected (α, β) / R

Corrections in pole figures


Transmission method for pole figures

Thin sample

X-rays should penetrate through

Good data at higher inclination

No errors for higher α (>45)

Use in combination with Schulz method to obtain complete pole figures


High brightness source micro-focus tube, liquid metal anode

Excellent beam optics to ensure parallel beam with small divergence and dimensions Gobel mirror, polycap

Position Sensitive detector 2θ range fast PF determination

Area detector 2θ and α range very fast PF determination

Well suited for in-situ heating (and straining) studies

State of the art


Synchrotron diffraction

Brighter than X-rays

Heterogeneity study (5µm width)

In-situ studies phase transformation, recrystallization

Excellent for weakly scattering materials like polymers and biological materials


Neutron diffraction

Only method to measure magnetic texture

Thermal neutrons (λ = 1-2Å) with higher penetration ensure no absorption and defocusing correction

Complete pole figures

Mono-chromator and Position Sensitive detector

Pulsed neutron source provide better statistics

Time of flight measurements


Other techniques

Ultrasonic measurement

Selected Area Diffraction

Kossel Diffraction

Electron Back Scatter Diffraction

Transmission Electron Microscopy-Orientation Imaging Microscopy


Summary

Crystallographic texture plays an important role in determining physical and mechanical properties of materials

X-ray diffraction offers a robust tool to study texture in a variety of materials

Schulz reflection geometry is widely used in laboratory X-ray diffractometers to determine texture

Data analysis and interpretation is very important to understand texture

Microstructure and Texture- Processing-Performance of materials can be appreciated



References

M. Humphreys and W. B. Heatherly, An introduction to texture in materials, Monograph No. 5, The Institution of Metallurgist London 1979.

H. J. Bunge, European Journal of Mineralogy, 9 (1997) 735. R. W. Cahn, Materials Science and Technology, 15 1991. U. F. Kocks, C. N. Tome amd H.-R Wenk, Texture and Anisotropy, Cambridge

University Press, 1998. H. J. Bunge, Texture Analysis in Materials Science-Mathematical

Methods, Buttersworth London 1982. V. Randle and O. Engler, Introduction to Texture Analysis

Macrotexture, Microtexture and Orientation Mapping, Gordon and Breach Science Publishers, 2000.

S. Suwas and N. P. Gurao, Crystallographic texture in Materials, Journal of the Indian Institute of Science 88 (2008) 151.

S. Suwas and R. K. Ray, Crystallographic Texture of Materials, book under preparation

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