Post on 30-Apr-2018
Introduction to Crystallographic Texture
Nilesh Prakash GuraoAssistant Professor
Materials Science and EngineeringIndian Institute of Technology Kanpur, Kanpur India
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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
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10%
25%
Initial
5%
35%50%
N. P. Gurao, S. Suwas, Materials Letters 2013
Rolling of Nickel at IISc
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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
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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
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Kanjeevaram Saree Rock
Morphological texture
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Crystallographic texture
Single orientation
Random orientation
Preferred orientation
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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
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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
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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
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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
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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
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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]
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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
+++==++
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rR ⋅
=
2tan θ
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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
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Pole figures
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Stereographic projection
RD
ND
TD
100
010001
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Cube {100}<001> orientation
011
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011 011 011
{100}<001> in different pole figures
RD
ND
TD
010
001100
100 pole figure 101 pole figure 111 pole figure
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Type of projection
Angle true
Area true
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Random texture
For a polycrystal
Single orientation Multiple spots
Deviation from exact orientation Scatter
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Contour levels
Density of poles
Compared with that of random sample
Smoothened contours
Multiples of random distribution (mrd)
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Inverse pole figure
FCC tension
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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.
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Euler angles and Orientation space
Roe Bunge Williamsφ ϕ2+π/2 αθ Φ ρψ ϕ1-π/2 β=ψ +tan-1(tanαcosρ)
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Every (hkl)[uvw] orientation has a unique (φ1, Φ, φ2) location in Euler space
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Volume of Orientation space
∫ ∫ ∫= = =
=π
φ
π
φ
π
φ
φφφφ2
0 0
2
021
1 2
sin dddVODF
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Volume element dV = dAdΦ 2 = sin Φ dΦ dΦ 1 dΦ 2
V = 8π2
Volume of Orientation space
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sc gOOg =′
Crystal and Sample Symmetry
Cubic 24 rotation matrices (432) Hexagonal 12 rotation matrices (62)
Tension/Compression (Axial symmetry) Rolling (222)
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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
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Cubic-axial
Initialtexture
Simulatedtexture
Experimentaltexture
Compression direction inverse pole figure of different FCC materials
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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
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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
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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
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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)
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Experimental ODF
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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
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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
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Volume fraction of different texture components in differently rolled
Copper
Beta fibre plot for Cu and Cu-Zn alloys
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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
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Diffraction and Texture
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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
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Polarized Light Micrograph
Nano-cellulose Magnesium
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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
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Riot of colours in α+γ steels
Gurao and Suwas, unpublished work
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Braggs Law
nλ = 2dsinθ
d d sinθ d sinθ
X-ray Diffraction for bulk texture
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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
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http://ww1.iucr.org/cww-top/crystal.index.html
Powder diffraction
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XRD of bulk electrodeposited nano-Ni
Lattice parameter, phase diagramsTexture, Strain (micro and residual)Size, microstructure (twins anddislocations)
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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
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Diffraction in TEM
Diffraction in polycrystal in Bragg-Brentano geometry
Source Detector
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Diffraction in polycrystal in Schulz reflection geometry
Source Detector
xy oscillation
β
α
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RD
TD
Schulz reflection geometry
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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)
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Pole figure measurement
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Continuous and discrete scans
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Limitation of Schulz method
De-focusing error at higher tilt
Incomplete pole figures α = 70-75◦
Pole figures not measured at θ < 10◦
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α path length of X-rays Irradiated volume
Absorption is balanced for thicker sample
Absorption correction for thinner samples
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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
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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
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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
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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
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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
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Other techniques
Ultrasonic measurement
Selected Area Diffraction
Kossel Diffraction
Electron Back Scatter Diffraction
Transmission Electron Microscopy-Orientation Imaging Microscopy
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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
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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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