Texture Analysis using OIM

Texture Analysis using OIM Stuart I. Wright Acknowledgements: David Field, Washington State University Karsten Kunze, ETH Zurich Outline What is crystallographic texture? Mathematical constructs Texture calculation parameters Analyzing Textures Volume Fractions Texture related functions (MDF, Scalar textures) 2 1

References For a description of the representation of orientation, especially in terms of Euler angles, and the mathematics behind the harmonic expansion of the ODF see: H. Bunge (1982). Texture Analysis in Materials Science. Butterworths: London. For a general overview of textures in metals see: I. Dillamore and W. Roberts (1965). Preferred orientation in Wrought and Annealed Metals. Metallurgical Reviews, 10, 271-380 For a general overview of textures in hexagonal materials see the first few chapters of: E. Tenckhoff (1988). Deformation Mechanisms, Texture and Anisotropy in Zirconium and Zircaloy. ASTM: Philadelphia. 3 A good place to become familiar with the general body of literature in texture analysis is in the proceedings of the International Conference on Texture of Materials (ICOTOM) held every three years. The next one is in June 2008 in Pittsburgh. Number of Papers 400 350 300 250 200 150 100 50 EBSD Papers ICOTOM EBSD Papers Automation 0 1980 1985 1990 1995 2000 2005 Year References Texture and Anisotropy Edited by Fred Kocks, Carlos Tomé and Rudy Wenk Cambridge University Press Introduction Heinz Mecking Part I Description of Textures and Anisotropies 1. Anisotropy and Symmetry Fred Kocks 2. The Representation of Orientations and Textures Fred Kocks 3. Determination of the Orientation Distribution from Pole Figures John Kallend 4. Pole Figure Measurement with Diffraction Techniques Rudy Wenk 5. Typical Textures in Metals Tony Rollett & Stuart Wright 6. Typical Textures in Geological Materials and Ceramics Rudy Wenk Part II. Anisotropic Mechanical Properties in Textured Polycrystals 7. Tensor Properties of Textured Polycrystals Carlos Tomé 8. Kinematics and Kinetics of Plasticity Fred Kocks 9. Simulation of Deformation Textures for Cubic Metals Fred Kocks 10. Effects of Texture on Plasticity Mike Stout & Fred Kocks 11. Self Consistent Modeling of Heterogeneous Plasticity Carlos Tomé & Fred Kocks 12. Finite-Element Modeling of Heterogeneous Plasticity Paul Dawson & Armand Beaudoin Part III. Some Applications 13. Finite-Element Simulation of Metal Forming Paul Dawson & Armand Beaudoin 14. Plasticity Modeling in Minerals and Rocks Rudy Wenk Appendix: The Elastic Inclusion Problem Carlos Tomé 4 2

References Electron Backscatter Diffraction in Materials Science edited by Adam J. Schwartz - Lawrence Livermore National Laboratory, CA, USA Mukul Kumar - Lawrence Livermore National Laboratory, CA, USA Brent L. Adams - Brigham Young University, Provo, UT, USA Kluwer Academic/Plenum Publishers, 2000 Introduction to Texture Analysis - Macrotexture, Microtexture and Orientation Mapping Valerie Randle - Department of Materials Engineering University of Wales, Swansea, UK Olaf Engler - Materials Science and Technology Division, Los Alamos National Laboratory, NM, USA Gordon Breach, 2000 - http://www.gbhap.com Part I Fundamental Issues: Descriptors of Orientation Applications of Diffraction to Texture Analysis Part II Macrotexture Analysis: Macrotexture Measurements Evaluation and Representation of Macrotexture Data Part III Microtexture Analysis: The Kikuchi Diffraction Pattern Scanning Electron Microscopy (SEM) Based Techniques Transmission Electron Microscopy (TEM) Based Techniques Evaluation and Representation of Microtexture Data Orientation Microscopy and Orientation Mapping Crystallographic Analysis of Interfaces, Surfaces and Connectivity Synchrotron Radiation, Nondiffraction Techniques and Comparisons Between Methods Part IV Case Studies 5 References Textured Structures David P. Field ASM Handbook Volume 9 Metallurgy and Microstructures (2004) pp. 215-226 Orientation Texture Stuart I. Wright Encyclopedia of Condensed Matter Physics (2005) Last But Not Least! www.edax.com Particularly the EBSD bibliography section which we try to keep up to date (www.edax.com/about/bib.cfm). It contains references to papers where EBSD has been used in a wide variety of applications as of June 2007, it has nearly 5500 entries! 6 3

What is Texture? Texture is a measure of the similarity in orientation of the crystal lattice within the constituent grains of a polycrystal. A single crystal would have the maximum texture, a material where all the grains have different orientation would have a random texture. Most materials fall somewhere between these two extrema depending on how the material was processed. 7 Microstructure Measurement Techniques (Micro-)Texture Measurement Techniques Bulk Techniques X-Ray Neutron Diffraction Spatially Specific Techniques Synchrotron TEM Kikuchi Patterns TEM Spot Patterns Polarized Light Kossel Patterns Channeling Patterns Electron Backscatter Diffraction (EBSD) 8 4

Traditional Texture Measurements X-Ray Diffraction (Pole Figures) - Area Measurement Neutron Diffraction (Pole Figures or Rietveld Techniques) - Volumetric Measurement In X-Ray and Neutron we only know the number of grains diffracting for a given pole, but not which ones. 9 Spatially Specific Texture Measurements Synchrotron Radiation (H. F. Poulson & D. Juul-Jensen) Low Spatial Resolution (5µm), moderate angular resolution (1-2 ), Good statistics, 3-d information, difficult mathematical reconstruction, poor availability Polarized Light (R. Heilbronner ) Not very quantitative, inexpensive, good accessibility, 2-d, limited materials TEM Diffraction (Kikuchi patterns, Spot patterns, CBED) High spatial resolution, good accuracy, extremely limited area, difficult sample preparation, as of yet - limited automation, some 3-d information, poor statistics Electron Channeling Poor spatial resolution (5-10µm), moderate accuracy (0.5 ), 2-d, no automation so poor statistics. Kossel X-Ray Diffraction Poor spatial resolution (10µm), good accuracy (0.1 ), 2-d, limited materials, no automation so poor statistics. Electron Backscatter Diffraction Good spatial resolution (~20nm), good angular resolution (~1 ), reasonable statistics with automation, good availability, 2-d 10 5

Texture Analysis via X-Ray vs. OIM Advantages Lots of orientations measured (Good Statistics) Easy sample preparation Fast Familiar texture analysis grew up in the era of X-Ray texture measurements. Disadvantages No information on spatial arrangement of orientations Somewhat convoluted mathematics in pole figure inversion ( Ghost Correction ). Lots of corrections in determining pole figure distribution function Relative to electron diffraction methods 11 Texture Analysis via X-Ray vs. OIM X-Ray OIM Absorption Correction, Random Sample Normalization Texture Calculation Texture Calculation Rolled Copper 12 6

Texture Analysis and EBSD EBSD Data is obtained in the form of discrete orientation measurements. While these data can be plotted in discrete pole figures and ODFs. These plots can be difficult to interpret. What is going on in this cluster? Are there many overlapping points or just enough to make it appear black. Are there sub-clusters of points in here? These are the kinds of question texture analysis can help resolve. 13 Discrete points to continuous distributions First it is important to understand what is the ODF (Orientation Distribution Function) as it is the basis of texture analysis. The orientation distribution function (f(g)) or ODF is a probability density function describing the probability of finding a grain with an orientation g within a given angular distance, g, of a specified orientation g o in a polycrystal; or, alternatively, the volume fraction of material oriented within g of g o. V( g0 + g) = f ( g) dg V g ( g0 + g) In OIM, individual orientations are collected. The aim of texture analysis is to essentially create a smooth curve fit of the discrete orientations. 14 7

Discrete points to continuous distributions Imagine a simple case where orientations are 1- dimensional. They could then be plotted on a 1-d bounded number line. The objective analysis is to essentially come up with a curve fit which characterizes the statistical distribution. One way to do this would be to bin the data and then fit a curve to the bins. How large should the bins be? How should data near the edges of the bins contribute to the bins, should the neighboring bin also be incremented by some amount as well? 15 Binning OIM uses the direct space approach as suggested by Matthies (Matthies, S., and Vinel, G. W., "On some Methodical Development Concerning Calculations Performed Directly in the Orientation Space", Materials Science Forum (Proceedings of ICOTOM-10), 157-162, pp. 1641-1646.). In OIM the full Euler angle space is divided up into bins. For each orientation in the data set the corresponding bin is incremented. The crystal and sample symmetries are applied so that for a single orientation there may be many bins which are incremented. After the binning is completed for all orientations. An bins are converted to an ODF based on the volume of the bin and the number of votes the bin has received. The ODF is then normalized according to: f ( g) dg = 1 The ODF is then smoothed. 16 8

Harmonics The so-called harmonic analysis is analogous to a Fourier series expansion. However, texture analysis is performed using generalized spherical harmonic expansion (GSHE). The GSHE analysis used in OIM follows the formulation of Bunge (H. Bunge (1982). Texture Analysis in Materials Science. Butterworths: London.). In this method, the ODF can be expanded into a series of generalized spherical harmonic functions: l = 0 l mn mn f ( g) = C T ( g) l m= l n= l The T l mn (g) functions are generalized spherical harmonics and are known. The C l mn coefficients are determined from the orientation measurements using the following: C mn l 2l + 1 = N N i= 1 l KT mn l l ( g ) Where the g i are the individual orientation measurements, N is the number of measurements and K is a factor based on the amount of smoothing to be applied. i 17 Harmonics vs. Binning GSHE Well established technique in the texture community. Property models have been developed based on the C coefficients. Because the series can t be expanded to infinity there will be truncation error. Lower order of the series expansion correspond to the green curve (i.e. l = 4) and higher truncations correspond to the orange curve (i.e. l = 34) BINNING Artifacts can arise due to the boundaries between bins. The bins are not of equal volume which may introduce some errors. (Assuming 5 x 5 x 5 bins: dv(φ=2.5 ) = 0.044 and dv(φ=87.5 ) = 0.999) Larger bins (i.e. 15 ) create ODFs corresponding to the green curve and smaller bins (i.e. 3 ) correspond to the orange curve. Which is correct? It depends on the property of interest. For example, elastic anisotropy arising from texture is dependent only on the l=4 coefficients, piezoelectricity is dependent on the 9 th order coefficients. Plasticity is even higher order and is generally modeled with discrete orientations. 18 9

GSHE Order L=16 L l l mn mn f ( g) = Cl Tl ( g) l= 0 m= l n= l L=22 L=34 19 Gaussian Half-Width C mn l 2l + 1 = N N i= 1 KT mn l ( g ) i exp( l ω / 4) exp( ( l + 1) ω / 4 K = 2 1 exp( ω / 4) ω is the half width. Several values have been proposed. However, my opinion is that most of them are fairly arbitrary. One idea is that it should reflect the accuracy of the measurement another that it should scale with the number of measurements, another with the sharpness of the texture. l = 8 l = 16 l = 32 2 2 2 2 ω = 0.01 ω = 5 20 10

Sample Symmetry The sample symmetry may be more pronounced in some samples than others. This could be real the processing isn t as symmetric as assumed or could be due to a lack of statistics in the sampling. To check the statistics, obtain measurements from another sample from the same material. OIM is ideal for examining statistical symmetry and determining at what length scale the sample processing symmetry is exhibited. Sample symmetry can be enforced in the texture calculations. This is especially useful in the harmonic calculations as it reduces the number of coefficients. However, in the binning approach it will actually increase the time. Enforcing the symmetry is useful when comparing to texture results obtained by X- Ray diffraction where sample symmetry is almost always enforced. Orthotropic (rolled sheet) symmetry enforced. 22 Sample Symmetry However, the sample symmetry must be applied with the measurements in the correct reference frame. Consider drawn wire. No sample symmetry enforced Symmetry Axis Axial sample symmetry enforced 23 11

Sampling [100] directions 24 Origin of Texture Texture arises from almost every forming process (except perhaps powder metallurgy processes) Deformation Rolling Forging Extrusion Casting Solidification Deposition Electrodeposition Vapor depostion Heat Treating Recrystallization Phase Transformations Welding 25 12

Deformation Textures In deformed materials. Texture or preferred orientation exists due to the anisotropy of slip. While slip in bcc metals generally occurs in the <111> type direction, it may be restricted to {110} planes or it may involve other planes. After T. H. Courtney, Mechanical Behavior of Materials, McGraw-Hill, New York, 1990. 26 Deformation Textures In deformed materials. Texture or preferred orientation exists due to the anisotropy of slip. Forged Tantalum I. L. Dillamore and H. Katoh, Met. Sci., 8 (1974), p. 73. 27 13

Recrystallization Textures C. Necker (1992), Recrystallization in copper. M.S. Thesis, Drexel University. 28 Solidification Textures C.-A Gandin, M. Rappaz, D. West and B. L. Adams, Met. Mat. Trans., 26A (1995), 1543-1551. 29 14

Fiber Textures Some materials like thin films exhibit a strong fiber texture. We use the term fiber texture to describe a texture where a particular crystal axis is aligned with a particular sample direction (typically the sample normal) and there is no in-plane alignment. OIM has tools for looking at the details of fiber textures 111 Pole Figure and corresponding OIM Map from an Aluminum Thin Film 30 Case Study Cu Thin Film Consider an example of a copper thin film. (111) Pole Plot 32 15

Copper Thin Film An inverse pole figure for the sample revealed some other significant peaks that weren t obvious in the pole plot. 33 Copper Thin Film A quick approach to verify that the texture was not a simple 111 fiber texture is an inverse pole figure map. 34 16

Multiple Fibers We examined the pole plot in more detail to look at the minor peaks. The 115 fits the secondary peaks well but the 325 doesn t fit as well. There appear to be a set of near 325 peaks in the cluster at 325. 35 Volume Fractions Once we ve identified the peaks of interest we can calculate the volume fraction of each component by looking at each individual measurement and seeing if it is within a given tolerance of any of the identified fiber components. The tolerances used here were derived from the pole plot analysis. (Crystal Direction Map) (111) 27.5% [9.0 ] (325) 8.8% [6.5 ] (115) 39.6% [9.5 ] 36 17

Texture vs. Volume Fraction So what is going on? Why does the (111) peak appear so strong in both the pole plot calculations and in the harmonic texture calculations? In both of these representations, a few points closely oriented produce larger peaks than a large cluster of peaks more broadly related. This shows up as the tolerance value shrinks in the volume fraction measurements. 37 Fibers in Multiple Directions Interconnect lines formed by the copper damascene process often produce microstructures having fibers in multiple directions. Copper Damascene Process Overburden Copper After Electropolishing 38 18

Fibers in Multiple Directions The two fibers could be analyzed individually using the approach described for the fibers in a single direction. Alternatively they can be analyzed simultaneously using a crystal direction map. The red grains have (111) planes parallel to the sample normal and the blue grains have (111) planes normal to the transverse direction. 39 Texture Component Analysis For some complicated textures, the texture can be described in terms of a set of ideal components. This simplifies texture analysis to a single scalar value or small set of scalar values (i.e. the volume fractions of the given components) instead of an ODF or pole figure analysis. Rolled FCC Materials Recrystallized FCC Materials Rolled BCC Materials Rolled HCP Materials c/a>1.63 c/a 1.63 c/a<1.63 40 19

Component Analysis Components simplify tracking texture changes over a matrix of experiments. {111} Pole Figures for Copper-Zinc alloys rolled to 96% reduction 6 5 4 3 2 Cu 1 Brass 0 0 5 10 15 20 25 30 Zn Content [%] Stephens, A.S. (1968). Texture and mechanical anisotropy in the copper-zinc system. Ph.D. Thesis, University of Arizona. 41 Texture Component Analysis Screen shot of OIM Analysis showing a discrete (111) pole figure and a crystal orientation map overlaid on an image quality map. 42 20

Texture Component Analysis Care should be taken when looking at texture components. The sample symmetry should generally be enforced when studying materials with sample symmetry such as rolled sheet. Alternatively. You can enter the symmetric variants manually. The symmetry utility can be used to identify them. 43 Texture Component Analysis If the tolerances on the components are large such that they overlap then a point in the scan will be assigned to the closest component. 15 Tolerance 14.8% Direction <111> ND <001> ND <115> ND <325> ND 36.2% Fraction 29.8% 6.8% 16.4% 44.5% 25.3% 50.6% 44 21

Texture Related Functions 45 GB Texture: MDF f ( g) = f ( ω, α, β ) f ( g) = f ( ω, α, β ) 32 16 8 4 2 1 0.5 Misorientation Angle Distribution MDF plotted in Axis/Angle Space 46 22

Conclusions 47 23