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Crystallographic reconstruction methods to study phase transformations by EBSD Cyril CAYRON

Crystallographic reconstruction methods to study phase transformations by EBSD Cyril CAYRON CEA-Grenoble Laboratory of Innovation for New Energy Technologies and Nanomaterials , France. Introduction: Phase transitions, EBSD .

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Crystallographic reconstruction methods to study phase transformations by EBSD Cyril CAYRON

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  1. Crystallographic reconstruction methods to study phase transformations by EBSD Cyril CAYRON CEA-Grenoble Laboratory of Innovation for New Energy Technologies and Nanomaterials, France

  2. Introduction: Phase transitions, EBSD. • Theory 1: Crystallography of phase transitions, groups and beyond • Application 1: Reconstruction of parent grains • Theory 2: Crystallography of multiple twinning, fractals, strings, number theory. • Application 2: Reconstruction of twin related domains • Conclusions and perspectives

  3. MATHEMATICS Phase transitions Cosmology, Elementary particles Experiments Theory Crystallography Microstrutural Characterization RX, Neutrons, TEM, Kossel, EBSD AF Gourgues, Int. Mater. Rev. 52 (2007) 65. Raman, AFM, Calorimetry, Dilatometry … • Structural Phase transformations • 1st order, 2d order • Displacive, Order-Disorder, Reconstructive • Diffusive, diffusionless Thermodynamics Physical properties Ferromagnetism, Ferroelectricity, Multiferroics Strength, Conductivity ’ precipitates in AlMgSiCu alloy,C. Cayron, Thesis EPFL, Switzerland Widmanstatten  in Ti alloy, JP Blank Ohio State, USA

  4. 0 : Identity 1: 90° [1 10 0], 180° [52 91 49], etc.. minimum angle = 90° / [1 10 0] 2: 10.5° [0 0 1], 180° [1 10 0], etc.. minimum angle = 10.5° / [0 0 1] 3: 120° [3 30 25], 90° [52 91 49], etc.. minimum angle = 60.8° / [1 10 1] 4: 120° [1 0 0], 180° [-8 8 5], etc.. minimum angle = 60° / [1 0 0] 5: 120° [1 5 0], 180° [48 16 15], etc.. minimum angle = 63.2° / [0 5 1] Classical approach : identify the special misorientations (operators) Application to the  (BCC)  (HCP) phase transformation in Ti and Zr alloys Burgers OR, 12 variants, 6 operators  grains in a TA6V alloy, Cayron, Briottet, Jouneau, Proc. Channel Users Meeting (2004)

  5. Metz group (France): Classical approach + calculation of the parent orientations TheoryM. Humbert, F. Wagner, C. Esling, J. Appl. Cryst. 25 (1992) 724. M. Humbert, N. Gey, J. Appl. Cryst. 35 (2002) 401. EBSD N. Gey, M. Humbert, J. Mater. Sci. 38 (2003) 1289. L. Germain,S.R. Dey, M. Humbert, N. Gey, J. Microsc. 227 (2007) 284. Ti alloy Reconstructed  grains  grains Retained  grains L. Germain, N. Gey, M. Humbert, Ultramicroscopy 107 (2007) 1129.

  6. The limits of the classical method Steel • Problems with steels because: • High symmetry of the daughter phase • High level of intragranular deformation • the probability that two randomly oriented grains are linked accidentally by an operator is not negligible! Reconst. 1° Reconst. 3°

  7. THEORY 1: Groupoids of orientational variants = Operators = Class of misorientation between variants = Double-Cosets Variants = Cosets V. Janovec, Czech. J. Phys B22 (1972) 974. U. Dahmen, Phase Transformations, Encyclopedia of Physical Science and Technology, 1987, pp.319-354. T. Hahn, V. Janovec, H. Klapper, Ferroelectrics 222 (1999) 11. C. Cayron, Acta Cryst. A 62 (2006) 21.

  8. About the number of variants Direct Transition: Inverse Transition : But: And: The number of variants is the number of cosets. It is given by the Lagrange formula: Ex: 8/2 = 4 Ex: 4x6 = 3x8

  9. About the number of operators Ex: 4 = 1+2+1 Avec 2 et 1 /2 Class equation Burnside formula The number of operators is the number of double-cosets.

  10. Groupoid of orientational variants Definition: X set of objects  Set of arrows (i, ik, ik)

  11. Groupoid composition table The operators can be expressed by set of variants / 1 Groupoid Composition Table

  12. Computer program: GenOVa 3D representation Pole figures Groupoid tables C.Cayron, J. Appl. Cryst. 40 (2007) 1179. Electron diffraction

  13. APPLICATION 1: Reconstruction of parent grains from EBSD data Idea : Use the groupoid composition table to check the coherency • Two Steps: • Nucleation with low tolerance angle (<5°) • Growth with high tolerance angle (8°-25°) C. Cayron, et al.Mater. Charact.57 (2006) 386.

  14. Computer program: ARPGE C. Cayron, J. Appl. Cryst. 40 (2007) 1183.

  15. Applications: Ti alloys <111>  <111>  <110>  <001>  Titanium alloy for Aerospace (from A. Ambard, CEA-Grenoble, now EdF)  grains  grains reconstruction

  16. Applications: Steels and Fe alloys Martensitic iron from meteorite (from Joe Michael, USA) Only one parent  grain  grain  grains reconstruction Calculated orientation of the  grain(ARPGE) Simulated  grain + 24 variants (GenOVa) The calculated orientation of the  grain is correct !

  17. Bainitic steel (from PH Jouneau, INSA, now CEA, France)  grains  grains reconstruction

  18. Martensitic steel (from Laurent Legras, EdF, France)  grains  grains reconstruction

  19. Martensitic steel for nuclear application (from F. Barcelo, CEA, France) • grains 2 and 3 are 3 twins  grains  grains reconstruction

  20. Digression : Thermal cycling, entropy and time’s arrow We consider a phase transition material that follows a cycle of transitions Open systems, not isolated (Prigogine, Physics Today, 1972) Landau transitions: G = H N = G/ G N = G/ G=1 Time does not exist material = crystal = « death » (Fedorov) 1 Reconstructive transitions: Time is flowing The material self-organizes Stabilization after N cycles ?

  21. Grain Boundary Engineering and multiple twinning Nowadays, GBE in CFC materials = Increase the number of 3 (111) twin grain boundaries = Understand and Control the multiple twinning during recrystallization Copper thin film Lei Lu et al. Science 304 (2004) 422. 316 austenitic steel • but : • How recognize a 3n grain boundary? • How define unambiguously the Twin Related Domains (TRD)? Twin Related Domain (where are the frontiers?)

  22. THEORY 2: Fractal & String representations of multiple twinning 81d  string 1111 81+a  = string 1112 • Variants of 1st generation = coset of matrices, 3 operator = double-cosets, 3n operators = multiple double-cosets. • The global algebraic structure is a groupoid : • Object = variants, Arrows = misorientations between variants, Operators = types of arrows. • The equivalent paths that links the variants constitute the 3n operators • A string can be associated to any path (Reed’s idea). • A 3n operator an be encoded by a set of equivalent strings. • The encoding algebraic structure depends on the choice of representatives in the cosets: • Free group with 4 equivalent letters (Reed) or Semi group with 3+1 letters (Cayron) or… BW Reed et al. Acta Cryst A60 (2004) 263. C Cayron Acta Cryst A63 (2006) 11.

  23. Calculation and representation of the 3n operators Schematic representation of the operators = Cayley graph

  24. Composition of the3n operators

  25. APPLICATION 2: Reconstruction of the TRDs Copper thin film Daughter grains Twin Related Domains reconstruction Nb of grains / TRD LLC / TRD LLC = Longest Length of Chain

  26. Ammann-Beenker Tiling Space groupoids Polytypes: Quasicrystals : Digression 2: The link with tilings Cycle of transformations with special variant selection rules = tiling

  27. Conclusion & Perspectives /Theory • Geometric/algebraic theory to describe the variants • (Landau transition, reconstructive transitions, precipitates, twinning) • Variants = cosets • Operators = double-cosets • Importance of the normality of the intersection group • Cycles of phase transformation  unify many fundamental problems • Groupoids • Complexity • Tiling • Entropy, time’arrow • Fractals • Strings • Number theory. • A unique mathematical basis: non-commutative algebra! Alain Connes • Atomic spectroscopy • ij = Rc(1/i2 – 1/j2) • ij = i - j ik = jj+ jk (groupoid law)

  28. Conclusion & Perspectives /EBSD applications • Reconstruction of the parent grains based on groupoid • - Automatic (only 2 tolerance angles). • Treats any phase transition materials (cubic, hexagonal, …) • Adapted for highly deformed materials. • Gives statistics on the variants and operators (variant selection). • But (a) still slow (10-30 min per map) • (b) still in 2D (3D should not be a problem) • (c) does not yet treat the interface planes / operators. • Reconstruction of the 3ntwin related domains • Automatic (only 1 tolerance angle). • Gives new parameters such as nb of grains per TRD and LLC. • Can be an interesting statistical tool for GBE studies. Information: GenoVa and ARPGE are home-made programs (written in Python). Commercialization expected for end of 2009. Thanks: Drs. B. Artaud, PH Jouneau, JM Gentzbittel, L. Briottet, J. Michael, D. Galy, F. Barcelo, Y. De Carlan, A. Gholinia, T. Epicier, L. Legras; Profs. Michéa, Weinstein, Litvin

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