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Henning Friis Poulsen Materials Research Department Risø National Lab., Dk-4000 Roskilde

New Mathematics and Algorithms for 3-D Image Analysis, Minneapolis January 2006. Grain maps and grain dynamics – a reconstruction challenge. Henning Friis Poulsen Materials Research Department Risø National Lab., Dk-4000 Roskilde henning.friis.poulsen@risoe.dk. Polycrystals.

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Henning Friis Poulsen Materials Research Department Risø National Lab., Dk-4000 Roskilde

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  1. New Mathematics and Algorithms for 3-D Image Analysis, Minneapolis January 2006 Grain maps and grain dynamics – a reconstruction challenge Henning Friis Poulsen Materials Research Department Risø National Lab., Dk-4000 Roskilde henning.friis.poulsen@risoe.dk

  2. Polycrystals • Bravais lattice • Group symmetry • Basis (atoms) • Orientation • Elastic Strain Phase Grain morphology

  3. 4D vision • Bulk penetration (1 mm – 1 cm). • 3D characterisation on a micron scale: • morphology • orientation • phase • plastic and elastic strain • Maps of100-1000 grains • In-situ studies ------------------------------------ H.F. Poulsen: Three-Dimensional X-ray Diffraction Microscopy (Springer, 2004).

  4. ESRF, Grenoble

  5. Risø: J.R. Bowen, C. Gundlach, K. Haldrup, B. Jacobsen, D. Juul Jensen, E. Knudsen, E.M. Lauridsen, L. Margulies,S.F. Nielsen, W. Pantleon, S. Schmidt, H.O. Sørensen, J. Wert, G. Winther ESRF, ID11:A. Goetz, Å. Kvick, G. Vaughan ESRF, ID15: T. Buslaps, V. Honkimäki APS: U. Lienert, J. Almer GKSS: F. Beckmann, R.V. Martins IMSA, Lyon: W. Ludwig City Uni of N.Y.: A. Alpers, G.T. Herman, L. Rodek

  6. Sampling strategy Position: 3D Orientation: 3D Elastic strain: 6D Plastic strain 8D Phase ? Serial data acquisition: B.C. Larson et al. (2002). Nature415, 887-890. Tomographic reconstruction: 3DXRD

  7. Diffraction Diffraction spots: Where: Position of voxel + Symmetry + Orientation + Elastic strain Intensity: ~ volume Finite number

  8. 3DXRD set-up Area detector Detector I L = 5-10 mm Position and Orientation Detector II L = 40 cm Orientation and Strain

  9. Grain maps • Simplifications: • Monophase • No strain • Undeformed material Morphology + Orientation CMS + Volume + Orientation Full field Layer-by- layer

  10. W fr r 3 r 2 r l n z z’ f r y’ y x x’ Orientation space Rodrigues vector: r = n tan(j/2) Rodrigues space: Each grain: a point

  11. W fr r 3 r 2 r l • For > 1000 grains: • Orientation • Volume • CMS Position GRAINDEX Blob-finding in orientation space: ------------------------------------------ E.M. Lauridsen, S. Schmidt, R.M. Suter, H.F. Poulsen. J. Appl. Cryst. (2001) 34, 744 .

  12. Ferrite – Austenite: N g q g g g a a q a a dN/dt a b Phase Transformations in Carbon Steel Work with T.U. Delft ----------------------------------- S.E. Offermanet al.(2002). Science298, 1003. S.E. Offermanet al.(2004). Acta Mater.52, 4757.

  13. Growth curves for individual grains Grain radius (mm) Annealing time (sec) Standard Avrami type models are gross simplifications

  14. Grain Maps: grain by grain Grain map algorithms: Filtered back-projection Algebraic Reconstruction (ART)

  15. ART for tomography Solve: Ax = b x: density of voxel b: detector pixel intensitites A: geometry of set-up Solve iteratively by Kaczmark routine: xi bj l --------------- H.F. Poulsen & X. Fu. J. Appl. Cryst 36, 1062 (2003).

  16. Constraint on probability 0  xj 1 ART for 3DXRD Solve: Ax = b x: prob. of voxel belonging to grain b: detector pixel intensitites A: geometry of set-up Solve iteratively by Kaczmark routine: xi bj l --------------- H.F. Poulsen & X. Fu. J. Appl. Cryst 36, 1062 (2003).

  17. Dependence on number of projections FBP ART 5 projections: 49 projections:

  18. 2D-ART: Results mm 5 min acquisition time Resolution 5 mm H.F. Poulsen, X. Fu. J. Appl.Cryst36, 1062 (2003)

  19. Video of growth of an internal grain Recrystallization of 42% deformed pure Al during annealing at ~200 C. -------------------------- S. Schmidt, S. F. Nielsen, C. Gundlach, L. Margulies, X. Huang, D. Juul Jensen. Science305, 229 (2004)

  20. Grain growth ------------------- Work in progress by S. Schmidt, J. Driver et al.

  21. Hierachial solution GRAINDEX ART Discrete Monte Carlo (*) ----------------------- (*) A. Alpers, H.F. Poulsen, E. Knudsen, G.T. Herman Electron. Notes Discrete Math. 20, 419-437 (2005).

  22. Deformation 0% 11% Grain maps in deformed case: Spot overlap

  23. r3 r x r1 y z r Reconstruction of deformed materials: Density in 6D space: Vectorfield Eulerian space x SO(3) Challenges: Dimension Curvature Crystal symmetry Finite # projections

  24. Detector plane Envelope surface (L, ydet, zdet) Wfr 4q r3 r2 rl xl zl Sample yl Projection lines Projection surface in 6D space Position space: Orientation space:

  25. Reconstruct density in 6D space A: 0  xj r0 ; j, B: ; jkl. Challenge: Dimensionality size of A: 1010 x 1010 Extremely sparse H.F. Poulsen. Phil. Mag. 83, 2761 (2003).

  26. Properties of grains • Discrete objects. • Simply-connected space filling objects • Similarity of grain maps • The grain boundaries are smooth. • Near convex • Approx. polyhedra

  27. Tomography ID19 – ID15 3DXRD Spatial resolution 0.6 – 2.8 mm 5 mm 3DXRD + Tomography Resolving power 0.4 – 2 mm 0.1 mm Time resolution 1 min – 2 sec 0.3 sec – 1 h Multiphase materials

  28. 3DRXD + Tomography Misorientations Tomography 41.5 3.7 (a) (b) (c) 37.7 52.3 53.5 49.5 45.1 (d) (e) Ex: Grain boundary wetting Challenge: Combined reconstruction --------------------------------------------------------- Collaboration w/ W. Ludwig, D. Bellet S.F. Nielsen et al. Proc. 21st Risø Int. Symp. Mat. Science p 473 (2000)

  29. kH k0 G 100 µm Detectors Extinction contrast tomography INSA-Lyon: W. Ludwig; Risø: E.M. Laridsen, S. Schmidt, H.F. Poulsen

  30. W fr Extinction contrast tomography Work in progress: • + Potential for 100 nm resolution • - 1000 projections => slow • Only near-perfect grains • Fewer grains

  31. Plastic flow in 3D by tomography Work with F. Beckmann at BW2, HASYLAB Trace position ofmarkers: + Universal + Large strains - Artifical markers Future: internal markers 1 mm markers => 1% strain resolution with 20 mm spatial resolution --------------------- S.F. Nielsen, H.F. Poulsen, F. Beckmann, F. Thorning, J.A. Wert. Acta Mater. (2003) 51, 2407.

  32. Effect of material geometry: Displacement field: Simple deformation theory: --------------- K. Haldrup, S.F. Nielsen, F. Beckmann, J.A. Wert. Mater. Sci.Techn., 2005, in print.

  33. Maps that completely describe the fundamental plastic flow mechanism in a 3D, bulk sample Measuring slip activity Tomography: Local plastic flow 3DXRD: Local orientation change

  34. Total Crystallography + Grain map Phase • Examples: • Identification of new drugs • Drug distribution in tablets • Rocks, meteorites • Approach: • ”Orthogonal data” • Bootstrapping Project partners: Risø, ESRF, CUNY, Novo, Oxford, MPIbpc, IP-Prague

  35. Summary Mission: Map {phase, orientation, elastic strain, plastic strain, …} in 4D MShard Approach: x-rays, tomographic reconstruction, 3D detector Challenges:High dimensional space; extremely sparse Gray value/discrete parameters Number of projections Strategy ?:Discrete properties Hierachial approach Hybrid models

  36. ESRF Current beamline 50 m Spatial Resolution Present: 1 x 5 x 5 mm3 New detector (2006): 1 x 2 x 2 mm3 Nanoscope: 0.1 x 0.1 x 0.1 mm3 Operation mid 2007

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