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

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

3. Bulk penetration (1 mm – 1 cm). 3D characterisation on a micron scale: morphology orientation phase plastic and elastic strain Maps of 100-1000 grains In-situ studies 4D vision ------------------------------------ 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 Serial data acquisition: B.C. Larson et al. (2002). Nature 415, 887-890. Tomographic reconstruction: 3DXRD Position: 3D Orientation: 3D Elastic strain: 6D Plastic strain 8D Phase ?

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 Full field Layer-by- layer Morphology + Orientation CMS + Volume + Orientation

10. Orientation space x r n x’x’ z y z’z’ y’y’  Rodrigues vector: r = n tan(  /2) r l r 2 r 3  fr  Rodrigues space: Each grain: a point

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

12. g a g a q a b      Ferrite – Austenite: N dN/dt ----------------------------------- S.E. Offerman et al. (2002). Science 298, 1003. S.E. Offerman et al. (2004). Acta Mater. 52, 4757. Phase Transformations in Carbon Steel Work with T.U. Delft

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

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

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

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

17. Dependence on number of projections FBPART 5 projections: 49 projections:

18. H.F. Poulsen, X. Fu. J. Appl.Cryst 36, 1062 (2003) 5 min acquisition time Resolution 5  m 2D-ART: Results mm

19. Video of growth of an internal grain -------------------------- S. Schmidt, S. F. Nielsen, C. Gundlach, L. Margulies, X. Huang, D. Juul Jensen. Science 305, 229 (2004) Recrystallization of 42% deformed pure Al during annealing at ~200 C.

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

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

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

23. x y z rr rr r  Density in 6D space: VectorfieldEulerian space x SO(3) Reconstruction of deformed materials: Challenges: Dimension Curvature Crystal symmetry Finite # projections

24. xlxl ylyl zlzl (L, y det, z det ) Sample Detector plane 44 rlrl r2r2 r3r3  fr Envelope surface Projection lines Projection surface in 6D space Position space:Orientation space:

25. Challenge: Dimensionality size of A: 10 10 x 10 10 Extremely sparse H.F. Poulsen. Phil. Mag. 83, 2761 (2003). Reconstruct density in 6D space A: 0  x j    ;  j, B: ;  jkl.

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. Multiphase materials 3DXRD + Tomography Tomography ID19 – ID15 3DXRD Spatial resolution 0.6 – 2.8  m5  m Resolving power 0.4 – 2  m0.1  m Time resolution1 min – 2 sec0.3 sec – 1 h

28. Ex: Grain boundary wetting --------------------------------------------------------- Collaboration w/ W. Ludwig, D. Bellet S.F. Nielsen et al. Proc. 21st Risø Int. Symp. Mat. Science p 473 (2000)        (a)(b) (d)(e) (c) Tomography 3DRXD + Tomography Misorientations Challenge: Combined reconstruction

29. Extinction contrast tomography G k0k0 kHkH Detectors 100 µm INSA-Lyon: W. Ludwig; Risø: E.M. Laridsen, S. Schmidt, H.F. Poulsen

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

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

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

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

34. Total Crystallography + Examples: Identification of new drugs Drug distribution in tablets Rocks, meteorites Grain map Phase 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

37. Spatial Resolution Present: 1 x 5 x 5  m 3 New detector (2006): 1 x 2 x 2  m 3 Nanoscope:0.1 x 0.1 x 0.1  m 3 Operation mid 2007 50 m ESRF Current beamline