Henning Friis PoulsenMaterials 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

Polycrystals

Bravais lattice• Group symmetry• Basis (atoms)

Orientation

Elastic Strain

Phase

Grain morphology

• Bulk penetration (1 mm – 1 cm).

• 3D characterisation on a micron scale:morphologyorientation 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).

ESRF, Grenoble

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. HonkimäkiAPS: U. Lienert, J. AlmerGKSS: F. Beckmann, R.V. MartinsIMSA, Lyon: W. LudwigCity Uni of N.Y.: A. Alpers, G.T. Herman, L. Rodek

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 ?

Diffraction

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

3DXRD set-up

Area detector

Detector IL = 5-10 mmPosition and Orientation

Detector IIL = 40 cmOrientation and Strain

Grain maps

Simplifications:• Monophase• No strain• Undeformed material

Full field

Layer-by-layer

Morphology +Orientation

CMS + Volume +Orientation

Orientation space

x

r

n

x’

z

y

z’

y’

Rodrigues vector:

r = n tan(/2)

rl r2

r3

fr

Rodrigues space:

Each grain: a point

GRAINDEX

------------------------------------------E.M. Lauridsen, S. Schmidt, R.M. Suter, H.F. Poulsen. J. Appl. Cryst. (2001) 34, 744 .

rl r2

r3

fr

Blob-finding in orientation space:

For > 1000 grains:• Orientation• Volume• CMS Position

a

b

600 650 700 750 800 850 9000.0

0.2

0.4

0.6

0.8

1.0 Measurement

CNT

(dN

/dt)

/(d

N/d

t) ma

x

T (oC)

0

20

40

60

80

100

Nto

tal

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

Growth curves for individual grains

Standard Avrami type models are gross simplifications

Grain radius (m)

Annealing time (sec)

Grain Maps: grain by grain

Grain map algorithms:Filtered back-projectionAlgebraic Reconstruction (ART)

ART for tomography

xi

bj

Solve: Ax = bx: density of voxel b: detector pixel intensititesA: geometry of set-up

Solve iteratively by Kaczmark routine:

M

jkj

M

j

kjkjk

kk

A

xAb

xx

1

2

11

---------------

H.F. Poulsen & X. Fu. J. Appl. Cryst 36, 1062 (2003).

ART for 3DXRD

xi

bj

Solve: Ax = bx: prob. of voxel belonging to grainb: detector pixel intensititesA: geometry of set-up

Solve iteratively by Kaczmark routine:

M

jkj

M

j

kjkjk

kk

A

xAb

xx

1

2

11

---------------

H.F. Poulsen & X. Fu. J. Appl. Cryst 36, 1062 (2003).

Constraint on probability

0 xj 1

-30 -20 -10 0 10 20 30

-20

-15

-10

-5

0

5

10

15

20

Dependence on number of projections

FBP ART

-30 -20 -10 0 10 20 30

-20

-15

-10

-5

0

5

10

15

20

-30 -20 -10 0 10 20 30

-20

-15

-10

-5

0

5

10

15

20

-30 -20 -10 0 10 20 30

-20

-15

-10

-5

0

5

10

15

20

5 projections:

49 projections:

H.F. Poulsen, X. Fu. J. Appl.Cryst 36, 1062 (2003)

5 min acquisition timeResolution 5 m

2D-ART: Results

m

Video of growthof 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.

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

Grain growth

Hierachial solution

GRAINDEX ART Discrete Monte Carlo (*)

-----------------------(*) A. Alpers, H.F. Poulsen, E. Knudsen, G.T. HermanElectron. Notes Discrete Math. 20, 419-437 (2005).

Grain maps in deformed case:

Deformation

0% 11%

Spot overlap

x

y

z

r

r

r

Density in 6D space: Vectorfield Eulerian space x SO(3)

Reconstruction of deformed materials:

Challenges: DimensionCurvatureCrystal symmetryFinite # projections

xl

yl

zl

(L, ydet, zdet)

Sample

Detector plane

4

rl

r2

r3

fr

Envelope surface

Projection lines

Projection surface in 6D space

Position space: Orientation space:

Challenge:

Dimensionality size of A: 1010 x 1010

Extremely sparse

H.F. Poulsen. Phil. Mag. 83, 2761 (2003).

Reconstruct density in 6D space

0mnp

jklmnpx

A: 0 xj ; j,

B:

ijklmnpijklmnp bxA

; jkl.

Properties of grains

• Discrete objects. • Simply-connected space filling objects • Similarity of grain maps

• The grain boundaries are smooth.• Near convex• Approx. polyhedra

Multiphase materials

3DXRD + Tomography

  TomographyID19 – ID15

3DXRD

Spatial resolution 0.6 – 2.8 m 5 m

Resolving power 0.4 – 2 m 0.1m

Time resolution 1 min – 2 sec 0.3 sec – 1 h

Ex: Grain boundary wetting

---------------------------------------------------------Collaboration w/ W. Ludwig, D. BelletS.F. Nielsen et al. Proc. 21st Risø Int. Symp. Mat. Science p 473 (2000)

(a) (b)

(d) (e)

(c)

Tomography3DRXD +Tomography Misorientations

Challenge: Combined reconstruction

Extinction contrast tomography

G k0

kH

Detectors100 µm

INSA-Lyon: W. Ludwig; Risø: E.M. Laridsen, S. Schmidt, H.F. Poulsen

Extinction contrast tomography

Work in progress:

+ Potential for 100 nm resolution

- 1000 projections => slow- Only near-perfect grains- Fewer grains

fr

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.

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.

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

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

Summary

Mission:

Map {phase, orientation, elastic strain, plastic strain, …} in 4D MShard

Approach: x-rays, tomographic reconstruction, 3D detector

Challenges: High dimensional space; extremely sparseGray value/discrete parameters

Number of projections

Strategy ?: Discrete propertiesHierachial approachHybrid models

Spatial Resolution

Present: 1 x 5 x 5 m3

New detector (2006): 1 x 2 x 2 m3

Nanoscope: 0.1 x 0.1 x 0.1 m3

Operation mid 2007

50 m

ESRF Current

beamline