Descriptor-based Microstructure Characterization

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Citation information: Xu, Hongyi, et al. "Descriptor-based methodology for statistical characterization and 3D reconstruction of microstructural materials.
McCormick Robert R. McCormick School of Engineering and Applied Science

Descriptor-based methodology for statistical characterization and 3D reconstruction of microstructural materials Hongyi Xu

Citation information: Xu, Hongyi, et al. "Descriptor-based methodology for statistical characterization and 3D reconstruction of microstructural materials." Computational Materials Science 85 (2014): 206-216. 1

Overview of Characterization and Reconstruction Image pre-processing

2D microstructure characterization

πœƒ

3D microstructure descriptor prediction

𝑅2𝐷

3D microstructure reconstruction 2

Descriptor-based Microstructure Characterization  Low computational cost

 High accuracy

 Clear physical meaning

Initialization: define target descriptor values Dispersion

Composition

Geometry Phase II

Composition Phase I

 Volume fraction of each phase, VF

VF %

Dispersion  Mean of Nearest Distances, 𝒓𝒅  Variance of Nearest Distances, π’“π’…π―πšπ« rd

Geometry  Average Radius, rc  Elongation ration, el  Orientation (random)

π‘Ÿπ‘ = π‘Žπ‘ π‘Ž 𝑒𝑙 = 𝑏

a

b

: orientation 3

Microstructure Characterization (1)

Original SEM/TEM image (1)

Composition Dispersion

Pre-defined volume fraction VF

Aggregate’s mean radius rc

Geometry

(2) Binary image (2)

(3)

Matrix-shielded grey scale image (3)

Aggregates counting Nearest distance nd

(4)

Mark aggregate’s centers (4) Distribution of aggregates’ aspect ratio (ρ)

(5)

Isolated-aggregate image (5)

Distribution of aggregates’ radius rc

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Composition: VF  Binary image is obtained by setting a greyscale threshold  Binary image corresponds to a thin top layer of the material, with thickness of 𝐷 = 2 Γ— π‘Ÿπ‘ , D is filler diameter *

2rc

* Jean, AurΓ©lie, et al. "A multiscale microstructure model of carbon black distribution in rubber." Journal of Microscopy 241.3 (2011): 243-260.

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Dispersion: Nearest Center Distances  Screen out the unnecessary information (greyscale values of the matrix phase), get matrix-shielded image

Original Image

x binary image matrixshielded image

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Finding Aggregates’ Centers

 Calculate the dispersion status (distribution of nearest center distance). 7

Geometry: Aspect Ratio & Equivalent Radius  Objective: characterize aggregates’ geometry accurately  Method: pick out the β€œsingle aggregate cluster”(marked by green circles), to calculate the particle shape descriptors.

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Summary of Microstructure Descriptors Category

Descriptors

Predefined descriptors

𝑽𝑭 𝒏𝐝_πŸπ‘«

2D descriptors characterized from 2D images

π†πŸπ‘« 𝑨𝒆

π‘΅πŸπ‘« 𝒏𝐝_πŸ‘π‘« 3D descriptors to be predicted

π†πŸ‘π‘« 𝑽𝒆 π‘΅πŸ‘π‘« 9

Predicting Aggregate Number in 3D Space

Num = 𝑁2𝐷 Γ—

𝐿 𝐷

L: side length of the 3D cube D: depth of the binary 2D image (D)

D

D

L

D: diameter Observed Particle Centers

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Predicting Aspect Ratio in 3D Space 𝑅3𝐷

π‘Ÿ3𝐷

90Β°

D 𝑅2𝐷 π‘Ÿ2𝐷

0Β° 2D projection: 2Γ— 𝑅2𝐷

Assumptions: 1. Isotropic material 2. Ellipsoidal Geometry. The two short semi-axis is equal length. 3D-to-2D projection:

𝜌2𝐷 = 𝑓 𝜌3𝐷 , πœƒ, π‘Ÿ2𝐷 =

π‘₯ βˆ™ π‘π‘œπ‘ πœƒ + 𝑦 βˆ™ π‘ π‘–π‘›πœƒ MAX π‘₯ 𝜌3𝐷2

2

+ βˆ’π‘₯ βˆ™ π‘ π‘–π‘›πœƒ + 𝑦 βˆ™ π‘π‘œπ‘ πœƒ

2

= π‘Ÿ2𝐷2

π‘Ÿ2𝐷 11

Cont. Predicting Aspect Ratio in 3D Space

90Β°

πœƒmax 𝜏

𝑦

πœ€

D

π‘Ÿ 𝑅

0Β° 2D projection: 2Γ— 𝑅2𝐷

2D-to-3D prediction:

π†πŸ‘π‘«_π’‘π’“π’†π’…π’Šπ’„π’•

 π‘₯

𝑅2𝐷

1 = βˆ— πœƒ

πœƒβˆ—

𝑓 βˆ’1 𝜌3𝐷 , πœƒ, π‘Ÿ2𝐷 π‘‘πœƒ

Where πœ½βˆ— is from:

0

Consideration of layer thickness constraints & randomness of spatial location:

π‘₯ βˆ™ π‘π‘œπ‘ πœƒ + 𝑦 βˆ™ π‘ π‘–π‘›πœƒ MAX 𝑦 𝑓′(𝜌3𝐷 , πœƒ, π‘Ÿ2𝐷 )2

2

+ βˆ’π‘₯ βˆ™ π‘ π‘–π‘›πœƒ + 𝑦 βˆ™ π‘π‘œπ‘ πœƒ

2

𝐷 = π‘Ÿ2𝐷2 = 𝜏~U(0, ) 2 12

Predicting Nearest Distance in 3D Space 𝑁𝑑_3𝐷 πœƒ1βˆ—

𝑁𝑑_2𝐷

πœƒ2βˆ—

D

πœ€ 𝐷 πœ€~U(0, ) 2

𝑁𝑑_3𝐷

 Minor influence on long distances observed in 2D image  Adjustment may be needed for short 2D distances

𝑛𝑑_3𝐷 =

πœƒ1βˆ— 𝑛𝑑_2𝐷 π‘‘πœƒ βˆ’πœƒ2βˆ— cosπœƒ πœƒ1βˆ— + πœƒ2βˆ—

𝐷 βˆ’πœ€ 2 βˆ— πœƒ1 = arctan 𝑛𝑑_2𝐷 𝐷 +πœ€ 2 βˆ— πœƒ2 = arctan 𝑛𝑑_2𝐷

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Microstructure Reconstruction Descriptor Pre-specification Target dispersion descriptors

Target geometry descriptors

Target composition descriptors

Dispersion Reconstruction: Optimization-based dispersion generator (based on Simulated Annealing algorithm) Geometry Reconstruction: Aggregate’s geometry generator

Composition Adjustment: Overlap elimination (optional) VF compensation

3D microstructure reconstructed images 14

Optimization-based dispersion generator

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Verification 1: Reconstruction of 3D structure

Target

2D descriptors (predict 3D descriptors)

2D cuts

Compare

Reconstruction

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Reconstruction & Verification Process 1. Create artificial 3D target structure

2. Take a thin layer (mimic the scan depth of SEM imaging)

3. Project the layer on the X-Y plane

Z Y Z X

0

0

1

0

1

0

0

1

1

0

0

1

1

1

0

1

3

2

1

X(Y)

0

Compare 6. Reconstruct 3D structure

5. Characterize 2D image & predict 3D descriptors

4. Binarize the grey scale image

Get grey scale image (mimic SEM image)

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CDF

Comparison of Predicted Descriptors vs. Real Descriptors

Nearest distance nd (voxel)

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Comparison of Reconstructions vs. Real Image 2D cross-section of target

Target

2-point correlation

𝐒2 (x1 , x2 ) = 𝐈 π‘₯1 𝐈(π‘₯2 )

2D cross-section of reconstruction

Reconstruction

Surface correlation 𝐅2 (x1 , x2 ) = 𝐌 π‘₯1 𝐌(π‘₯2 )

r

r Fillers

r r Fillers 19

Percent error 0.8%

Distance (voxel)

Surface correlation function

2-point correlation function

Comparison of Correlation Functions

Percent error 1.9%

Distance (voxel)

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CDF

Verification 2: 3D Polymer Composites

SEM image

Binary image

Nearest distance nd (voxel)

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Verification 2: Property Comparison Target:500X500

(1) Correlation Comparison

Reconstruction: 500X500X500

2D cuts

(2) Property Comparison: tan 𝜹

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Comparison of 2-point & Surface Correlation

23

tan 𝛿

Verification 2: Property Comparison

Frequency 24

Summary  Developed methods of characterization based on 2D SEM images  Developed fast 3D descriptor-based microstructure reconstruction algorithm and applied it on polymer nanocomposites

 Verified the accuracy of the developed method using two case studies  Reconstruct 3D polymer nanocomposites microstructures based on only one 2D microscopic image

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McCormick Robert R. McCormick School of Engineering and Applied Science

NU-GT Carbon Black Project

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Matrix-shielded Image for Aggregate Center Marking 2. Binary image

1. Grey scale image

3. Matrix-shielded image

Element-by-element multiplication

99

98

87

89

0

0

0

0

0

0

0

0

86

198

98

88

0

1

0

0

0

198

0

0

90

220

215

91

0

1

1

0

0

220

215

0

91

214

210

92

0

1

1

0

0

214

210

0

90

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93

0

0

0

0

0

0

0

0

.Γ—

=

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