Descriptor-based Microstructure Characterization

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

𝜃

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, 𝒓𝒅𝐯𝐚𝐫 rd

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

𝑟𝑐 = 𝑎𝑏 𝑎 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

4

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 × 𝑟𝑐 , D is filler diameter *

2rc

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

5

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

6

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 “single aggregate cluster”(marked by green circles), to calculate the particle shape descriptors.

8

Summary of Microstructure Descriptors Category

Descriptors

Predefined descriptors

𝑽𝑭 𝒏𝐝_𝟐𝑫

2D descriptors characterized from 2D images

𝝆𝟐𝑫 𝑨𝒆

𝑵𝟐𝑫 𝒏𝐝_𝟑𝑫 3D descriptors to be predicted

𝝆𝟑𝑫 𝑽𝒆 𝑵𝟑𝑫 9

Predicting Aggregate Number in 3D Space

Num = 𝑁2𝐷 ×

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

10

Predicting Aspect Ratio in 3D Space 𝑅3𝐷

𝑟3𝐷

90°

D 𝑅2𝐷 𝑟2𝐷

0° 2D projection: 2× 𝑅2𝐷

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

𝜌2𝐷 = 𝑓 𝜌3𝐷 , 𝜃, 𝑟2𝐷 =

𝑥 ∙ 𝑐𝑜𝑠𝜃 + 𝑦 ∙ 𝑠𝑖𝑛𝜃 MAX 𝑥 𝜌3𝐷2

2

+ −𝑥 ∙ 𝑠𝑖𝑛𝜃 + 𝑦 ∙ 𝑐𝑜𝑠𝜃

2

= 𝑟2𝐷2

𝑟2𝐷 11

Cont. Predicting Aspect Ratio in 3D Space

90°

𝜃max 𝜏

𝑦

𝜀

D

𝑟 𝑅

0° 2D projection: 2× 𝑅2𝐷

2D-to-3D prediction:

𝝆𝟑𝑫_𝒑𝒓𝒆𝒅𝒊𝒄𝒕

 𝑥

𝑅2𝐷

1 = ∗ 𝜃

𝜃∗

𝑓 −1 𝜌3𝐷 , 𝜃, 𝑟2𝐷 𝑑𝜃

Where 𝜽∗ is from:

0

Consideration of layer thickness constraints & randomness of spatial location:

𝑥 ∙ 𝑐𝑜𝑠𝜃 + 𝑦 ∙ 𝑠𝑖𝑛𝜃 MAX 𝑦 𝑓′(𝜌3𝐷 , 𝜃, 𝑟2𝐷 )2

2

+ −𝑥 ∙ 𝑠𝑖𝑛𝜃 + 𝑦 ∙ 𝑐𝑜𝑠𝜃

2

𝐷 = 𝑟2𝐷2 = 𝜏~U(0, ) 2 12

Predicting Nearest Distance in 3D Space 𝑁𝑑_3𝐷 𝜃1∗

𝑁𝑑_2𝐷

𝜃2∗

D

𝜀 𝐷 𝜀~U(0, ) 2

𝑁𝑑_3𝐷

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

𝑛𝑑_3𝐷 =

𝜃1∗ 𝑛𝑑_2𝐷 𝑑𝜃 −𝜃2∗ cos𝜃 𝜃1∗ + 𝜃2∗

𝐷 −𝜀 2 ∗ 𝜃1 = arctan 𝑛𝑑_2𝐷 𝐷 +𝜀 2 ∗ 𝜃2 = arctan 𝑛𝑑_2𝐷

13

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

15

Verification 1: Reconstruction of 3D structure

Target

2D descriptors (predict 3D descriptors)

2D cuts

Compare

Reconstruction

16

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)

17

CDF

Comparison of Predicted Descriptors vs. Real Descriptors

Nearest distance nd (voxel)

18

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

Target

2-point correlation

𝐒2 (x1 , x2 ) = 𝐈 𝑥1 𝐈(𝑥2 )

2D cross-section of reconstruction

Reconstruction

Surface correlation 𝐅2 (x1 , x2 ) = 𝐌 𝑥1 𝐌(𝑥2 )

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)

20

CDF

Verification 2: 3D Polymer Composites

SEM image

Binary image

Nearest distance nd (voxel)

21

Verification 2: Property Comparison Target:500X500

(1) Correlation Comparison

Reconstruction: 500X500X500

2D cuts

(2) Property Comparison: tan 𝜹

22

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

25

McCormick Robert R. McCormick School of Engineering and Applied Science

NU-GT Carbon Black Project

26

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

92

94

93

0

0

0

0

0

0

0

0

.×

=

27