A New Approach for Microstructure

Characterization of Porous Fuel Cell Materials:

“Selection of Representative Volume Elements”

9th ASME International Conference on Fuel Cell Science,

Engineering and Technology – Washington, D.C.

Monday, 8th August 2011

E. A. Wargo, A. C. Hanna, A. Cecen,

S. R. Kalidindi and E. C. Kumbur

Electrochemical Energy Systems Laboratory

Mechanics of Microstructures Group

Department of Mechanical Engineering

Drexel University, Philadelphia PA

www.mem.drexel.edu/energy 2 µm3

Pore Body

Pore Throat

Solid

Solid

Void Space

Gostick et al, J. Power Sources, 2010

• Select the best possible small representative volume

elements (RVEs) that accurately reflect the structural

features and transport properties of the microstructure.

GOAL

A. Pore Network Modeling (PNM)

Limitations:

• Idealized and simplified

geometry

• May misrepresent true

material structure

B. Pore-scale modeling applied

to real microstructures

(e.g., LB or VOF methods)

Limitations:

• Computationally expensive

• Limited to only small volumes

• Accuracy depends on volume

selected

Material Structure

Full Dataset

Volume

Random

Volume

Selection of pore scale modeling domain is very important

Pore-Scale Modeling Efforts

• RVE should ideally capture:

– material structure in a smaller volume

– key structural and transport properties

Method of Approach

Dataset

Acquisition

Image Processing

& Segmentation

Full Material

Dataset

RVE Selection

from Full Dataset

Representative Volume Elements (RVEs)

RVE

WSVE Set

SVE

SVE

SVE

Micro-porous layer (MPL) selected as a test case.

Full Dataset

Volume

Statistical Volume

Element (SVE)

• We optimally select a set of weighted

statistical volume elements

(WSVE Set approach)

Adapted from: Iwai et al., J. of P. Sources, 2010

• 1.5 µm protective Pt layer

• SEM resolution = 9.8nm/pixel

• Slice thickness = 20nm

• 150 slices total

2 µm3

Dataset 1

Dataset 3 Dataset 4

Dataset 2

x

y

z

Adapted from:

Iwai et al., J. of P. Sources, 2010

Platinum

coating

Imaged

surface

Target

volume

Adapted from: Iwai et al., J. of P. Sources, 2010

Platinum coating

Imaged surface

Target volume

Data Acquisition via FIB-SEM (MPL Sample)

1. Image pre-processing

2. Remove gradients

• Biparabolic surface fit

3. Segment images

• Expected porosity, ε = 0.4 – 0.6 1, 2

1) Shoulder thresholding

(ε = 0.22 ± 0.12)

2) ISODATA method (ε = 0.42 ± 0.02)

3) Otsu’s method (ε = 0.43 ± 0.02)

4. Remove material islands

Raw Image Gradient Gradient-free

PoreFull Dataset of MPL1 Tang et al., J. Power Sources, 20072 Cho et al., J. Power Sources, 2010

shoulder

Image Processing & 3-D Reconstruction

5x8x2 µm

1x1x1 µm

• Diffusivity Coefficient

– Finite volume approximation

– Steady-state Fickian model

• Porosity

Structural Diffusivity Coefficient

Microstructure analysis tools were developed to determine

the key structural properties of the full MPL dataset.

• Internal surface area

– Count voxel faces of internal pore structure

• Connected Internal surface area– Excludes isolated pores

Pore Voxel

Metrics (Property) Analysis Tools

• Tortuosity, τ– Utilized shortest path

search algorithm

– Between start and end surfaces

pathdirect

path computed

Computed Shortest Tortuous Paths

in the Measured Microstructure

1 1.2 1.4 1.6 1.8 2 2.20

0.5

1

1.5

2

2.5

3

3.5

Pro

babili

ty D

ensity

Tortuosity

τeff = τavg = 1.42

St. Dev. = 0.37

Tortuosity,τ

Pro

babili

ty D

ensity

Single Random

Volume

1x1x1 µm

We can determine

spatial distribution of

tortuous paths, not

just one value.

Metrics (Property) Analysis Tools

– Detailed tortuosity

statistics

Metric Average Units

Porosity 0.4115 (fraction)

Internal Surface Area 23.77 (µm2/µm3)

Connected Internal Surface Area 22.30 (µm2/µm3)

Tortuosity 1.34

Structural Diffusivity Coefficient 0.225

Select 300 Random Volumes

Volume 1

Volume 2

Volume 300

Each 100 pixels3

1x1x1 µm

1x1x1 µm

1x1x1 µm

Apply metrics analysis tools to 300 random volumes

Calculated Metrics of Full Dataset

5x8x2 µm

Selection of RVEs

• N-point statistics

– Capture the structure

– Shape/size/spatial distribution

of phases

– Used 2-point statistics for

tested MPL (pore, solid phase)

2-Phase Image 2-pt Statistics Plot

Kalidindi et al. 2009

100x100x100 pixels

1 million 2-point

statistics (dimensions)

To select RVEs, we used the WSVE Set approach which is based

on n-point statistics and principal component analysis.

RVEs selected to

best reflect the 2-

point statistics

• Principal component analysis (PCA)

– Adequately represents in a lower dimensional space

– Identifies the most important statistics

– Speeds up RVE selection

Challenging Task

0

0.1

0.2

0.3

0.4

0.5

0 50 100 150 200Scala

r S

tati

sti

cal

Err

or

Window Size (pixels3)

Selected RVE Configuration:• Window size = 100 pixels3

• 5-member RVE Set0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 2 4 6 8 10Scala

r S

tati

sti

cal

Err

or

# Members in RVE Set

Exact

Location

of 5-Member

RVE Set

• Small size, computationally efficient

• 5 members, high accuracy

RVE Window Size & Members Selection

Full Dataset Volume

SVE5

SVE4

SVE3

SVE1

SVE2

• RVE Weights

– Weights allow the RVE Set to better reflect the full dataset

Selected RVE Sets and Weights

1-member

RVE Set3-member

RVE Set

5-member

RVE SetFull Dataset

Exact Location

of RVE member

– Any number of members can form an RVE Set

– Weighted SVE Set approach identifies the best set of volumes

– Optimally assigned to each SVE during the PCA procedure

Set RVESVE1

i

n

i

i w

SVE1

SVE1 SVE2

SVE3

SVE1

SVE2

SVE3

SVE4

SVE5

Weight

Full Dataset 0.4115 23.77 22.30 1.34 0.225

Weighted AveragesPorosity

(fraction)

Surface Area

(µm2/µm3)

Connected SA

(µm2/µm3) Tortuosity

Structural

Diff. Coeff.

5-member

RVE Set 0.4108 24.23 22.75 1.36 0.218

• Compare RVE Set metrics to the Full Dataset

Apply Metric Algorithms to Each Volume

RVE Set Validation & Comparison

Member 1 Member 2

Weight = 0.200 Wt. = 0.316 Wt. = 0.192 Wt. = 0.130 Wt. = 0.161

Member 3 Member 4 Member 55-member

RVE Set

%Error 0.18 1.93 2.04 1.13 3.28

Random selection of volumes will very likely yield a less accurate RVE

RVE Set Validation

0.35 0.4 0.45 0.50

50

100

150

200

250

5000 random sets

Normal fit

5-member WSVE Set

Ensemble average

Porosity

Fre

quency

Full Dataset

Volume 1

Volume 2

Volume 300

. . .

300 Random Volumes

5000 random

sets composed

of 5 members

. . . .n = 1

• Determine property averages for

each of the 5000 random sets

97% of the random sets

are less accurate

. . . .

. . . .

. . . .

n = 2 n = 5000

Vol 1

Vol 2

Vol 5

• Each of the 5 members is equally

weighted

Implementation Strategies for Different Problems

• For effective structural/transport properties

– RVE Set composed of a few small volumes

– Cubic RVEs

• For directionally specific information

– e.g., temperature, reactant or liquid water distribution

– Make the RVE longer in the direction of interest

Size and shape of the RVEs can be tailored to provide an

appropriate model domain

RVEs selected by this approach provide a small

(computationally efficient) and reliable model domain

for pore scale analysis of fuel cell materials.

Th

rou

gh

-pla

ne

• Use of “weighted statistical volume element” (WSVE) approach enables selection of small RVE Sets, which accurately capture the prominent features of a much larger material dataset

• RVE Set metrics match full MPL dataset very well– Less than 5% error

– Small volumes, only 100 pixels3 (~1 µm3)

• Random selection of volumes will very likely yield less accurate RVEs

• Matching 2-pt statistics ensures that key structural and transport properties of the material will be captured by the RVE Set

• RVEs selected by this approach provide a computationally efficient and reliable model domain for pore scale analysis

Conclusions

16

Acknowledgements

Future Work

• Dr. Craig L. Johnson (Centralized Research Facilities, Drexel) and

David M. Turner (Mechanics of Microstructures Group, Drexel)

• NSF Grant #1066623

• NSF Grant #DMR-0722845

• ED Award #P200A100145

• Investigation of a bi-layered GDL sample

• Pore scale modeling of transport in GDL using the selected RVEs

• Characterization of catalyst layer of PEM fuel cells

17

THANK YOU!

18

Spatial Variance of Microstructure in MPL

-1.30E-18

4.00E-04

8.00E-04

1.20E-03

1.60E-03

2.00E-03S

cala

r S

tati

sti

cal

Err

or

Dataset of Particular Window Size (pixels3)

Average Error for Dataset #1 (D1)

Average Error for Dataset #2 (D2)

Average Error for Dataset #3 (D3)

Average Error for Dataset #4 (D4)

100 pixels3

Window Size

= 50 pixels3

75 pixels3

• Four MPL datasets were obtained via FIB-SEM

• A comparison of their 2-point statistics shows that all four datasets

have very similar structures

Comparison of RVE Sets

Weighted AveragesPorosity

(fraction)

Surface Area

(µm2/µm3)

Connected SA

(µm2/µm3) Tortuosity

Structural

Diff. Coeff.

Full Dataset 0.4115 23.77 22.30 1.34 0.225

RVE Set:

1-member

(%Error)

0.4081

(0.84)

27.52

(15.76)

25.91

(16.19)

1.35

(0.74)

0.221

(2.13)

3-member

(%Error)

0.4110

(0.13)

24.91

(4.76)

23.35

(4.70)

1.37

(1.88)

0.222

(1.66)

5-member

(%Error)

0.4108

(0.18)

24.23

(1.93)

22.75

(2.04)

1.36

(1.13)

0.218

(3.28)

0

5

10

15

20

0 1 2 3 4 5 6

% E

rro

r

# Members in WSVE Set

Internal Surface Area (ISA)

Connected Internal Surface Area (CISA)

Porosity (ε)

Comparison of RVE Sets

0

1

2

3

4

5

0 1 2 3 4 5 6

% E

rro

r

# Members in WSVE Set

Structural Diffusivity Coefficient (K)

Tortuosity (τ)

Case #1 (low τ, high K ): Case #2 (high τ, low K ):

Pore

region

Material

A

B

A

B

Compare WSVE Sets and Random Sets

0.35 0.4 0.45 0.50

50

100

150

200

250

5000 random sets

Normal fit

5-member WSVE Set

Ensemble average

18 19 20 21 22 23 24 25 26 270

50

100

150

200

250

5000 random sets

Normal fit

5-member WSVE Set

Ensemble average

Connected Surface Area (µm2/µm3)

b)a)

Porosity

Fre

quency

1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.60

50

100

150

200

250

5000 random sets

Normal fit

5-member WSVE Set

Ensemble average

Tortuosity

c)

Fre

quency

0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.320

50

100

150

200

250

5000 random sets

Normal fit

5-member WSVE Set

Ensemble average

Structural Dif fusivity Coef ficient

d)

Down-shift

Stretch

Down-shift

)sin(/1 factor

dy

Inkson et al, Scripta Materialia 2001

dz

Ion

beam

Imaging

directionθ

θdy

Raw ImageAlignment )cos( dzdy

Cropped ImageStretch

Crop

Image

Image Processing