Towards PC Methods for the Characterization of Randomly ...cadmus.usc.edu/PCE-Workshop08/presentations/Knio.pdf · Final Product: NiAl intermetallic Al. ... CA 7 Modeling of Transient

USC UQ Workshop • August 21-22, 2008

Towards PC Methods for theCharacterization of Randomly Structured

NanoComposites

Omar M. KnioJohns Hopkins University

USC UQ Workshop• LA, CA 2

Caveat Lector

• “Best effort” to solve a very difficult engineeringproblem

• “Toward PCE” is not redundant• Best == “guess” the solution (“because you’re

compelled to check your answer”)• Emphasis on ideas; no theory, no algorithms

(yet…)

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Outline

• Introduction– Uniformly layered material– Randomly layered composites

• Modeling approaches:– Deterministic model for uniformly layered materials– Stochastic model for randomly layered composites

• Results• Discussion and Conclusions

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Uniformly Layered Material

• typically vapor deposited• controllable reaction properties: propagation and ignition

Ni

Initial reactants: Ni and Al

Final Product: NiAl intermetallic

Al

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Control of Reaction Properties: Bilayer

0

5

10

15

20

10 100 1000

Barbee and Weihs - LLNL

Rea

ctio

n V

elo

city

(m

/s)

Multilayer Period (nm)

JHU

Barbee and Weihs @ LLNL, 1996

Weihs et al.@ JHU, 1997

Bilayer Thickness (nm)

Rea

ctio

n Ve

loci

ty (

m/s

)

Control throughcomposition,annealingdiluents, etc… isalso possible

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Randomly Layered Composites

Examples of rolled Ni/Al structures

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Modeling of Transient Reaction Properties

• Elementary continuum description!• Model is deceptively simple -- based on (coupled)

evolution equations of concentration and energy:

)(

)(

CDdt

dC

dt

dQTk

dt

dh

!•!=

+!•!="

x

y

• In the case of uniformly layered systems, computationaldomain is simple as well!

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Assumptions; Experimental inputs• Fast, diffusion-limited reaction• Thermal conductivity is independent of

temperature and composition• Fickian diffusion between reactants, described

using Arrhenius diffusivity:D = Do exp(-E/RT)

• Do, E, reaction heat are obtained fromexperimental observations

• Other thermophysical properties obtained fromfundamental measurements or calculations

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Predictions: Effect of Bilayer and Annealing

0

2

4

6

8

10

12

0 50 100 150 200 250

Experiments: As-deposited

Experiments: 6hrs @ 150°C

Predictions: 6hrs @ 150°C

Predictions: As-deposited

Rea

ctio

n V

eloci

ty (

m/s

)

Multilayer Period (nm)

2w = 2.4nm

2w = 6.3nm

Annealingincreases

intermixingand slows the

reaction

Bilayer Thickness (nm)

Reac

tion

Vel

ocity

(m/s

)

Kinetics

Ther

mo

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Modeling randomly (non-uniformly) layeredcomposites

Mechanically-processed

Bilayer Frequency vs. Bilayer Thickness

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 2000 4000 6000 8000 10000 12000 14000 16000 18000

Bilayer Thickness (nm)

Bil

ay

er

Fre

qu

en

cy

n=4

n=6

n=8

Controllable distributions

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Approach• Probabilistic approach based on exploiting

measured pdfs of bilayer distribution. Assumptions:– include underlying assumptions as in deterministic model

for uniformly layered RMs– assume thermal profile is uniform in the front, which is

normal to propagation direction• Computational model:

– based on discretizing the pdf using an appropriatenumber of bins (need to worry about statisticalconvergence);

– solve for transient reaction in coupled layers with bilayerscorresponding to discretized pdf, and averaged energyequation that accounts for heat release in representativebilayer and for the corresponding density.

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Illustration

Reaction Front

- Molecular diffusion process issimulated for representativelayer in each bin independently

- Solutions are coupled throughsection averaged energyequation that accounts for theprobability density of individualbins

- KL spectrum: crushed!Justification: asymptotics

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Consistency of Stochastic Model

Discretization of experimental pdfs for rolled Al-Pd nanolaminates

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Stochastic Convergence

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Stochastic Predictions:Rolled Al-Pd Nanolaminates

• Reasonable agreement with experimental observations• Construction of simple design correlations for material behavior

– Random by design!

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Dual Bilayer Ni-Al Composites

• Fabricated bydepositing two stakcsof different layering

• Use as a limitingexample of non-uniformly layered RM

Small Bilayers (~20nm)

Large Bilayers (~110nm)

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Predictions and Data

Coarse

Fine

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Discussion

• Reasonable agreement between modelpredictions and fine composite; disagreement forcoarse structure

• Origin of disagreement can be traced to theunderlying assumption in the probabilistic modelthat the temperature is uniform across layers

• Analysis consequently points to importance ofmulti-dimensional propagation regimes thatdepend on lengthscales of KL spectrum

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Acknowledgments

Work supported by NIST Advanced Technology Programand ONR MURI

Contributors: T. Weihs, T. Hufnagel

M. Snyder (USNA), J. Trenkle, G. Fritz, R. Knepper, N.Walker, D. van Heerden, A. Mann, E. Besnoin, A. Gavens,S. Jayaraman, J. Spey