towards pc methods for the characterization of randomly...
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USC UQ Workshop • August 21-22, 2008
Towards PC Methods for theCharacterization of Randomly Structured
NanoComposites
Omar M. KnioJohns Hopkins University
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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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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 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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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