P1249945246LydhX

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Stress induced instabilities in material science
and biology
C. Misbah, CNRS and Univ. J. Fourier Grenoble I
Los Angeles October. 2005
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Uniaxial stress
Biaxial stress
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Questions
Melt, vapor
1) Front growth or recession?
Solid
2) Planar front stable?
Solid
3) Ultimate stage?
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time
Coarsening or fixed ?
Stranski-Krastanov?
Size selection?
Perpetual coarsening?
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ATG Intability (Asaro-Tiller,1972 -Grinfeld, 1986)
Quantum dots formation
pyramid-shaped quantum dots grown from indium,
gallium, and arsenic. Each dot is about 20
nanometers wide and 8 nanometers in height.
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Swelling or shrinkage of gels
Clamped at bottom
Gel swelling
T. Tanaka, H. Tanaka, Kawasaki, Sekimoto, Onuki
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Misbah C., Renard F., Gratier J.P., Kassner K.,
Geoph. Res. Lett., 31, L6618 (2004). J.
Schmittbuhl, F. Renard, J. P. Gratier, and R.
Toussaint Phys. Rev. Lett. 93, 238501 (2004)
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Misbah C., Renard F., Gratier J.P., Kassner K.,
Geoph. Res. Lett., 31, L6618 (2004). J.
Schmittbuhl, F. Renard, J. P. Gratier, and R.
Toussaint Phys. Rev. Lett. 93, 238501 (2004)
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Stylolithes formation
(exlimestone, queensland, Australia)
dissolution
Stress- induced corrugation
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ATG instability comes to life!
Actin-assisted cell motility
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L. A. Cameron et al., PNAS, 96, 4908 (1999).
J. van der Gucht et al. PNAS, 102, 7847 (2005)
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Actin Polymerization at bead/layer interface
P. Peyla, C. Misbah, preprint (2005)
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Physical picture of the instability
solid
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Physical picture of the instability
solid
Instability unavoidable
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Typical lengthscales of the pattern
Strain due to protuberance
Strain energy gain
Surface energy loss
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(No Transcript)
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For
For
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Yang and Srolovitz (1993) Kassner and Misbah
(1994)
Spencer Meiron (steady-states, 1994)
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Surface tension effect
Stress effect
Close to a crack
Sound speed, finite interface width
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Simple ansatz
Cycloid (Chui,Gao, 1993)
Double cycloid (conformal mapping, Kassner,
Misbah, 2001)
, derivation of groove velocities
Multicycloids (Kohlert, Kassner, Misbah, 2003)
(good agreement with numerics with few modes)
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Phase-field approach singularity?
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Phase field models
Sharp interface
Diffuse interface
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Reference state
Reference state 1 stress0 when strain0
Ref. state 2
Zero strain is no stress free
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If equilibrium
If ref. state strain is zero when when stress is
BC
For z0
There is no stress a all!
Because
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If equilibrium
If ref. state strain is zero when when stress is
BC
Plays the role of a unixial stress
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Viewed as the work of external force
Gas (liq., vac.)
Diffuse
Solid
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gas
solid
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Sharp interface limit, asymptotics
Is a Singular perturbation
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Outer solution (regular)
Inner solution (singular)
Matching inner-outer solutions
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Lamé
Outer solutions
Inner solutions
(1) Zeroth order
(at all orders)
And BC
(2) First order
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Main Results
1) Increase of the amplitude without bound
2) Phase-field supresses finite time singularity
3) Perpetual coarsening
4) A finite interface width
 Yield stress 
5) Final groove velocity
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1D dynamics
Stress accumulation in the grooves, fracture?
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Coarsening
Mass flux
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Coarsening
Mass flux
If non conserved
Exceptions in 1d with no noise!
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Contact coalescence
Volume variation surface
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Coarsening dominated by elasticity
conserved
nonconserved
(cycloids show driving force independent of R)
In progress
Agreement with the experiment of Koehn et
al.,Geochimica and Cosmochimica Acta, 2004
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Heteroepitaxy
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Partial relaxation
Stressed film
Oswald ripening?
Narrow size observed InGaAs/GaAs, InAs/inP,
InAS/InGaAs, ...
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Monoatomic films
substrate
Elastic monopoles
step energy/elastic energy
coverage
Marchenko-Parshin (1980) and Marchenko
1992 Alerhand et al. 1989.
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(Tersoff, Villain, Müller, Kern.)
Coasening should be inevitable!
Gain in elastic energy and in surface energy
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Open questions
1) In the pure thermodynamical limit, does
coarsening persist? 2) Is it thermodynamical or
kinetical? In the first case which ingredients
would supress coarsening? Coarsening is subtle
(Politi, Misbah, Phys. Rev. Lett. 2004) 3) In
Dynamical simulations coarsening stops or
slowed down? 4) Some systems QD form under
compression but not under tension! 5) Cell
motility bead-gel friction decisive?