stress induced instabilities in material science and biology

Oct. 18th 2005 IPAM Oct. 2005

Stress induced instabilities inmaterial science and biology

Los Angeles October. 2005

C. Misbah, CNRS and Univ. J. Fourier Grenoble I

Oct. 18th 2005 IPAM Oct. 2005

Uniaxial stress

Biaxial stress

Oct. 18th 2005 IPAM Oct. 2005

Solidxx xx

Solid

QuestionsMelt, vapor

1) Front growth or recession?

2) Planar front stable?

3) Ultimate stage?

Oct. 18th 2005 IPAM Oct. 2005

timeCoarsening or fixed ?

t ?

Stranski-Krastanov?

Size selection?Perpetual coarsening?

Oct. 18th 2005 IPAM Oct. 2005

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.

Oct. 18th 2005 IPAM Oct. 2005

Swelling or shrinkage of gels

Clamped at bottom Gel swelling

T. Tanaka, H. Tanaka, Kawasaki, Sekimoto, Onuki

Oct. 18th 2005 IPAM Oct. 2005

Misbah C., Renard F., Gratier J.P., Kassner K., Geoph. Res. Lett., 31, L6618 (2004).J. Schmittbuhl, F. Renard, J. P. Gratier, and R. ToussaintPhys. Rev. Lett. 93, 238501 (2004)

Oct. 18th 2005 IPAM Oct. 2005

Misbah C., Renard F., Gratier J.P., Kassner K., Geoph. Res. Lett., 31, L6618 (2004).J. Schmittbuhl, F. Renard, J. P. Gratier, and R. ToussaintPhys. Rev. Lett. 93, 238501 (2004)

Oct. 18th 2005 IPAM Oct. 2005

Stylolithes formation

dissolution

Stress-inducedcorrugation

(ex:limestone, queensland,Australia)

Oct. 18th 2005 IPAM Oct. 2005

ATG instability comes to life!

Actin-assisted cell motility

Oct. 18th 2005 IPAM Oct. 2005

L. A. Cameron et al., PNAS, 96, 4908 (1999).

J. van der Gucht et al. PNAS, 102, 7847 (2005)

Oct. 18th 2005 IPAM Oct. 2005

Actin Polymerization at bead/layer interface

1 E

E2

P. Peyla, C. Misbah, preprint (2005)

1 E

Oct. 18th 2005 IPAM Oct. 2005

Physical picture of the instability

solidxxxx

zz

)sin(dSdSx

)cos(dSdS z

tdF

Oct. 18th 2005 IPAM Oct. 2005

Physical picture of the instability

solidxxxx

zz

)sin(dSdSx

)cos(dSdS z

tdF

][

])[sin()cos(

)sin()cos(

xxzz

xxzz

zzzxxxt

dS

dS

dSdSdF

Instability unavoidable

Oct. 18th 2005 IPAM Oct. 2005

Typical lengthscales of the pattern

h

Strain due to protuberance h

Ee 0

zzxx)0()0(

0

Strain energy gain hh

E

2

0

Surface energy loss hh

LL

20 )(

0L

L

hhhhE

G

][

2

20

Oct. 18th 2005 IPAM Oct. 2005

hgE

][

2

20

hhgE lsg )(

0g

0g

c mmg

c

12231120 101011010 gE cc

bar 10 c

Oct. 18th 2005 IPAM Oct. 2005

hgE

][

2

20

hhgE lsg )(

0g

0g

c mmg

c

222

0

)//()/(

aabaaE

E

GeSi /For 50/ab

%4/ aa

nm 20 GaAsInGaAs /For

%7/ aa

Oct. 18th 2005 IPAM Oct. 2005

Yang and Srolovitz (1993)Kassner and Misbah (1994)

Spencer Meiron (steady-states, 1994)

Oct. 18th 2005 IPAM Oct. 2005

][2

2

2

1nnttE

Surface tension effectStress effect

RRRE

20

Close to a crack

R

R1

Sound speed, finite interface width

Oct. 18th 2005 IPAM Oct. 2005

Simple ansatz

Cycloid (Chui,Gao, 1993)

Double cycloid (conformal mapping, Kassner, Misbah, 2001)

Multicycloids (Kohlert, Kassner, Misbah, 2003)

, derivation of groove velocities

(good agreement with numerics with few modes)

Oct. 18th 2005 IPAM Oct. 2005

Phase-field approach: singularity?

Oct. 18th 2005 IPAM Oct. 2005

Phase field models

1 2 1 2

Phase field

Sharp interface Diffuse interface

Oct. 18th 2005 IPAM Oct. 2005

Reference state

uu ijijkkij 2

Reference state 1 : stress=0 when strain=0

Ref. state 2: uij

)0(

uuu ijijij

)0(

'

)'')0()0( (2)( uuuu ijijijkkkkij

Zero strain is no stress free

Oct. 18th 2005 IPAM Oct. 2005

)2) ( ( uuuu eq

ijijij

eq

kkkk

eq

ijij

If equilibrium: ij

eq

ij p0

ij

eq

ij

pu 32

0

If ref. state: strain is zero when when stress is

ijsp

0 ij

eq

ij

ppu s

320 0

BC:

0u For z=0 0uxx

00p

0uzz

Because 0 zz There is no stress a all!

Oct. 18th 2005 IPAM Oct. 2005

)2) ( ( uuuu eq

ijijij

eq

kkkk

eq

ijij

If equilibrium: ij

eq

ij p0

ij

eq

ij

pu 32

0

If ref. state: strain is zero when when stress is

ijsp

0 ij

eq

ij

ppu s

320 0

BC: 0' u xx

2

0p

u szz

0 zz

22

0p

sxx Plays the role of a unixial stressp

s0

Oct. 18th 2005 IPAM Oct. 2005

uuuuu iijjiiijijpf

2

sppp 00 Viewed as the work ofexternal force

]21),([ )(

22 ijufrdF

Solid

Gas (liq., vac.)

Diffuse

1

0

Oct. 18th 2005 IPAM Oct. 2005

ff hhf01

)](1[)(

]21),([ )(

22 ijufrdF

0)0( ,1)1( , )23(2

hhh

f1

f0solid gas

F

t 0

iuF

uuuuu iijjiiijijpf

2

)1(22

Oct. 18th 2005 IPAM Oct. 2005

Sharp interface limit, asymptotics

0

Is a Singular perturbation ]

21),([ )(

22 ijufrdF

Oct. 18th 2005 IPAM Oct. 2005

......),,(),,(),,( 10 tsrtsrtsr

......),,(),,(),,( 10 tststs

Outer solution (regular)

Inner solution (singular)

rMatching inner-outer solutions

Oct. 18th 2005 IPAM Oct. 2005

Outer solutions: constant + Lamé

Inner solutions:

(1) Zeroth order: )1(2' 000

)]tanh(1[210 (at all orders)

And BC 0 , ntnnp

(2) First order: )()L(01

G

][2

2

2

1nnttE

kV n

r

Oct. 18th 2005 IPAM Oct. 2005

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 »

0y

5) Final groove velocity 1V

Oct. 18th 2005 IPAM Oct. 2005

1D dynamics

Stress accumulation in the grooves, fracture?

Oct. 18th 2005 IPAM Oct. 2005

CoarseningMass flux

jvdt

dR

2/1)/1( RRj

.2 Constdt

dRR 3/1tR

Oct. 18th 2005 IPAM Oct. 2005

CoarseningMass flux

jvdtdR

Rj /1

.2 Constdt

dRR 3/1tR

If non conserved:

1/Rvdt

dR

2/1tR

Exceptions in 1d with no noise!

Oct. 18th 2005 IPAM Oct. 2005

Contact coalescence

tRCtedt

dRR

dt

dRS

dt

dV , , , 2

3

Volume variation ~ surface

Oct. 18th 2005 IPAM Oct. 2005

Coarsening dominated by elasticity

2/1tR conserved

tR nonconserved

In progress

Agreement with the experiment of Koehn et al.,Geochimica and Cosmochimica Acta, 2004

(cycloids show driving force independent of R)

Oct. 18th 2005 IPAM Oct. 2005

Heteroepitaxy

Oct. 18th 2005 IPAM Oct. 2005

Oswald ripening?

Narrow size observed InGaAs/GaAs, InAs/inP, InAS/InGaAs, …...

Stressed filmPartial relaxation

Oct. 18th 2005 IPAM Oct. 2005

substrate

Monoatomic films

Marchenko-Parshin (1980) and Marchenko 1992Alerhand et al. 1989.

eCCae 21/

)sin(

coverage 21/CC =step energy/elastic energy

Elastic monopoles

Oct. 18th 2005 IPAM Oct. 2005

(Tersoff, Villain, Müller, Kern….)

Coasening should be inevitable!

Gain in elastic energy and in surface energy

Oct. 18th 2005 IPAM Oct. 2005

Open questions

1) In the pure thermodynamical limit, does coarseningpersist? 2) Is it thermodynamical or kinetical? In the first casewhich 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?