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


Uniaxial stress

Biaxial stress


Solidxx xx

Solid

QuestionsMelt, vapor

1) Front growth or recession?

2) Planar front stable?

3) Ultimate stage?


timeCoarsening or fixed ?

t ?

Stranski-Krastanov?

Size selection?Perpetual coarsening?


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.


Swelling or shrinkage of gels

Clamped at bottom Gel swelling

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


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)


Stylolithes formation

dissolution

Stress-inducedcorrugation

(ex:limestone, queensland,Australia)


ATG instability comes to life!

Actin-assisted cell motility


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

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


Actin Polymerization at bead/layer interface

1 E

E2

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

1 E


Physical picture of the instability

solidxxxx

zz

)sin(dSdSx

)cos(dSdS z

tdF


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


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


hgE

][

2

20

hhgE lsg )(

0g

0g

c mmg

c

12231120 101011010 gE cc

bar 10 c


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


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

Spencer Meiron (steady-states, 1994)


][2

2

2

1nnttE

Surface tension effectStress effect

RRRE

20

Close to a crack

R

R1

Sound speed, finite interface width


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)


Phase-field approach: singularity?


Phase field models

1 2 1 2

Phase field

Sharp interface Diffuse interface


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


)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!


)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


uuuuu iijjiiijijpf

2

sppp 00 Viewed as the work ofexternal force

]21),([ )(

22 ijufrdF

Solid

Gas (liq., vac.)

Diffuse

1

0


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


Sharp interface limit, asymptotics

0

Is a Singular perturbation ]

21),([ )(

22 ijufrdF


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

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

Outer solution (regular)

Inner solution (singular)

rMatching inner-outer solutions


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


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


1D dynamics

Stress accumulation in the grooves, fracture?


CoarseningMass flux

jvdt

dR

2/1)/1( RRj

.2 Constdt

dRR 3/1tR


CoarseningMass flux

jvdtdR

Rj /1

.2 Constdt

dRR 3/1tR

If non conserved:

1/Rvdt

dR

2/1tR

Exceptions in 1d with no noise!


Contact coalescence

tRCtedt

dRR

dt

dRS

dt

dV , , , 2

3

Volume variation ~ surface


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)


Heteroepitaxy


Oswald ripening?

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

Stressed filmPartial relaxation


substrate

Monoatomic films

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

eCCae 21/

)sin(

coverage 21/CC =step energy/elastic energy

Elastic monopoles


(Tersoff, Villain, Müller, Kern….)

Coasening should be inevitable!

Gain in elastic energy and in surface energy


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?

stress induced instabilities in material science and biology

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