Stress induced instabilities in material science and biology C. Misbah, CNRS and Univ. J. Fourier Grenoble I Los Angeles October. 2005 Oct. 18th 2005 IPAM Oct. 2005

Uniaxial stress Biaxial stress Oct. 18th 2005 IPAM Oct. 2005

Questions 1) Front growth or recession? Solid 2) Planar front stable? Melt, vapor 1) Front growth or recession? Solid 2) Planar front stable? Solid 3) Ultimate stage? Oct. 18th 2005 IPAM Oct. 2005

time Coarsening or fixed ? 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. Toussaint Phys. 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. Toussaint Phys. Rev. Lett. 93, 238501 (2004) Oct. 18th 2005 IPAM Oct. 2005

Stylolithes formation (ex:limestone, queensland, Australia) dissolution Stress- induced corrugation Oct. 18th 2005 IPAM Oct. 2005

ATG instability comes to life! Actin-assisted cell motility Oct. 18th 2005 IPAM Oct. 2005

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

at bead/layer interface Actin Polymerization at bead/layer interface P. Peyla, C. Misbah, preprint (2005) Oct. 18th 2005 IPAM Oct. 2005

Physical picture of the instability solid Oct. 18th 2005 IPAM Oct. 2005

Physical picture of the instability solid Instability unavoidable Oct. 18th 2005 IPAM Oct. 2005

Typical lengthscales of the pattern Strain due to protuberance Strain energy gain Surface energy loss Oct. 18th 2005 IPAM Oct. 2005

Oct. 18th 2005 IPAM Oct. 2005

For For Oct. 18th 2005 IPAM Oct. 2005

Spencer Meiron (steady-states, 1994) Yang and Srolovitz (1993) Kassner and Misbah (1994) Spencer Meiron (steady-states, 1994) Oct. 18th 2005 IPAM Oct. 2005

Sound speed, finite interface width Surface tension effect Stress effect Close to a crack Sound speed, finite interface width Oct. 18th 2005 IPAM Oct. 2005

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) Oct. 18th 2005 IPAM Oct. 2005

Phase-field approach: singularity? Oct. 18th 2005 IPAM Oct. 2005

Phase field models Sharp interface Diffuse interface Oct. 18th 2005 IPAM Oct. 2005

Reference state 1 : stress=0 when strain=0 Ref. state 2: Zero strain is no stress free Oct. 18th 2005 IPAM Oct. 2005

If ref. state: strain is zero when when stress is If equilibrium: If ref. state: strain is zero when when stress is BC: For z=0 There is no stress a all! Because Oct. 18th 2005 IPAM Oct. 2005

If ref. state: strain is zero when when stress is If equilibrium: If ref. state: strain is zero when when stress is BC: Plays the role of a unixial stress Oct. 18th 2005 IPAM Oct. 2005

Gas (liq., vac.) Diffuse Solid Viewed as the work of external force Oct. 18th 2005 IPAM Oct. 2005

gas solid Oct. 18th 2005 IPAM Oct. 2005

Sharp interface limit, asymptotics Is a Singular perturbation Oct. 18th 2005 IPAM Oct. 2005

Outer solution (regular) Inner solution (singular) Matching inner-outer solutions Oct. 18th 2005 IPAM Oct. 2005

Outer solutions: + Lamé Inner solutions: (1) Zeroth order: (at all orders) And BC (2) First order: 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 » 5) Final groove velocity Oct. 18th 2005 IPAM Oct. 2005

Stress accumulation in the grooves, fracture? 1D dynamics Stress accumulation in the grooves, fracture? Oct. 18th 2005 IPAM Oct. 2005

Coarsening Mass flux Oct. 18th 2005 IPAM Oct. 2005

Exceptions in 1d with no noise! Coarsening Mass flux If non conserved: Exceptions in 1d with no noise! Oct. 18th 2005 IPAM Oct. 2005

Volume variation ~ surface Contact coalescence Volume variation ~ surface Oct. 18th 2005 IPAM Oct. 2005

Coarsening dominated by elasticity conserved nonconserved In progress (cycloids show driving force independent of R) Agreement with the experiment of Koehn et al.,Geochimica and Cosmochimica Acta, 2004 Oct. 18th 2005 IPAM Oct. 2005

Heteroepitaxy Oct. 18th 2005 IPAM Oct. 2005

Partial relaxation Stressed film Oswald ripening? Narrow size observed InGaAs/GaAs, InAs/inP, InAS/InGaAs, …... Oct. 18th 2005 IPAM Oct. 2005

=step energy/elastic energy Monoatomic films substrate Elastic monopoles coverage =step energy/elastic energy Marchenko-Parshin (1980) and Marchenko 1992 Alerhand et al. 1989. 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

1) In the pure thermodynamical limit, does coarsening persist? 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? Oct. 18th 2005 IPAM Oct. 2005