Non-equilibrium systems

External flux

self-organization

d ~ characteristic size

(D1/2~ characteristic size

electro convection …

…

ocean currents

10-4 104

Desert vegetation patterns

Chemical Turing patterns (Swinney)

Striped & hexagonal patterns Labyrinthine pattern

Experimental cell

Animal coats & Turing patterns

Simulated byRD equations

Zebra & leopard

Spiral patterns in range (CO oxidation on Pt, Imbihl & Ertl, 1995)

Polycrystalline surface 110 surface

STM image of Pt(110) – (1x2) showing the corrugated-iron structure; the inset shows a line scan across that structureK. Swamy, E. Bertel and I. Vilfan Surface Science, 425 L369 (1999)

Dewetting pattern J.Klein et al, PRL 86 4863 (2001)

I.Leizerson & S.G.Lipson

Patterns of crystal growth

The crystal growth sequence on an (001) cleavage plane in a BaSO4 solution

Pina et al, Nature 395, 483 (1998)

Colloidal assembly

G. Subramania et al, Phys. Rev. B 63 235111 (2001)

J.E.G. Wijnhoven and W.L. Vos, Science 281, 802 (1998)

Nanoscale deposition pattern

STM image of a periodic array of Fe islands nucleated on the dislocation network of a Cu bilayer on Pt(111)

Nanocluster arrays on interfaces

STM images of In nanoclusters on Si(111)

J.-L.Li et al, PRL 88 066101 (2002)

Molecular self-assembly on interfaces

Rows of pentacene on Cu(110) produced by a substrate-mediated repulsion

S.Lucas et al, PRL 88 028301 (2002)

Devil’s Causeway

Rayleigh–Bénard convection

Rayleigh–Bénard convection rolls,squares, hexagons, etc. Spiral defect chaos

Patterns of vibrating sand (Swinney)

Development of Turing pattern

Activator excited locally

Long-range inhibitor excited

Activator suppressed at neighboring locations

Periodic pattern starts to develop

activators & inhibitors

convection buoyancy heat transfer

optical cavity refractive index light intensity

solid film elastic stress surface tension

neuron membrane potential

ionic conductance

epidemics infectious agent immunity

Taylor column centrifugal force viscosity

Crystals & patternsEquilibrium systems Non-equilibrium systems

Short-range repulsionLong-range attraction

Short-range activatorLong-range inhibitor

Crystal Turing pattern

Evolution to equilibriumFrozen defects

Non-potential effects:Dynamic regimes are possible

Hexagonal & striped Turing patterns

0-hex -hexstripe

Double triplet: quasicrystal

Two-wavelength Turing patterns

L. Yang, M. Dolnik, A.M.Zhabotinsky, and I.R.Epstein, PRL 88 208303 (2002)

A two-layer system with different diffusivities

Two-wavelength superposition patterns

A two-layer system with strongly different diffusivities

L. Yang, M. Dolnik, A.M.Zhabotinsky, and I.R.Epstein, PRL 88 208303 (2002)

Resonant superlattice patterns

G. Dewel et al, 2001

Superlattice patterns: convection in vibrated layer

W. Pesch et al, PRL 85 4281 (2000)

Rayleigh–Bénard convection: complex patterns

Nucleation of hexagons in a defect core

Rolls, up- and down- hexagons Experiments of V.Steinberg

Two-frequency forced parametric waves

H.Arbell and J.Fineberg, PRE 65 036224 (2002)

Dynamics of spots in the plane

C.P.Schenk,M.Or-Guil,M.Bode,and H.-G.Purwins, Phys.Rev.Lett.78,3781 (1997)

Spirals and labyrinth patterns in BZ reaction

Action of incoherent light:

A spiral wave forms in the upper half of the same reactor, which is in the dark

A labyrinthine standing-wave pattern forms in the lower half of the reactor, which is illuminated with light pulsed at twice the natural frequency of the reaction

Chemical waves in the BZ reaction. Top: target patterns in a thin film of reagent (1.5 mm). Bottom: spiral waves in reagent similar to above except less acidic. Both sequences from left to right are at 60 s intervals. Reprinted with permission from: Winfree, A. T. Prog. Theor. Chem.

1978, 4, 1.

Spiral wave patterns in CGLE

Frustrated pattern

Turbulent pattern

P. G. Kevrekidis, A. R. Bishop, and K. Ø. Rasmussen Phys. Rev. E 65, 016122 (2002)

Spiral wave and its break-up

M. Baer, M. OrGuil, PRL 82 1160 (1999)

Instability of a reaction front

Boundary dynamics: cn= cn(v) + f() (Meron et al)

Labirynthine pattern develops from a single stripe when the inhibitor is fast

Spiral turbulence develops from a single stripe when the inhibitor is slow

3D instabilities in surface growth

Snowflakes

Dendritic patterns in electrodeposition

Bacterial colony

Multiple-exposure photograph of a dendrite advancing downwards Huang and Glicksman Acta Metall.29 717 (1981)