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Non-equilibrium systems External flux self-organization d ~ characteristic size ( D 1/2 ~ characteristic size electro convection …… ocean currents 10 -4 10 4

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Desert vegetation patterns

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Chemical Turing patterns (Swinney) Striped & hexagonal patterns Labyrinthine pattern Experimental cell

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Animal coats & Turing patterns Simulated by RD equations Zebra & leopard

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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 structure K. Swamy, E. Bertel and I. Vilfan Surface Science, 425 L369 (1999)

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Dewetting pattern J.Klein et al, PRL 86 4863 (2001) I.Leizerson & S.G.Lipson

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Patterns of crystal growth The crystal growth sequence on an (001) cleavage plane in a BaSO4 solution Pina et al, Nature 395, 483 (1998)

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Colloidal assembly G. Subramania et al, Phys. Rev. B 63 235111 (2001) J.E.G. Wijnhoven and W.L. Vos, Science 281, 802 (1998)

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

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Nanocluster arrays on interfaces STM images of In nanoclusters on Si(111) J.-L.Li et al, PRL 88 066101 (2002)

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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)

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Devil’s Causeway Rayleigh–Bénard convection

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

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Patterns of vibrating sand (Swinney)

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Development of Turing pattern Activator excited locally Long-range inhibitor excited Activator suppressed at neighboring locations Periodic pattern starts to develop

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activators & inhibitors convectionbuoyancyheat transfer optical cavityrefractive indexlight intensity solid filmelastic stresssurface tension neuronmembrane potentialionic conductance epidemicsinfectious agentimmunity Taylor columncentrifugal forceviscosity

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Crystals & patterns Equilibrium systemsNon-equilibrium systems Short-range repulsion Long-range attraction Short-range activator Long-range inhibitor CrystalTuring pattern Evolution to equilibrium Frozen defects Non-potential effects: Dynamic regimes are possible

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Hexagonal & striped Turing patterns 0-hex -hex stripe

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Double triplet: quasicrystal

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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

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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)

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Resonant superlattice patterns G. Dewel et al, 2001

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Superlattice patterns: convection in vibrated layer W. Pesch et al, PRL 85 4281 (2000)

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Rayleigh–Bénard convection: complex patterns Nucleation of hexagons in a defect core Rolls, up- and down- hexagons Experiments of V.Steinberg

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Two-frequency forced parametric waves H.Arbell and J.Fineberg, PRE 65 036224 (2002)

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Dynamics of spots in the plane C.P.Schenk,M.Or-Guil,M.Bode,and H.-G.Purwins, Phys.Rev.Lett.78,3781 (1997)

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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

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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.

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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)

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Spiral wave and its break-up M. Baer, M. OrGuil, PRL 82 1160 (1999)

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Instability of a reaction front

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Boundary dynamics: c n = c n (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

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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)

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