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Non-equilibrium systems External flux d ~ characteristic size electro convection 10-4 self-organization (Dt 1/2~ ocean …… currents 104 characteristic size Desert vegetation patterns Chemical Turing patterns (Swinney) Experimental cell Striped & hexagonal patterns Labyrinthine pattern Animal coats & Turing patterns Simulated by RD equations Zebra & leopard Spiral patterns in m 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) 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 J.E.G. Wijnhoven and W.L. Vos, Science 281, 802 (1998) G. Subramania et al, Phys. Rev. B 63 235111 (2001) 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) Rayleigh–Bénard convection Devil’s Causeway 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 & patterns Equilibrium systems Non-equilibrium systems Short-range repulsion Long-range attraction Short-range activator Long-range inhibitor Crystal Turing pattern Evolution to equilibrium Frozen defects Non-potential effects: Dynamic regimes are possible Hexagonal & striped Turing patterns 0-hex stripe p-hex Double triplet: quasicrystal Two-wavelength Turing patterns A two-layer system with different diffusivities L. Yang, M. Dolnik, A.M.Zhabotinsky, and I.R.Epstein, PRL 88 208303 (2002) 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 Experiments of V.Steinberg Rolls, up- and down- hexagons Nucleation of hexagons in a defect core 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 P. G. Kevrekidis, A. R. Bishop, and K. Ø. Rasmussen Phys. Rev. E 65, 016122 (2002) Turbulent pattern 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(k) Labirynthine pattern develops from a single stripe when the inhibitor is fast (Meron et al) Spiral turbulence develops from a single stripe when the inhibitor is slow 3D instabilities in surface growth Snowflakes Multiple-exposure photograph of a dendrite advancing downwards Huang and Glicksman Acta Metall.29 717 (1981) Dendritic patterns in electrodeposition Bacterial colony