non-equilibrium systems external flux self-organization d ~ characteristic size ( d 1/2 ~...
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Non-equilibrium systems
External flux
self-organization
d ~ characteristic size
(D1/2~ characteristic size
electro convection …
…
ocean currents
10-4 104
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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 byRD 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 structureK. 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
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
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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
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Hexagonal & striped Turing patterns
0-hex -hexstripe
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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: 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
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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)