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Patterns in a vertically oscillated granular layer: (1) lattice dynamics and melting, (2) noise-driven hydrodynamic modes, (3) harvesting large particles – Experiments Dan Goldman (now Berkeley) Mark Shattuck ( now City U. New York) Harry Swinney University of Texas at Austin – Simulations Sung Jung Moon (now Princeton) Jack Swift Southern Workshop on Granular Materials Pucón, Chile 10-13 December 2003

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Particles in a vertically oscillating container light f = frequency (10-200 Hz) = (acceleration amplitude)/g = 4 2 f 2 /g (2-8)

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Square pattern f = 23 Hz acceleration = 2.6g Particles: bronze, d=0.16 mm layer depth = 3d 1000d

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OSCILLONS peak crater localized oscillatory: f /2 nonpropagating stable Umbanhowar, Melo, & Swinney, Nature (1996)

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Oscillons: no interaction at a distance

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Oscillons: building blocks for molecules each molecule is shown in its two opposite phases dimer tetramer polymer chain

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Oscillons: building blocks of a granular lattice? each oscillon consists of 100-1000 particles

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Dynamics of a granular lattice 18 cm Goldman, Shattuck, Moon, Swift, Swinney, Phys. Rev. Lett. 90 (2003) = 2.90, f = 25 Hz, lattice oscillation 1.4 Hz snapshot snapshot: close up time evolution

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Coarse-graining of granular lattice: frequency at edge of Brillouin zone A lattice of balls connected by Hooke’s law springs? Then the dispersion relation would be: where k is wavenumber and a is lattice spacing

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Compare measured dispersion relation with lattice model lattice model f Lattice (Hz) = 2.75 k Brillouin Zone (for (1,1) T modes) From space-time FFT I (k x,k y,f L )

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Create defects: make lattice oscillations large = 2.9 FFT apply FM 52 cycles later 235 cycles later DEFECTS modulation rate = 2 Hz 32 Hz container position: Resonant modulation: FM at lattice frequency:

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Frequency modulate the container, and add graphite to reduce friction MELTING = 2.9, f = 32 Hz, f mr (FM) = 2 Hz add graphite by 175 cycles: melted 56 cycles later

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MD simulation: reduce friction to zero crystal melts (without adding frequency modulation) = 0.5 = 0 22 cycles later 100 cycles later: melted = 3.0, f = 30 Hz

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Lindemann criterion for crystal melting Lindemann ratio: where u m and u n are displacements from the lattice positions of nearest neighbor pairs, and a is the lattice constant. Simulations of 2-dimensional lattices in equilibrium show lattice melting when Bedanov, Gadiyak, & Lozovik, Phys Lett A (1985) Zheng & Earnshaw, Europhys Lett (1998)

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Lindemann criterion = 0.5: no melting Test Lindemann criterion on granular lattice MD simulations lattice melts = 0.1 melting threshhold Goldman, Shattuck, Moon, Swift, Swinney, Phys. Rev. Lett. 90 (2003)

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Conclude: granular lattice is described well by discrete lattice picture. How about a continuum description? Granular patterns: as in continuum systems -- vertically oscillated liquids, liquid crystals, … --- squares, stripes, hexagons, spiral defect chaos Instabilities as in Rayleigh-Bénard convection --- skew-varicose, cross-roll

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Spiral defect chaos Rayleigh-Bénard convection Granular oscillating layer deBruyn, Lewis, and Swinney Phys. Rev. E (2001) Plapp and Bodenschatz Physica Scripta (1996)

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Skew-varicose instabililty observed in granular expt: same properties as skew-varicose instability of Rayleigh-Bénard convection rolls wavelength increases deBruyn et al., Phys. Rev. Lett. (1998) 1 2 34

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wave- length decreases de Bruyn, Bizon, Shattuck, Goldman, Swift, and Swinney, Phys. Rev. Lett. (1998) Cross-roll instability observed in granular experiment: same properties as cross-roll instability in convection

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Continuum models of granular patterns Tsimring and Aranson, Phys. Rev. Lett. (1997) Shinbrot, Nature (1997) Cerda, Melo, & Rica, Phys. Rev. Lett. (1997) Sakaguchi and Brand, Phys. Rev. E (1997) Eggers and Riecke, Phys. Rev. E (1998) Rothman, Phys. Rev. E (1998) Venkataramani and Ott, Phys. Rev. Lett. (1998)

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Convecting fluids: thermal fluctuations drive noisy hydrodynamic modes below the onset of convection Theory: Swift-Hohenberg eq., derived from Navier-Stokes Swift & Hohenberg, Phys Rev A (1977) Hohenberg & Swift, Phys Rev A (1992) Experiments : convecting fluids and liquid crystals: Rehberg et al., Phys Rev Lett (1991) Wu, Ahlers, & Cannell, Phys Rev Lett (1995) Agez et al., Phys Rev A (2002) Oh & Ahlers, Phys. Rev. Lett. (2003) Granular systems are noisy. Can hydrodynamic modes be seen below the onset of patterns?

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Noise below onset of granular patterns 6.2 cm snapshot time evolution 170 m stainless steel balls (e 0.98) time (T) = 2.6, f = 30 Hz x

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Increase toward pattern onset at c = 2.63 : S max (k) increases 0 15 30 45 60 Hz |k| P(f) S(k x,k y )

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Emergence of square pattern with long-range order S(k x,k y ) P(f) frequency of square pattern container frequency S(k) = 2.8 k

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Swift-Hohenberg model for convection: from Navier-Stokes eq. with added noise If no noise ( F = 0) (“mean field”), pattern onset is at But if F 0, onset of long-range (LR) order is delayed, Xi, Vinals, Gunton, Physica A (1991); Hohenberg & Swift, Phys Rev A (1992)

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Compare granular experiment to Swift-Hohenberg model Experiment Swift-Hohenberg DISORDERED SQUARES Granular noise is: -- 10 4 times the k B T noise in Rayleigh-Bénard convection [Wu, Ahlers, & Cannell, Phys. Rev. Lett. (1995)] --10 times the k B T noise in Rayleigh-Bénard convection near T c [ Oh & Ahlers, Phys. Rev. Lett. (2003 )] Goldman, Swift, & Swinney Phys. Rev. Lett. (Jan. 2004) = ( – c )/ c

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Segregation: separate particles of different sizes

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f* = f x [(layer depth)/g] 1/2 Kink: boundary between regions of opposite phase -- layer on one side of kink moves down while other side moves up flat with kinks OSCILLONS

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Kink: a phase discontinuity 3-dimensional MD simulation =6.5 container x/d 0 100 200 kink Moon, Shattuck, Bizon, Goldman, Swift, Swinney Phys. Rev. E 65, 011301 (2001)

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Convection toward a kink falling rising This is NOT a snapshot: the small black arrows show the displacement of a particle in 2 periods (2/f )

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Larger particles rise to top (Brazil nut effect) and are swept by convection to the kink ® this segregation is intrinsic to the dynamics (not driven by air or wall interaction) glass particles dia. = 4d bronze particles dia. = d

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Moon, Goldman, Swift, Swinney, Phys. Rev. Lett. 91 (2003) kink particle trajectory oscillating kink EXPERIMENT: controlled motion of the kink harvests the larger particles black glass dia. = 4d bronze d = 0.17 mm 247 cycles 566 cycles t = 0

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Dynamics of a granular lattice Granular lattice: like an equilibrium lattice of harmonically coupled balls and springs Lindemann melting criterion supports the coupled lattice picture Question: Would continuum pattern forming systems, e.g., Faraday waves in oscillating liquid layers, Rayleigh-Bénard convection patterns, falling liquid columns, Taylor-Couette flow, viscous film fingers, … exhibit similar lattice dispersion and melting phenomena?

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Noise Near the onset of granular patterns, noise drives hydrodynamic-like modes, which are well described by the Swift-Hohenberg equation.

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Harvesting large particles Segregation of bi-disperse mixtures has been achieved for particles with Diameter ratios: 1.1 – 12 Mass ratios: 0.4 - 2500

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