Patterns in a verticallyoscillated 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 SwinneyUniversity of Texas at Austin

– Simulations• Sung Jung Moon (now Princeton)

• Jack Swift

Southern Workshop on Granular MaterialsPucón, Chile 10-13 December 2003

Particles in a vertically oscillating container

light

f = frequency (10-200 Hz) = (acceleration amplitude)/g = 42f2/g (2-8)

Square pattern

f = 23 Hzacceleration = 2.6g

Particles:bronze, d=0.16 mm

layer depth = 3d

1000d

OSCILLONS

peak

crater

• localized• oscillatory: f /2• nonpropagating• stable

Umbanhowar, Melo,& Swinney, Nature (1996)

Oscillons:no

interactionat a

distance

Oscillons: building blocks for moleculeseach molecule is shown in its two opposite phases

dimer tetramer

polymerchain

Oscillons:building blocks of a granular lattice?

each oscillon consists of

100-1000 particles

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 uptime evolution

Coarse-graining of granular lattice:

2 2

| sin( ) |lattice BZ

kaff

frequency at edgeof 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

Compare measured dispersion relation with lattice model

lattice model

fLattice

(Hz)

= 2.75

kBrillouin Zone(for (1,1)T modes)

From space-time FFT I(kx,ky,fL)

Create defects: make lattice oscillations large

= 2.9

FFT FFT FFTapply FM 52 cycles later 235 cycles laterDEFECTS

( ) sin[2 sin(2 )]msmr

mr

fy t A ft f t

f

modulationrate = 2 Hz

32 Hz

containerposition:

Resonant modulation: FM at lattice frequency:

Frequency modulate the container, and

add graphite to reduce friction MELTING

= 2.9, f = 32 Hz, fmr(FM) = 2 Hz

add graphite by 175 cycles: melted56 cycles later

MD simulation: reduce friction to zerocrystal melts (without adding frequency modulation)

= 0.5 = 0 22 cycles later 100 cycles later: melted

= 3.0, f = 30 Hz

Lindemann criterion for crystal melting

Lindemann ratio:2

2

| |m nu ua

where um and un are displacements from the lattice positions of nearest neighbor pairs, and a is the lattice constant.

Simulations of 2-dimensional lattices in equilibriumshow lattice melting when

0.1

Bedanov, Gadiyak, & Lozovik , Phys Lett A (1985)Zheng & Earnshaw, Europhys Lett (1998)

Lindemann

criterion

= 0.5:no melting

Test Lindemann criterion on granular latticeMD simulations

latticemelts

= 0.1melting

threshhold

Goldman, Shattuck, Moon, Swift, Swinney, Phys. Rev. Lett. 90 (2003)

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

Spiral defect chaos

Rayleigh-Bénard convection Granular oscillating layer

deBruyn, Lewis, and SwinneyPhys. Rev. E (2001)

Plapp and BodenschatzPhysica Scripta (1996)

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

3 4

wave-length

decreases

de Bruyn, Bizon, Shattuck, Goldman, Swift, and Swinney, Phys. Rev. Lett. (1998)

Cross-roll instabilityobserved in granular experiment:

same properties as cross-roll instability in convection

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)

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?

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

Increase towardpattern onset at c = 2.63 :

Smax(k) increases

0 15 30 45 60 Hz|k|

P(f)S(kx,ky)

Emergence of square pattern with long-range order

S(kx,ky) P(f)

frequency ofsquare pattern

containerfrequency

S(k)

= 2.8

k

Swift-Hohenberg model for convection:from Navier-Stokes eq. with added noise

( , ) ( ', ') 2 ( ') ( ') where x t x t F x x t t

If no noise (F = 0) (“mean field”), pattern onset is at

0MFc

But if F 0, onset of long-range (LR) order is delayed,

2/30LR LRc cwhere F

Xi, Vinals, Gunton, Physica A (1991); Hohenberg & Swift, Phys Rev A (1992)

Compare granular experiment to Swift-Hohenberg model

Experiment

Swift-Hohenberg

DISORDERED

SQUARES

Granular noise is:

-- 104 times the kBT noise in Rayleigh-Bénard

convection [Wu, Ahlers, & Cannell, Phys. Rev. Lett. (1995)]

--10 times the kBT noise in Rayleigh-Bénard

convection near Tc [Oh & Ahlers,

Phys. Rev. Lett. (2003)]

Goldman, Swift, & SwinneyPhys. Rev. Lett. (Jan. 2004)

= ( – c)/c

Segregation:

separate particlesof different sizes

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

Kink: a phase discontinuity3-dimensional MD simulation

=6.5

container

x/d

x/d

0 100 200

kink

Moon, Shattuck, Bizon, Goldman, Swift, SwinneyPhys. Rev. E 65, 011301 (2001)

Convection toward a kink

fallingrising

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

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)

glassparticlesdia. = 4d

bronze particles dia. = d

Moon, Goldman, Swift, Swinney,Phys. Rev. Lett. 91 (2003)

kink

particle trajectory

oscillating kink

EXPERIMENT:controlled motion of

the kink harveststhe larger particles

black glassdia. = 4d

bronzed = 0.17 mm

247 cycles

566 cycles

t = 0

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?

Noise

Near the onset of granular patterns,noise drives

hydrodynamic-like modes, which are well described by

the Swift-Hohenberg equation.

Harvesting large particles

Segregation of bi-disperse mixtures

has been achieved for particles with

• Diameter ratios: 1.1 – 12

• Mass ratios: 0.4 - 2500

END