experiments dan goldman (now berkeley) mark shattuck ( now city u. new york )
DESCRIPTION
Southern Workshop on Granular Materials Puc ón, Chile 10-13 December 2003. 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) - PowerPoint PPT PresentationTRANSCRIPT
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