EXPANDED VERSION OF TALK GIVEN AT SOUTHERN WORKSHOP ON GRANULAR MATERIALS, VINA DEL MAR, CHILE 2006...
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EXPANDED VERSION OF TALK GIVEN AT SOUTHERN WORKSHOP ON GRANULAR MATERIALS, VINA DEL MAR, CHILE 2006
Daniel I. Goldman*University of California Berkeley
Department of Integrative BiologyPoly-PEDAL Lab
*starting Assistant Professor at Georgia Tech, January 2007
CONTACT: [email protected]://socrates.berkeley.edu/~digoldma/
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Signatures of glass formation and jamming in a fluidized bed of hard spheres
Daniel I. Goldman*University of California Berkeley
Department of Integrative BiologyPoly-PEDAL Lab
*starting Assistant Professor at Georgia Tech, January 2007
Harry L. SwinneyUniversity of Texas at Austin
Physics DepartmentCenter for Nonlinear Dynamics
Thanks to Mark Shattuck, Matthias Schröter, David
Chandler, Albert Pan, Juan Garrahan, and Eric Weeks
Phys. Rev. Lett. 96, 145702 (2006)
water
2 cm
100x100x700250±10 m
glassspheres
Q (0-100 mL/min)
v<0.3 cm/sec
Fluidized bed allows:• Uniform bulk excitation 2. Fine control of system
parameters (like solid volume fraction by control of flow rate Q
Question: how do grains stop moving as flow is reduced?
1 mm
Support: Welch, DOE, IC Postdoc Fellowship, Burroughs Wellcome Fund
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Fluidized beds: relevance to locomotionGoldman, Korff, Wehner, Berns, Full, 2006
5 cm
5 cm
Mojave desert
Outer Banks, NC
UC Berkeley, Dept of Integrative Biology
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Relevance of fluidized bedsCat cracker:$200 billion/yearLaboratory
fluidized bed
50 m10 cm
Goldman & Swinney, UT Austin
Texaco
Fossil fuel refinement
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Physics of fluidizationsingletmanyt
v50
vP KQ
1P 1P 1P
h
manyfQ
f tAv v
permeability
1
Kozeny-Carman
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Height ~
Increasing flow leads to “fluidization” at Qf
Decreasing flow leads to “defluidization”: independent of Q
Fluidized bed basics (cohesionless particles)
Final state is independent of particle size, aspect ratio, container shape,≈ 0.59
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Experimental apparatus 100 to 1000 m glass beads
Goldman & Swinney, Phys. Rev. Lett., 2006
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h
1 cm
Volume fraction & pressure measurement
5 m resolution
( )s f
PP
Agh
P
Volume fraction
Ah
m
s
s
Sensitivity:0.6 Pa
Bottom of bed
Top of bed
Side view of bed
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flow pulses
a
Fluidized bed basics
In slow fluidization cycle, initial state is not unique, final state is.
a≡volume fraction no longer changes with changes in Q
Bed height
Pressure drop
--Goldman, Shattuck Swinney, 2002--Schröter, Goldman & Swinney 2005
defluidization
fluidization
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a≈0.59 achieved after defluidization is independent of particle size, aspect ratio, cross-sectional area
Ojha, Menon and Durian (2000)
Gas-fluidized bed
(or hydrodynamic forces)
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Growing time-scale
Glotzer (2000)
Weeks et al (2000)
Dynamical Heterogeneity
Phenomena associated with glass formation (large literature, many types of systems)
Rate dependence
Pan, Garrahan, Chandler (2004)
NMR: Sillescu, 1999, Ediger, 2000
REVIEW ARTICLE: Ediger, Angell, Nagel (1996)
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Glass formation* in hard
spheres occurs near g ≈ 0.58
• Colloids: Pusey 1987, van Megen 1993, Weeks 2000…
• Simulation: Speedy 1998, Heuer 2000…
Beyond g spheres can no longer move greater than a particle diameter
Speedy 1998
Heuer 2000
Pusey 1987van Megen 1993 Speedy 1998Weeks 2000
Dynamical heterogeneity observed in hard disks
Deviation from ideal gas PV/NkT
*rapid slowing of dynamics with no apparent change in static structure
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a depends on rate of decrease of Q
Goldman & Swinney, Phys. Rev. Lett., 2006
Ramp rate, dQ/dt
mL/min2
“defluidization” = no visible particle motion
a
Water-fluidized bed
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Dynamical Heterogeneity
camera
60 PD
t+T t
=
Goldman & Swinney, Phys. Rev. Lett., 2006
1 PD= 250 m
Particle motion is spatially correlated for characteristic correlation time.
=0.57
Moved in T
Immobile
Difference of images taken T=0.3 sec apart
3x speed
Side view of bed
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Heterogeneity observed at surface of bed
cameramirror
Indicates that the dynamics in the interior are also heterogeneous
~0.56
~0.59
Difference of images taken T=0.3 sec apart
3x speed
1 mm
Top view of bed
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Time evolution of heterogeneity
=0.568 =0.590
Heterogeneity persists for characteristic time
Goldman & Swinney, Phys. Rev. Lett., 2006
snapshot
40 PD
space
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Measure correlation time,
1. For each pixel, perform autocorrelation of I(t)2. measure 1/e point for each correlation curve =
…
I(x,y,t)
Side view
Particle motion causes pixel intensity fluctuations
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Increasing average correlation time Goldman & Swinney, Phys. Rev. Lett., 2006
eg. lattice model of Pan et al 2004
Distribution of correlation times increases as well
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Length-scale of heterogeneity, increases with increasing
250 m glass spheresGoldman & Swinney, Phys. Rev. Lett., 2006
40 PD
Difference of images taken T=0.3 sec apart
Side view of bed
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Determine correlation length1. Perform 2D spatial autocorrelation on single difference image, for fixed T2. Measure length at which correlation function has decayed by 1/e (We find xy=3. Average over independent images at fixed
T=0.3 sec
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Increasing dynamic correlation length
Loss of mobility on particle diameter scale occurs near g
Weeks et al, Science 2000.Goldman & Swinney, PRL, 2006
g
COLLOIDSFLUIDIZED BED
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--loss of mobility on particle diameter scale occurs near g
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Scaling of correlation length and time
Pan, Garrahan, Chandler (2004)
2/3~ 4/1max ~
For <g
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Hard sphere glass physics
• In the fluidized bed, we observe:– Rate dependence– Increasing time-scale– Dynamical heterogeneity
• Does this relate to hard sphere glass formation?
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Change in curvature near g ≈ 0.58
Inflection point
Goldman & Swinney, Phys. Rev. Lett., 2006
Ramp rate:1.82 mL/min2
CURVATURE CHANGE
Pusey 1987van Megen 1993 Speedy 1998Weeks 2000
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g a
Inflection point near g
Goldman & Swinney, Phys. Rev. Lett., 2006
As g is approached, system can no longer pack sufficiently in response to changes in Q
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Pressure drop vs. Q
Goldman & Swinney, Phys. Rev. Lett., 2006
fluidized
defluidized
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P can no longer remain near unityg a
Speedy 1998
Goldman & Swinney, Phys. Rev. Lett., 2006
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Diffusing Wave Spectroscopy (DWS) to probe the interior at short length and timescales
Resolution estimate: 532 nm/100 particles across ≈ 5 nm particle displacements, microsecond timescales
Use DWS theory, from g(t) obtain
Pine, Weitz, Chaikin,
Herbolzheimer PRL 1988
)( 2 tr
I(t) : intensity of interfering light
at point
2.5 cm
Laser light
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Correlation time of multiply scattered light
1/e point
DWS
)exp( 2tBasically ~
Goldman & Swinney, Phys. Rev. Lett., 2006
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Divergence and arrest
a
g?
Goldman & Swinney, Phys. Rev. Lett., 2006
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Decoupling macro and microscopic motions
SOLID LINE: measured by camera imaging scaled by 3x105
Same functional forms below g
DWS
g aGoldman & Swinney, Phys. Rev. Lett., 2006
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Fit region
Ballistic motion between collisionsCaging
Short time plateau indicates particles remain in contact
Motion on short time and length scales Particles move < 1/1000 of
their diameter
Doliwa 2000
0.58
0.5
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Loss of ballistic motion between collisions at g
Exponent of fit
~)(r 2
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Our picture
• We propose that at g, the bed undergoes a glass transition
• Many spheres must now move cooperatively for any sphere to move so the system begins to undergo a structural arrest
• can no longer change adequately with changes in Q so P can no longer be maintained close to 1.
• P drops rapidly effectively freezing the system—particle motion is arrested at a
The bed thus defluidizes and arrests ~ ≈0.59 because of glass formation ~ ≈0.58
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Conclusions on defluidization
• Dynamics of fluidized bed similar to supercooled liquids becoming glasses
• Glass formation explains a independent of particle size, etc.
• Nonequilibrium steady state suspension shows similar features of glass transition as seen in “equilibrium” hard spheres
Multiple lines of evidence indicate a transition at g=0.585±0.005 results in arrest of particle motion at a=0.593±0.004
Goldman & Swinney, Phys. Rev. Lett., 2006
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Arrested state continues to slowly decrease as Q decreases
a
g
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Multiple scattered laser light imaged on CCD resolves
motions of <1 nm
5
“Speckle” pattern
Each pixel receives randomly scattered light that has combined
from all paths through bed
Integrate over 1/30 sec
Laser light probes short length and timescale motion
Crude estimate: light to dark=change in path length of 532 nm, 100 particles across, if each moves 532/100=5 nm per particle, 256 grayscales=5/255=0.02 nm motions
=532 nm R=1 cm
z=50 cm
CCD arrayIncoherent illumination
Particles visible under incoherent illumination
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Microscopic motion persists in defluidized state
g
Laser off Laser on
The particles appear to arrest but the speckle does not indicating microscopic motion persists
Look at time evolution of row of pixels
Turn flow off suddenly: Free sedimentation
250 m
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Decrease Q through the glass & arrest transitions
Slight increase in Q jams the grains
300Time (sec)
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Jamming creates hysteresis
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Jammed state doesn’t respond to small changes in flow rate
Q increasing Q decreasing
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Summary• Decreasing flow to fluidized bed displays
features of a supercooled liquid of hard spheres becoming a glass
• Hard sphere glass formation governs transition to defluidized bed
• In arrested state, microscopic motion persists until state is jammed
USE WELL CONTROLED FB TO STUDY HARD SPHERE GLASSES & GLASSES CAN
INFORM FB
Fluidized bed allows:• Uniform bulk excitation 2. Fine control of system
parameters (like solid volume fraction by control of flow rate Q
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END