Daniel I. Goldman* University of California Berkeley Department of Integrative Biology

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: digoldma@berkeley.eduhttp://socrates.berkeley.edu/~digoldma/

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

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

Relevance of fluidized bedsCat cracker:$200 billion/yearLaboratory

fluidized bed

50 m10 cm

Goldman & Swinney, UT Austin

Texaco

Fossil fuel refinement

Physics of fluidizationsingletmanyt

v50

vP KQ

1P 1P 1P

h

manyfQ

f tAv v

permeability

1

Kozeny-Carman

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

Experimental apparatus 100 to 1000 m glass beads

Goldman & Swinney, Phys. Rev. Lett., 2006

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

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

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)

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)

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

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

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

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

Time evolution of heterogeneity

=0.568 =0.590

Heterogeneity persists for characteristic time

Goldman & Swinney, Phys. Rev. Lett., 2006

snapshot

40 PD

space

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

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

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

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

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

--loss of mobility on particle diameter scale occurs near g

Scaling of correlation length and time

Pan, Garrahan, Chandler (2004)

2/3~ 4/1max ~

For <g

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?

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

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

Pressure drop vs. Q

Goldman & Swinney, Phys. Rev. Lett., 2006

fluidized

defluidized

P can no longer remain near unityg a

Speedy 1998

Goldman & Swinney, Phys. Rev. Lett., 2006

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

Correlation time of multiply scattered light

1/e point

DWS

)exp( 2tBasically ~

Goldman & Swinney, Phys. Rev. Lett., 2006

Divergence and arrest

a

g?

Goldman & Swinney, Phys. Rev. Lett., 2006

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

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

Loss of ballistic motion between collisions at g

Exponent of fit

~)(r 2

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

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

Arrested state continues to slowly decrease as Q decreases

a

g

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

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

Decrease Q through the glass & arrest transitions

Slight increase in Q jams the grains

300Time (sec)

Jamming creates hysteresis

Jammed state doesn’t respond to small changes in flow rate

Q increasing Q decreasing

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

END