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 Biology Poly-PEDAL Lab *starting Assistant Professor at Georgia Tech, January 2007 CONTACT:

Signatures of glass formation and jamming in a fluidized bed of hard spheres Daniel I. Goldman* University of California Berkeley Department of Integrative Biology Poly-PEDAL Lab *starting Assistant Professor at Georgia Tech, January 2007 Harry L. Swinney University of Texas at Austin Physics Department Center for Nonlinear Dynamics Thanks to Mark Shattuck, Matthias Schröter, David Chandler, Albert Pan, Juan Garrahan, and Eric Weeks Phys. Rev. Lett. 96, (2006) water 2 cm 100x100x ±10  m glass spheres 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 locomotion Goldman, Korff, Wehner, Berns, Full, cm Mojave desert Outer Banks, NC UC Berkeley, Dept of Integrative Biology

Relevance of fluidized beds Cat cracker: $200 billion/year Laboratory fluidized bed 50 m 10 cm Goldman & Swinney, UT Austin Texaco Fossil fuel refinement

Physics of fluidization permeability 1 Kozeny-Carman

Height ~ Increasing flow leads to “fluidization” at Q f Decreasing flow leads to “defluidization”:  independent of Q water 100x100x  m glass spheres height 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 Volume fraction Sensitivity:0.6 Pa Goldman & Swinney, submitted to Phys. Rev. Lett., 2005 Bottom of bed Top of bed Side view of bed

flow pulses a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, 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 1987 van Megen 1993 Speedy 1998 Weeks 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/min 2 “defluidization” = no visible particle motion aa Water-fluidized bed

Dynamical Heterogeneity camera 60 PD t+  T t = Goldman & Swinney, Phys. Rev. Lett., 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 camera mirror 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 spheres Goldman & Swinney, Phys. Rev. Lett., PD Difference of images taken  T=0.3 sec apart Side view of bed  

Determine correlation length 1. Perform 2D spatial autocorrelation on single difference image, for fixed  T 2. Measure length  at which correlation function has decayed by 1/e ( We find  x  y =  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 Goldman & Swinney, PRL, 2006 gg COLLOIDS FLUIDIZED BED

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

Scaling of correlation length and time Pan, Garrahan, Chandler (2004) 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/min 2 CURVATURE CHANGE Hard sphere systems undergo glass transition at  g ≈ 0.58 Pusey 1987 van Megen 1993 Speedy 1998 Weeks 2000 “defluidization”

gg a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 unity gg a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 I(t) : intensity of interfering light at point 2.5 cm Laser light

Correlation time of multiply scattered light 1/e point  DWS Basically ~ Goldman & Swinney, Phys. Rev. Lett., 2006

Divergence and arrest aa g?g? Goldman & Swinney, Phys. Rev. Lett., 2006

Decoupling macro and microscopic motions SOLID LINE:  measured by camera imaging scaled by 3x10 5 Same functional forms below  g  DWS gg aa Goldman & Swinney, Phys. Rev. Lett., 2006

Fit region Ballistic motion between collisions Caging Short time plateau indicates particles remain in contact Motion on short time and length scales Particles move < 1/1000 of their diameter Doliwa

Loss of ballistic motion between collisions at  g Exponent of fit

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 This explains observation of Ojha et al, that all non- cohesive fluidized beds achieve same final volume fraction  ≈0.59

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a g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 array Incoherent 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 Liu, Nagel 1998 Slight increase in Q jams the grains 300Time (sec)

Jamming creates hysteresis

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