Random-Packing Dynamics in Granular Flow

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Random-Packing Dynamics in Granular Flow
Martin Z. Bazant Department of Mathematics, MIT

• The Dry Fluids Laboratory _at_ MIT
• Students
• Chris Rycroft, Ken Karmin, Jeremie Palacci,
• Jaehyuk Choi (PhD 05)
• Collaborators
• Arshad Kudrolli (Clark University, Physics)
• Andrew Kadak (MIT Nuclear Engineering)
• Gary Grest (Sandia National Laboratories)
• Ruben Rosales (MIT Applied Math)
• Support U. S. Department of Energy,
• NEC, Norbert Weiner Research Fund

Voronoi tesselation of a dense granular flow
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MIT Modular Pebble-Bed Reactor
http//web.mit.edu/pebble-bed

• Reactor Core
• D 3.5 m
• H 8-10m
• 100,000 pebbles
• d 6 cm
• Q 1 pebble/min

MIT Technology Review (2001)
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Experiments and Simulations
MIT Dry Fluids Lab http//math.mit.edu/dryfluids
Half-Reactor Model (coke bottle) Kadak Bazant
(2004) Plastic or glass beads
3d Full-Scale Discrete- Element
Simulations (molecular dynamics) Rycroft,
Bazant, Landry, Grest (2005) Sandia parallel code
from Gary Grest Frictional, visco-elastic
spheres N400,000
Quasi-2d Silo Experiments Choi et al., Phys. Rev.
Lett. (2003) Choi et al., J. Phys. A Condens.
Mat. (2005) d3mm glass beads Digital-video
particle tracking near wall
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How does a dense random packing flow?

• Dilute random packing
• (gas)
• Boltzmanns kinetic theory sudden randomizing
  collisions
• Dense, ordered packing (crystal)
• Vacancy/interstitial diffusion
• Dislocations and other defects
• Dense, random packing
• (liquid, glass, granular,...)
• Long-lasting, many-body contacts
• Are there any defects?
• How to describe cooperative random motion?

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Experimental Apparatus Dry Fluids Lab (MIT
Applied Math)

• Quasi-2D silo
• W,D,L can be varied
• ?t 1ms (1000 frames/s)
• Diffusion and mixing
• is studied in nearly uniform flow
• W 8, 12, 16, , 32 mm
• vz ? (W-d)1.5
• Image processingcenteroid technique(d15
  pixels, ?x0.003d)

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Universal Crossover in Diffusion
J. Choi, A. Kudrolli, R. R. Rosales, M. Z.
Bazant, Phys. Rev. Lett. (2003).

• ??x2? ? ?t? ? 1.5 ? 1.0
• ??z2? ? ?t? ? 1.6 ? 1.0
• Super-diffusion for z ltlt d,
• Diffusion for z gtgt d
• Data collapses for all flowrates as a function
  ofdistance dropped (not time)
• Suggests that fluctuationshave the same
  physicalorigin as the mean flow(not internal
  temperature)

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The Mean Velocity in Silo Drainage
Kinematic Model
Choi, Kudrolli Bazant, J. Phys. A Condensed
Matter (2005).
Nedderman Tuzun, Powder Tech. (1979)
Diffusion equation
Green function point opening Parabolic
streamlines
but what is diffusing?
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The Paradox of Granular Diffusion
Particles diffuse much more slowly than free
volume.
Experiment by A. Samadani A. Kudrolli (3mm
glass beads)
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Microscopic Flow Mechanisms for Dense Amorphous
Materials
1. Vacancy mechanism for flow in viscous
liquids (Eyring, 1936) Also free volume
theories of glasses (Turnbull Cohen
1958, Spaepen 1977,)
2. Void model for granular drainage
(Litwiniszyn 1963, Mullins 1972)
3. Spot model for random-packing dynamics
(M. Z. Bazant, Mechanics of Materials 2005)
4. Localized inelastic transformations
(Argon 1979, Bulatov Argon 1994)
Shear Transformation Zones (Falk Langer
1998, Lemaitre 2003)
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Correlations Reduce Diffusion
Simplest example A uniform spot affects N
particles.

• Volume conservation (approx.)

• Particle diffusion length

Experiment 0.0025 DEM Simulation 0.0024
(some spot overlap)
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Direct Evidence for Spots
Spatial correlations in velocity fluctuations
(like dynamic hetrogeneity in glasses)
EXPERIMENTS
SIMULATIONS

• MIT Dry Fluids Lab
• 3mm glass beads, slow flow (mm/sec)
• particle tracking by digital video (at wall)
• 125 frames/sec, 1024x1024 pixels
• 0.01d displacements

• Discrete-element method (DEM), spheres
• Sandia parallel code on 32-96 processors
• Friction, Coulomb yield criterion
• Visco-elastic damping
• Hertzian or Hookean contacts

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DEM
Spot Model
Void Model
Simple spot model predicts mean flow and tracer
diffusion in silo drainage fairly well (with only
3 params), but does not enforce packing
constraints.
Simulations by Chris Rycroft
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Multi-scale Spot Algorithm1. Simple
spot-induced motion

• Apply the usual spot displacement first to all
  particles within range

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Multi-scale Spot Algorithm2. Relaxation by
soft-core repulsion

• Apply a relaxation step to all particles within a
  larger radius
• All overlapping pairs of particles experience a
  normal repulsive displacement (soft-core elastic
  repulsion)
• Very simple model - no physical parameters,
  only geometry.

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Multi-scale Spot Algorithm3. Net cooperative
displacement

• Mean displacements are mostly determined by basic
  spot motion (80), but packing constraints are
  also enforced
• Can this algorithm preserve reasonable random
  packings?
• Will it preserve the simple analytical features
  of the model?

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Multiscale Spot Algorithm in Two Dimensions

• Very strong tendency for crystal order
• (Artificial) flow by dislocations,
• grain boundaries, similar to crystals
• Fundamentally different from 3d

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Spot Model DEM

• Multiscale Spot Model
• In Three Dimensions
• Rycroft, Bazant, Grest, Landry (2005)
• Infer 3 spot parameters from DEM, as from expts
• radius 2.6 d
• volume 0.33 v
• diffusion length 1.39 d
• Relax particles each step with soft-core
  repulsion
• Time number of drained particles
• Very similar results as DEM, but gt100x faster!

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Spot Model vs. DEM Mean velocity profile
Excellent fit near bottom, but upper flow is more
plug-like in MD.
(Only one parameter, b, was fitted.)
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Spot Model vs. DEM Particle Diffusion
Diffusive regime
Superdiffusive regime

• Relaxation only slightly enhances diffusion
  (spot-induced rearrangements)
• Short-range (zltd) super-diffusion is not
  reproduced by the Spot Model.

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Spot Model vs. DEM Velocity correlations
Very similar structure of dynamical correlations.
(Only the decay length was fitted.)
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Spot Model vs. DEM bond lengths g(r)
t 1.0-1.4 sec
Identical flowing structure, different from the
initial condition, after substantial drainage
has occurred. (Steady state?)
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Spot Model vs. DEM bond angles
t 1.0-1.4 sec
Nearly identical flowing structure, different
from the initial condition.
Universal features of dense flowing hard spheres?
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Universal Flowing States in Spot Simulations

• Taller Silos, more particles,
• longer drainage times

g(r)

• Reaches statistical steady state
• Spot algorithm never breaks down
• Some local ordering compared to DEM
• (involving only 1 of particles)
• Seems to converge to DEM state
• (no ordering) as spot step size decreases
• Largely insensitive to other parameters

DEM flowing state Spot t 8 sec Spot t 16
sec
2. Periodic boundary conditions,
unbiased spot diffusion (glass)

• Also reaches statistical steady state
• Universal for fixed volume fraction
• (independent of initial conditions,
• mostly insensitive to parameters)
• New simulation method for
• (nearly) hard sphere systems

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A Mathematical Theory of Spot-Induced Particle
Motion
Bazant, Mechanics of Materials (2005)
Spot influence function
A non-local stochastic differential equation for
tracer diffusion
Interpret as the limit of a random (Riemann) sum
of random variables
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Mean-Field Approximation (for spots)

• Assume
• Poisson process for spot positions with (given)
  mean
• Independent spot displacements

Fokker-Planck equation
Fluid velocity (opposes spot drift climbs
gradient in spot density)
Diffusivity tensor
In this way, particle flow and diffusion are
expressed entirely in terms of spot dynamics,
which must be derived from mechanical principles
for a given material and geometry...
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Classical Mohr-Coulomb Plasticity

• 1. Theory of Static Stress in a Granular Material
• Assume incipient yield everywhere
• (t/s)max m
• m internal friction coefficient
• internal failure angle tan-1m

e p/4-f/2
2. Theory of Plastic Flow (only in a wedge
hopper!)
Levy flow rule / Principle of coaxiality Assume
equal, continuous slip along both incipient yield
planes (stress and strain-rate tensors have same
eigenvectors)
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Failures of Classical Mohr-Coulomb Plasticity to
Describe Granular Flow
1. Stresses must change from active to
passive at the onset of flow in a silo (to
preserve coaxiality).
Free surface
Active (gravity-driven stress)
Silo bottom
Passive (side-wall-driven flow)
Exit
2. Walls produce complicated velocity and stress
discontinuities (shocks) not seen in
experiments with cohesionless grains. 3. No
dynamic friction 4. No discreteness and
randomness in a continuum element
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Idea (Ken Kamrin) Replace coaxiality with an
appropriate discrete spot mechanism
d
Slip lines
2?
Spot
Stochastic Flow Rule
D
Spots random walk along Mohr-Coloumb slip
lines (but not on a lattice)
Similar ideas in lattice models for
glasses Bulatov and Argon (1993), Garrahan
Chandler (2004)
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A Simple Theory of Spot Drift
Spot localized failure where ? is replaced by
?k (static to dynamic friction)
(Assume quasi-static global mean stress
distribution is unaffected by spots.)
Net force on the particles affected by a spot
A spots random walk is biased by this force
projected along slip lines. Use the resulting
spot drift and diffusivity in the general theory
to obtain the fluid velocity and particle
diffusivity
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Towards a General Theory of Dense Granular Flow?
Gravity-driven drainage from a wide quasi-2d
silo Predicts the kinematic model
Slip lines and spot drift vectors
Mean velocity profile
Coette cell with a rotating rough inner wall
Predicts shear localization
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Particle Voronoi Volumes
DEM
Spot Model
Trouble in Paradise?
Chris Rycroft, PhD thesis

• Free volume peaks in regions of highest shear,
  not the center!
• Maybe somehow alter spot dynamics, or dont use
  spots everywhere

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General Theory of Dense Granular Materials
Stochastic Elastoplasticity?

• When forcing is first applied, the granular
  material
• responds like an elastic solid.
• Whenever the stability criterion is violated, a
  plastic liquid region at incipient yield is
  born.
• Stresses in elastic solid and plastic liquid
  regions evolve under load, separated by free
  boundaries.
• Solid regions remain stagnant, while liquid
  regions flow by stochastic plasticity.
  Microscopic packing dynamics is described by the
  Spot Model.

Classical engineering picture of silo flow with
multiple zones, without any quantitative theory.
(Nedderman 1991)
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Conclusion

• The Spot Model is a simple, general, and
  realistic mechanism for the dynamics of dense
  random packings

• Random walk of spots along slip lines yields a
  Stochastic Flow Rule for continuum plasticity
• Stochastic Mohr-Coulomb plasticity ( nonlinear
  elasticity?) could be a general model for dense
  granular flow

• Interactions between spots?
• Extend to 3d, other materials,?

For papers, talks, movies, http//math.mit.edu/
dryfluids
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Mohr-Coulomb Stress Equations (assuming incipient
yield everywhere)
Characteristics (the slip-lines)
e p/4-f/2
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• Spot drift opposes the net material force.
• Spots drift through slip-line field constrained
  only by the geometry of the slip-lines.
  Probability of motion along a slip-line is
  proportional to the component of Fnet in that
  direction. Yields drift vector
• For diffusion coefficient D2, must solve for
  the unique probability distribution over the
  four possible steps which generates the drift
  vector and has forward and backward drift aligned
  with the drift direction.