Random-Packing Dynamics in Granular Flow

The Dry Fluids Laboratory @ 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

Martin Z. BazantDepartment of Mathematics, MIT

Voronoi tesselation of a dense granular flow

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

MIT Technology Review (2001)

MIT Modular Pebble-Bed Reactorhttp://web.mit.edu/pebble-bed

Experiments and Simulations

Half-Reactor Model(“coke bottle”)Kadak & Bazant (2004)Plastic or glass beads

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Quasi-2d Silo ExperimentsChoi et al., Phys. Rev. Lett. (2003)Choi et al., J. Phys. A: Condens. Mat. (2005)d=3mm glass beadsDigital-video particle tracking near wall

MIT Dry Fluids Labhttp://math.mit.edu/dryfluids

3d Full-Scale Discrete-Element Simulations(“molecular dynamics”)Rycroft, Bazant, Landry, Grest (2005)Sandia parallel code from Gary GrestFrictional, visco-elastic spheresN=400,000

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How does a dense random packing flow?

• Dilute random “packing” (gas)

• Boltzmann’s 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?

Experimental ApparatusDry 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 processing:centeroid technique(d=15 pixels, x~0.003d)

z

x

y

W

L = 20cmD=2.5cm

H=1m

d=3mm

vz

Universal Crossover in Diffusion

x2 t : = 1.5 1.0 z2 t : = 1.6 1.0

• Super-diffusion for z << d, Diffusion for z >> 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”)

NormalDiffusion

SuperDiffusion

NormalDiffusion

SuperDiffusion

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

The Mean Velocity in Silo Drainage

2

2

x

vb

z

v ,

x

vbu

∂∂

=∂∂

∂∂

=

v + dvv

u

)4

exp(4

)()0,(2

bz

x

bz

Qv

xQzxv

−=

==

π

Green function = point openingParabolic streamlines

Nedderman & Tuzun, Powder Tech. (1979)

Choi, Kudrolli & Bazant, J. Phys. A: Condensed Matter (2005).

…but what is diffusing?

Kinematic Model

Diffusion equation…

“The Paradox of Granular Diffusion”

Experiment by A. Samadani & A. Kudrolli (3mm glass beads)

Particles diffuse much more slowly than free volume.

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)

Correlations Reduce Diffusion

• Volume conservation (approx.)

Simplest example: A uniform spot affects N particles.

• Particle diffusion length

Experiment: 0.0025 DEM Simulation: 0.0024 (some spot overlap)

Direct Evidence for SpotsSpatial correlations in velocity fluctuations

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

(like “dynamic hetrogeneity” in glasses)

DEM Spot Model Void Model

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Simulations by Chris Rycroft

Simple spot model predicts mean flow and tracer diffusion in silo drainagefairly well (with only 3 params), but does not enforce packing constraints.

“Multi-scale” Spot Algorithm

1. Simple spot-induced motion

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

• 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.

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

• 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?

“Multi-scale” Spot Algorithm3. Net cooperative displacement

Multiscale Spot Algorithm in

Two Dimensions

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• 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 DEMMultiscale 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 >100x faster!

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.)

Spot Model vs. DEM: Particle Diffusion

Diffusive regime

Superdiff

usive

regim

e

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

Spot Model vs. DEM: Velocity correlations

Very similar structure of dynamical correlations.(Only the decay length was fitted.)

Spot Model vs. DEM: “bond lengths” g(r)

Identical flowing structure, different from the initial condition, after substantial drainage has occurred. (Steady state?)

t = 1.0-1.4 sec

Spot Model vs. DEM: “bond angles”

Nearly identical flowing structure, different from the initial condition.

Universal features of dense flowing hard spheres?

t = 1.0-1.4 sec

Universal Flowing States in Spot Simulations

1. Taller Silos, more particles, longer drainage times

2. Periodic boundary conditions, unbiased spot diffusion (“glass”)

g(r)

DEM flowing stateSpot t = 8 secSpot t = 16 sec

• 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

• 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

A Mathematical Theory of Spot-Induced Particle Motion

Interpret as the limit of a random (Riemann) sum of random variables:

Spot influence function

Bazant, Mechanics of Materials (2005)

A non-local stochastic differential equation for tracer diffusion:

Mean-Field Approximation (for spots)

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

Fluid velocity (opposes spot drift + climbs gradient in spot density)

Diffusivity tensor

Fokker-Planck equation

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...

Classical Mohr-Coulomb Plasticity1. Theory of Static Stress in a Granular Material

Assume “incipient yield” everywhere:

max

internal friction coefficient= internal failure angle = tan-1

2. Theory of Plastic Flow (only in a wedge hopper!)

Levy flow rule / Principle of coaxiality:

Assume equal, continuous slipalong both incipient yield planes(stress and strain-rate tensors have same eigenvectors)

Silo bottom

Free surface

Exit

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).

“Passive” (side-wall-driven flow)

“Active” (gravity-driven stress)

2. Walls produce complicated velocity and stress discontinuities (“shocks”) not seen in experiments with cohesionless grains.3. No dynamic friction4. No discreteness and randomness in a “continuum element”

Spot

Slip lines

Stochastic “Flow Rule”

2

d

D

Spots random walk along Mohr-Coloumb slip lines(but not on a lattice)

Idea (Ken Kamrin):Replace coaxiality withan appropriate discretespot mechanism

Similar ideas in lattice models for glasses:Bulatov and Argon (1993), Garrahan & Chandler (2004)

A Simple Theory of Spot Drift

Spot = localized failure where is replaced by k (static to dynamic friction)

Net force on the particlesaffected by a spot

A spot’s 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…

(Assume quasi-static global mean stress distribution is unaffected by spots.)

Coette cell with a rotating rough inner wall: Predicts shear localization

Gravity-driven drainage from a wide quasi-2d silo: Predicts the kinematic model

Towards a General Theory of Dense Granular Flow?

Slip lines and spot drift vectors Mean velocity profile

Particle Voronoi Volumes

• “Free volume” peaks in regions of highest shear, not the center!• Maybe somehow alter spot dynamics, or don’t use spots everywhere…

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DEMSpot Model

Chris Rycroft, PhD thesis

Trouble in Paradise?

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 engineeringpicture of silo flow with multiple zones, without anyquantitative theory. (Nedderman 1991)

Conclusion

For papers, talks, movies,… http://math.mit.edu/dryfluids

• 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,…?

Characteristics (the slip-lines)

Mohr-Coulomb Stress Equations (assuming incipient yield everywhere)

• 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.

D1