Features of Jamming in Frictionless

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Features of JamminginFrictionless Frictional
Packings
Leo Silbert Department of Physics
Wednesday 3rd September 2008
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What Is Jamming?
Jamming is the transition between solid-like and
fluid-like phases in disordered systems
JAMMED
UNJAMMED
Many macroscopic and microscopic complex
phenomena associated with jammed states and the
transition to the unjammed phase
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Similarities

• Supercooled liquids and glasses
• Dense dispersions colloids, foams, emulsions
• Cessation of granular flows
• Mechanical properties of sand piles, polymeric
  networks, cells
• Fluffy Static Packings

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Foams Durian
Supercooled Liquids Glotzer
Sphere Packings
Colloidal Suspensions Weeks
Emulsions Brujic et al.
Grain Piles
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How to Study Jamming?

• Back To Basics
• What is the simplest system through which we can
  gain insight and develop our understanding of
  this range of phenomena?

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Why Does Granular Matter?

• Frictional Inelastic
• rolling/sliding contacts
• dissipative interactions on the grain
  miscroscopic scale
• Non-thermal and far from thermodynamic
  equilibrium
• static packings are metastable states
• Paradigm for non-equilibrium states
• similarities with other amorphous materials

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Granular Phenomena
Granular materials are ubiquitous throughout
nature
large-scale geological features
Natural phenomena
failure flows
Granular phenomena persist at the forefront of
many-body physics research demanding new concepts
applicable to a range of systems far from
equilibrium
avalanche.org
bbc.co.uk
Natural disasters
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Grain Piles
Duke Group

• Contact forces are highly heterogeneous
• force chains
• Distribution of forces
• wide distribution
• exponential at large forces

Chicago Group
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Jamming Transition in Static Packings

• Take a packing of spheres and jam them together
• Slowly release the confining pressure by
  decreasing the packing fraction
• Study how the system evolves
• At a critical packing fraction fc the packing
  unjams
• The properties of the packing are determined by
  the distance to the jamming transition

?f f - fc
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Jamming of Soft Spheres

• Monodisperse, frictionless, soft spheres
• finite range, repulsive, potential V(r) V0
  (1-r/d)2 r lt d
• 
  0 otherwise
• Transition between jammed and unjammed phases at
  critical packing fraction fc
• Frictionless Spheres
• critical packing fraction coincides with value of
  random close packing, fc 0.64 in 3D ( 0.84 2D)
  
• packings are isostatic at the jamming transition,
  coordination number zc 6 in 3D ( 4 in 2D)

Durian, Phys. Rev. Lett. 75, 4780 (1995) Makse
co-workers, Phys. Rev. Lett. 84, 4160 (2000)
Phys. Rev. E 72, 011301 (2005) Nature 453, 629
(2008) OHern et.al, Phys. Rev. Lett. 88, 075507
(2002) Phys. Rev. E 68, 011306 (2003) Kasahara
Nakanishi, Phys. Rev. E 70, 051309 (2004) van
Hecke and co-workers, Phys. Rev. Lett. 97, 258001
(2006) Phys. Rev. E 75, 010301 (2007) 75,
020301 (2007) Agnolin Roux, Phys. Rev. E, 76
061302-4 (2007)
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Static Packings Under Pressure
Compressed Frictionless Spheres (no gravity)
Very compressed high P, fgtgtfc
?f-gt0
Weakly compressed low P, ffc

• Force distributions and pressure
• Packing becomes more uniform with increasing
  pressure
• Exponential to Gaussian crossover with increasing
  pressure

Makse co-workers, Phys. Rev. Lett. 84, 4160
(2000) Silbert et al., Phys. Rev. E 73, 041304
(2006)
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Vibrational Density of States
Characteristic feature in D(?) signals onset of
jamming

boson peak
Packing become increasingly soft
D(?) constant at low ? Peak shifts to lower ?
Silbert et al. Phys. Rev. Lett. 95, 098301 (2005)
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Jamming and the Boson Peak

• Jamming transition accompanied by a diverging
  boson peak
• Two length scales characterize dynamical modes
• ?L (longitudinal correlation length)
• ?T (transversal correlation length)

Adding the Debye contribution The dispersion
relations read
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Jamming ? Critical Phenomena ?

• Some quantities behave like order parameters
• e.g. excess coordination number
• statistical field theory approach Henkes
  Chakraborty
• Correlation Lengths Characterizing the
  Transition
• Wyart et al length scales characterizing
  rigidity of the jammed network
• Schwarz et al percolation models and length
  scales
• Dynamic Length Scales on unjammed side
• Drocco et al. Phys. Rev. Lett. 95, 088001 (2005)
• How do we identify length scales in static
  packings?

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Low-k Behaviour of S(k) in Jammed Hard Soft
Spheres

• Hard Spheres Recent Molecular Dynamics (MD) has
  shown low-k behaviour of S(k) in a jammed system
  of hard spheres is linear, namely, S(k) ? k
• Note- systems of N gt 104 needed to resolve
  low-k region
• Donev et al. used 105-106 particles, f-0.64
• S(k) 1/Nlt?(k)?(-k)gt

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Hard Spheres
Donev et.al. Phys. Rev. Lett. 95 090604 (2005)

• Structure factor for a jammed N106 f0.642, and
  for a hard sphere liquid near the freezing point,
  f0.49, as obtained numerically and via PY theory

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Soft Sphere Liquid
T gt 0
Expected behaviour in the liquid state
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Jammed Soft Sphere Packings
T 0
Transition to linear behaviour Observed using
N256000, at f0.64
In jammed packings S(k) k, near jamming
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Phenomenology

• Second moment of dynamical structure factor
• Conjecture assume the dominant collective mode
  is given by dispersion relation ?B(k). Then
• and
• Assuming
• at small k then in the long wavelength limit
• transverse modes contribute to linear behaviour
  of S(k).

Silbert Silbert (2008)
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S(k) as a Signature of Jamming

• Linear behaviour in S(k) and the excess density
  of states are two sides of the same coin
• Suggest length scale where crossover to linear
  behaviour occurs
• Does this feature survive for polydispersity and
  2D?
• Does this feature survive with LJ interactions
• Can we see this in real glasses?
• Can we see this in s/cooled liquids where BP
  survives into liquid phase?

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Effect of Friction on S(k)
T 0
f0.64
At the same f low-k behaviour different
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Comparison Between Frictionless Frictional
Packings

• Frictional
• Stable over wide range of packing fractions
• 0.55 lt f lt 0.64
• Random Loose Packing
• fRLP 0.55
• Onada Liniger, PRL 1990
• Schroter et al. Phys. Rev. Lett. 101, 018301
  (2008)
• Are frictional packings isostatic?
• z(µgt0)iso D1

• Frictionless
• Random Close Packing
• fRCP 0.64
• Bernal, Scott, 1960s
• Jamming transition is RCP
• Frictionless packings at RCP are isostatic
• z(µ0)iso 2D

Abate Durian, Phys. Rev. E 74, 031308
(2006) Behringer co-workers, Phys. Rev. Lett.
98, 058001 (2007)
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Jamming Protocol

• Follow similar protocol used for frictionless
  studies
• N1024 monodisperse soft-spheres d 1
• particle-particle contacts defined by overlap
• Linear-spring dashpot model-
• stiffness kn kt 1 gt n 0
• fn kn(d-r) for rltd, fn 0 for rgtd
• ft kt?s for µgt0
• static friction tracks history of contacts
• All µ start from same initial fi0.65
• incrementally decrease f towards jamming
  threshold
• quench after each step

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Jamming of Frictional Spheres

• fc zc decrease smoothly with friction

• Identify jamming transition fc
• fcf(p0)
• Fit to
• p(f-fc)
• Find zc
• (z-zc) (f-fc)0.5
• Extract
• fc(µ) zc(µ)

fc
zc
µ
Random Loose Packing fRLP, emerges as high-µ
limit of isostatic frictional packing
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Scaling of Frictional Spheres
?z(f-fc)1/2
p(f-fc)
?f(µ)f-fc(µ)
?f(µ) measures distance to jamming transition
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Scaling with Friction
Frictional thresholds exhibit power law behaviour
relative to frictionless packing
zcµ0-zcµgt0
fcµ0-fcµgt0
µ
µ
(fcµ0-fcµgt0) µ0.5
(zcµ0-zcµgt0) µ0.5
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Structure
How different are packings with different µ?
f0.64
?f10-4
g(r)
µ increasing
r-d
r-d
Packings look the same at the same ?f, but not
at same f
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Second Peak in g(r)
T 0
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Universal Jamming Diagram
Makse and co-workers, arXiv0808.2196v1, Nature
453, 629 (2008)
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Frictional Frictionless Packings

• Random Close Packing is indentified with
  zero-friction isostatic state
• Random Loose Packing identified with
  infinite-friction isostatic state
• Each friction coefficient has its own effective
  RLP state
• ?f becomes friction-dependent
• Unusual scaling of jamming thresholds
• Packings at different µ can be mapped onto
  each other
• Method to study temperature in granular
  materials?

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Careful of History Dependence
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Dynamical Heterogeneities Characteristic
Length Scales in Jammed Packings

• How can we investigate these phenomena in jammed
  systems?
• Dynamic facilitation
• Response properties

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Displacement Field in D2, µ0
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Displacement Fields ? Low-Frequency Modes
µ0
Displacement Field
Low Frequency Mode
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2D Perturbations ?fgtgt0 with µ0
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2D Perturbations ?f0 with µ0
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2D Perturbations Particle Displacements
µ0
f gtgt fc
f gt fc
f fc
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Summary

• Frictionless packings exhibit anomalous low-k,
  linear behaviour near jamming transition, S(k)
  k
• Suppression of long wavelength density
  fluctuations are a result of large length scale
  collective excitations
• Frictional packings jam in similar way to
  frictionless packings
• Location of jamming transition sensitive to
  friction coefficient
• Random Loose Packing coincides with isostaticity
  of frictional system
• Other ways to probe length scales and dynamical
  facilitation in static packings

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Acknowledgements

• Gary Barker, IFR
• Bulbul Chakraborty, Brandeis
• Andrea Liu, U. Penn.
• Sidney Nagel, U. Chicago
• Corey OHern, Yale
• Matthias Schröter, MPI
• Moises Silbert, UEA/IFR
• Martin van Hecke, Leiden

SIU Faculty Seed Grant