Features of JammingFeatures of Jamminginin

Frictionless & Frictional PackingsFrictionless & Frictional Packings

Leo SilbertDepartment of Physics

Wednesday 3rd September 2008

JAMMEDJAMMED UNJAMMEDUNJAMMED

What Is Jamming?What Is Jamming?

Jamming is the transition between solid-like and fluid-like phases in disordered systems

Many macroscopic and microscopic complex phenomena associated with jammed states and the transition to the unjammed phase

Similarities…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

Supercooled Liquids[Glotzer ]

Foams[Durian]

Grain Piles

Colloidal Suspensions[Weeks] Emulsions

[Brujic et al.]

Sphere Packings

Back To Basics

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

How to Study Jamming?How to Study Jamming?

Why Does Granular Matter?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

Granular PhenomenaGranular PhenomenaGranular materials are ubiquitous throughout

nature

Natural phenomena

Natural disasters

avalanche.org bbc.co.uk

large-scale geological features

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

Grain PilesGrain Piles

Duke Group

Chicago Group

Contact forces are highly

heterogeneous

– “force chains”

Distribution of forces

– wide distribution

– exponential at large

forces

Jamming Transition in Static Jamming Transition in Static PackingsPackings

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 φc the packing unjams

The properties of the packing are determined by the distance to the jamming transition:

Δφ = φ - φc

Jamming of Soft SpheresJamming of Soft Spheres Monodisperse, frictionless, soft spheres :

finite range, repulsive, potential: V(r) = V0 (1-r/d)2 r < d

0 otherwise

Transition between jammed and unjammed phases at critical packing fraction φc

Frictionless Spheres: critical packing fraction coincides with value of random

close packing, φc ≈ 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)O’Hern 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)

Static Packings Under PressureStatic Packings Under Pressure

Force distributions and pressure– Packing becomes more ‘uniform’

with increasing pressure– Exponential to Gaussian crossover

with increasing pressure

Compressed Frictionless Spheres (no gravity)

Very compressed: high P, φ>>φc

Weakly compressed: low P, φ≈φc

Makse & co-workers, Phys. Rev. Lett. 84, 4160 (2000) Silbert et al., Phys. Rev. E 73, 041304 (2006)

Δφ->0

Vibrational Density of StatesVibrational Density of States

Debye

isostatic

Characteristic feature in D(ω) signals onset of jamming

“boson peak”

D(ω) ~ constant at low ωPeak shifts to lower ω

Packing become increasingly soft

Silbert et al. Phys. Rev. Lett. 95, 098301 (2005)

Jamming transition accompanied by a diverging boson peak Two length scales characterize dynamical modes

L (longitudinal correlation length)

T (transversal correlation length)

Jamming and the Boson PeakJamming and the Boson Peak

modes) transverseξω

modes) allongitudinξω2

TB

1LB

(

(

kβkcω LL 2

TT kαkcω

Adding the Debye contribution:

The dispersion relations read:

kcωD

Jamming Jamming Critical Phenomena ? 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?

Low-k Behaviour of S(k) Low-k Behaviour of S(k) in in

Jammed Hard & Soft SpheresJammed 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 > 104 needed to resolve low-k region Donev et al. used 105-106 particles, φ-≈0.64 S(k) = 1/N<ρ(k)ρ(-k)>

Hard SpheresHard Spheres

Structure factor for a jammed N=106; φ=0.642, and for a hard sphere liquid near the freezing point, φ=0.49, as obtained numerically and via PY theory

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

Soft Sphere LiquidSoft Sphere Liquid T > 0

Expected behaviour in the liquid state

Jammed Soft Sphere PackingsJammed Soft Sphere Packings

In jammed packings: S(k) ~ k, near jamming

Transition to linear behaviourObserved using N=256000, at φ+≈0.64

T = 0

PhenomenologyPhenomenology Second moment of dynamical

structure factor…

Conjecture: assume the dominant collective mode is given by dispersion relation ωB(k). Then…

…and

Assuming…

at small k; then in the long wavelength limit…

transverse modes contribute to linear behaviour of S(k).

220

2 kνω,kSωωd

kωωδS(k)ω,kS B

k2ωk

νkS 2B

220

...kakckω 2B

22

0 kOkca

212cv

kS

[Silbert & Silbert (2008)]

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?

S(k) as a Signature of JammingS(k) as a Signature of Jamming

Effect of Friction on S(k)Effect of Friction on S(k) T = 0

At the same φ low-k behaviour different

φ≈0.64

Comparison Between Comparison Between Frictionless & Frictional PackingsFrictionless & Frictional Packings

Frictional Stable over wide range

of packing fractions– 0.55 < φ < 0.64

Random Loose Packing– φRLP ≈ 0.55 – [Onada & Liniger, PRL 1990]– [Schroter et al. Phys. Rev. Lett.

101, 018301 (2008)]

Are frictional packings isostatic?– z(μ>0)iso = D+1

Frictionless Random Close Packing

– φRCP ≈ 0.64 – [Bernal, Scott, 1960’s]

Jamming transition is RCP

Frictionless packings at RCP are isostatic– z(μ=0)iso = 2D

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

Follow similar protocol used for frictionless studies– N=1024 monodisperse soft-spheres: d = 1– particle-particle contacts defined by overlap

Linear-spring dashpot model:-– stiffness: kn = kt = 1 => = 0– fn = kn(d-r) for r<d, fn= 0 for r>d– ft = ktΔs for μ>0– static friction tracks history of contacts

All μ: start from same initial φi=0.65– incrementally decrease φ towards jamming

threshold– quench after each step

Jamming ProtocolJamming Protocol

zc

Jamming of Jamming of FrictionalFrictional Spheres Spheres

Identify jamming transition φc

– φcc=φ(p=0) Fit to:

– p~(φ-φc) Find zc:

– (z-zc) ~ (φ-φc)0.5

Extract: – φc(μ) & zc(μ)

φc

μ

φc & zc decrease smoothly with friction

Random Loose Packing φRLP, emerges as high-μ limit of isostatic frictional packing

Scaling of Scaling of FrictionalFrictional Spheres Spheres

Δφ(μ)=φ-φc(μ)

p~(φ-φc)

Δφ(μ): measures distance to jamming transition

Δz~(φ-φc)1/2

Scaling Scaling withwith Friction Friction

μ

(φcμ=0-φc

μ>0) ~ μ0.5 (zcμ=0-zc

μ>0) ~ μ0.5

z cμ

=0-z

cμ>

0

φcμ

=0-φ

cμ>

0

Frictional thresholds exhibit power law behaviour relative to frictionless packing

μ

StructureStructure

φ=0.64 Δφ=10-4

r-d r-d

How different are packings with different μ?

Packings ‘look’ the same at the same Δφ, but not at same φ

g(r) μ increasing

Second Peak in g(r)Second Peak in g(r)

T = 0

Universal Jamming DiagramUniversal Jamming Diagram

Makse and co-workers, arXiv:0808.2196v1, Nature 453, 629 (2008)

Frictional & Frictionless PackingsFrictional & 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

Δφ 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?

Careful of History DependenceCareful of History Dependence

Dynamical Heterogeneities Dynamical Heterogeneities &&

Characteristic Length Scales Characteristic Length Scales in in

Jammed PackingsJammed Packings

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

Displacement Field in D=2, Displacement Field in D=2, μμ=0=0

Displacement Fields Displacement Fields Low-Frequency Low-Frequency ModesModesμμ=0=0

Displacement Field Low Frequency Mode

2D Perturbations: 2D Perturbations: ΔφΔφ>>0 with >>0 with μμ=0=0

2D Perturbations: 2D Perturbations: ΔφΔφ≈0 with ≈0 with μμ=0=0

2D Perturbations: Particle 2D Perturbations: Particle DisplacementsDisplacements

φ >> φc φ > φc φ ≈ φc

μμ=0=0

SummarySummary 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

AcknowledgementsAcknowledgements

Gary Barker, IFRGary Barker, IFRBulbul Chakraborty, BrandeisBulbul Chakraborty, BrandeisAndrea Liu, U. Penn.Andrea Liu, U. Penn.Sidney Nagel, U. ChicagoSidney Nagel, U. ChicagoCorey O’Hern, YaleCorey O’Hern, YaleMatthias Schröter, MPIMatthias Schröter, MPIMoises Silbert, UEA/IFRMoises Silbert, UEA/IFRMartin van Hecke, LeidenMartin van Hecke, Leiden

SIU Faculty Seed Grant