Features of Jamming in Frictionless & Frictional Packings Leo Silbert Department of Physics Wednesday 3 rd September 2008

JAMMED UNJAMMED 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…  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?

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 Phenomena Granular materials are ubiquitous throughout nature Natural phenomena Natural disasters avalanche.orgbbc.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 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 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 φ c the packing unjams   The properties of the packing are determined by the distance to the jamming transition: Δφ = φ - φ c

Jamming of Soft Spheres   Monodisperse, frictionless, soft spheres :   finite range, repulsive, potential: V(r) = V 0 (1-r/d) 2 r < d 0 otherwise   Transition between jammed and unjammed phases at critical packing fraction φ c   Frictionless Spheres:   critical packing fraction coincides with value of random close packing, φ c ≈ 0.64 in 3D (≈ D)   packings are isostatic at the jamming transition, coordination number z c = 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, (2005); Nature 453, 629 (2008) O’Hern et.al, Phys. Rev. Lett. 88, (2002); Phys. Rev. E 68, (2003) Kasahara & Nakanishi, Phys. Rev. E 70, (2004) van Hecke and co-workers, Phys. Rev. Lett. 97, (2006); Phys. Rev. E 75, (2007); 75, (2007) Agnolin & Roux, Phys. Rev. E, (2007)

Static 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>>φ c Weakly compressed: low P, φ≈φ c Makse & co-workers, Phys. Rev. Lett. 84, 4160 (2000) Silbert et al., Phys. Rev. E 73, (2006) Δφ->0

Vibrational Density of States Debye isostatic Characteristic feature in D(ω) signals onset of jamming “boson peak” D(ω) ~ constant at low ω Peak shifts to lower ω Packing become increasingly soft Silbert et al. Phys. Rev. Lett. 95, (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 Peak Adding the Debye contribution: The dispersion relations read:

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, (2005)   How do we identify length scales in static packings?

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 > 10 4 needed to resolve low-k region   Donev et al. used particles, φ - ≈0.64   S(k) = 1/N

Hard Spheres   Structure factor for a jammed N=10 6 ; 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 (2005)

Soft Sphere Liquid T > 0 Expected behaviour in the liquid state

Jammed Soft Sphere Packings In jammed packings: S(k) ~ k, near jamming Transition to linear behaviour Observed using N=256000, at φ + ≈0.64 T = 0

Phenomenology   Second moment of dynamical structure factor…   Conjecture: assume the dominant collective mode is given by dispersion relation ω 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)]

  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 Jamming

Effect of Friction on S(k) T = 0 At the same φ low-k behaviour different φ≈0.64

Comparison Between Frictionless & Frictional Packings Frictional   Stable over wide range of packing fractions – –0.55 < φ < 0.64   Random Loose Packing – –φ RLP ≈ 0.55 – –[Onada & Liniger, PRL 1990] – –[Schroter et al. Phys. Rev. Lett. 101, (2008)]   Are frictional packings isostatic? – –z(μ>0) iso = D+1 Frictionless  Random Close Packing –φ RCP ≈ 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, (2006) Behringer & co-workers, Phys. Rev. Lett. 98, (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: k n = k t = 1 => = 0 – –f n = k n (d-r) for r d – –f t = k t Δs for μ>0 – –static friction tracks history of contacts   All μ: start from same initial φ i =0.65 – –incrementally decrease φ towards jamming threshold – –quench after each step Jamming Protocol

zczc Jamming of Frictional Spheres   Identify jamming transition φ c – c –φ c =φ(p=0)   Fit to: – –p~(φ-φ c )   Find z c : – –(z-z c ) ~ (φ-φ c ) 0.5   Extract: – –φ c (μ) & z c (μ) φcφc μ  φ c & z c decrease smoothly with friction Random Loose Packing φ RLP, emerges as high-μ limit of isostatic frictional packing

Scaling of Frictional Spheres Δφ(μ)=φ-φ c (μ) p~(φ-φ c ) Δφ(μ): measures distance to jamming transition Δz~(φ-φ c ) 1/2

Scaling with Friction μ (φ c μ=0 -φ c μ>0 ) ~ μ 0.5 (z c μ=0 -z c μ>0 ) ~ μ 0.5 z c μ=0 -z c μ>0 φ c μ=0 -φ c μ>0 Frictional thresholds exhibit power law behaviour relative to frictionless packing μ

Structure φ=0.64Δφ=10 -4 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) T = 0

Universal Jamming Diagram Makse and co-workers, arXiv: v1, Nature 453, 629 (2008)

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   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 Dependence

Dynamical Heterogeneities & Characteristic Length Scales in Jammed Packings   How can we investigate these phenomena in jammed systems?   Dynamic facilitation   Response properties

Displacement Field in D=2, μ=0

Displacement Fields  Low-Frequency Modes μ=0 Displacement FieldLow Frequency Mode

2D Perturbations: Δφ>>0 with μ=0

2D Perturbations: Δφ≈0 with μ=0

2D Perturbations: Particle Displacements φ >> φ c φ > φ c φ ≈ φ c μ=0

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

Acknowledgements Gary Barker, IFR Bulbul Chakraborty, Brandeis Andrea Liu, U. Penn. Sidney Nagel, U. Chicago Corey O’Hern, Yale Matthias Schröter, MPI Moises Silbert, UEA/IFR Martin van Hecke, Leiden SIU Faculty Seed Grant