Features of Jamming in Frictionless & Frictional Packings

1. Features of JamminginFrictionless & Frictional Packings Leo Silbert Department of Physics Wednesday 3rd September 2008

2. Jamming is the transition between solid-like and fluid-like phases in disordered systems What Is Jamming? JAMMED UNJAMMED Many macroscopic and microscopic complex phenomena associated with jammed states and the transition to the unjammed phase

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

4. Foams [Durian] Supercooled Liquids [Glotzer ] Sphere Packings Colloidal Suspensions [Weeks] Emulsions [Brujic et al.] Grain Piles

5. 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?

7. 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

8. 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

9. Grain Piles Duke Group • Contact forces are highly heterogeneous • “force chains” • Distribution of forces • wide distribution • exponential at large forces Chicago Group

10. 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 φc the packing unjams • The properties of the packing are determined by the distance to the jamming transition: Δφ = φ - φc

11. Jamming of Soft Spheres • Monodisperse, frictionless, soft spheres : • finite range, repulsive, potential: V(r) = V0 (1-r/d)2r < 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)

12. Static Packings Under Pressure Compressed FrictionlessSpheres (no gravity) Force distributions and pressure • Packing becomes more ‘uniform’ with increasing pressure • Exponential to Gaussian crossover with increasing pressure Very compressed: high P, φ>>φc Δφ->0 Weakly compressed: low P, φ≈φc Makse & co-workers, Phys. Rev. Lett. 84, 4160 (2000) Silbert et al., Phys. Rev. E 73, 041304 (2006)

13. isostatic Debye Vibrational Density of States Characteristic feature in D(ω) signals onset of jamming “boson peak” Packing become increasingly soft D(ω) ~ constant at low ω Peak shifts to lower ω Silbert et al. Phys. Rev. Lett. 95, 098301 (2005)

14. 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:

15. 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?

17. 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 > 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)>

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

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

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

21. Phenomenology • 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). [Silbert & Silbert (2008)]

22. 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?

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

24. Comparison Between Frictionless & 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 isRCP • 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)

25. Jamming Protocol • 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 => n = 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

26. Identify jamming transitionφc φc=φ(p=0) Fit to: p~(φ-φc) Find zc: (z-zc) ~ (φ-φc)0.5 Extract: φc(μ) & zc(μ) Jamming of Frictional Spheres • φc & zc decrease smoothly with friction φc zc μ Random Loose Packing φRLP, emerges as high-μ limit of isostatic frictional packing

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

28. Scaling with Friction Frictional thresholds exhibit power law behaviour relative to frictionless packing zcμ=0-zcμ>0 φcμ=0-φcμ>0 μ μ (φcμ=0-φcμ>0) ~ μ0.5 (zcμ=0-zcμ>0) ~ μ0.5

29. Structure How different are packings with different μ? φ=0.64 Δφ=10-4 g(r) μ increasing r-d r-d Packings ‘look’ the same at the same Δφ, but not at same φ

30. Second Peak in g(r) T = 0

31. Universal Jamming Diagram Makse and co-workers, arXiv:0808.2196v1, Nature 453, 629 (2008)

32. 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?

33. Careful of History Dependence

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

35. Displacement Field in D=2, μ=0

36. Displacement Fields  Low-Frequency Modes μ=0 Displacement Field Low Frequency Mode

37. 2D Perturbations: Δφ>>0 with μ=0

38. 2D Perturbations: Δφ≈0 with μ=0

39. 2D Perturbations: Particle Displacements μ=0 φ >> φc φ > φc φ ≈ φc

40. 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

41. 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 Acknowledgements SIU Faculty Seed Grant