Frictional Jamming & MRJ/RLP crossover

Transcription

1 Frictional Jamming & MRJ/RLP crossover Stefanos Papanikolaou (Yale University) Ref: SP, C. S. O Hern, M. D. Shattuck, arxiv: (2012)

2 Outline Isostaticity in Jamming, Friction and the failure of isostaticity The Geometric Asperity model: A Hamiltonian way to model static friction -- Isostaticity for frictional jamming Vibrational Density of States at finite friction for truly mechanically stable (i.e. jammed) frictional packings Emergence of gearness at finite friction The Random-Close-Packing to Random-Loose- Packing crossover at finite friction as a competition of local particle cross-sections and the emergence of the dynamic Gear-Percolation transition

3 Jamming and Isostaticity Frictionless particle systems with a single curvature (disks/spheres) jam only when: (#forces) = (# degrees of freedom)+1 <Z c >~4 (2D) and <Z c >~6 (3D) The argument breaks down for tunably smooth shapes (ie. ellipses) Isostaticity at jamming provides a proof of marginality

4 Previously (1) Cundall-Strack model (CS) for studying jammed frictional packings static frictional forces develops with mutual contact sliding Large accumulated phenomenology for over 10 years. Facts: i) z~d+1 at large friction, ii) RCP to RLP crossover at ~0.1 (2D) and ~0.01(3D) for typical protocol. iii) A peak develops at mobility (ft/fn=μ) for low friction Mobility distribution Silbert ( ) φj and z vs. µμ (2D) Silbert (2010- review)

5 Previously (2) Attempts to generalize the isostaticity argument at finite friction for specially prepared packings. Argument depends on the model, numerical implementation and packing preparation protocol Vibrational Density of States literally in the air Vibrational DOS slipping contacts neglected Van Hecke 2010 Vibrational DOS slipping contacts kept

6 The f(r)ictional mysteries Is the frictionless isostaticity not existent at finite friction for generic protocols? What is the truth in frictionally jammed vibrational density of states? Can we build predictive intuition around it? Random Close Packing (RCP) to Random Loose Packing (RLP) crossover or transition? (Scott 1960)

7 A/ Model Static Friction in a Hamiltonian/geometric manner

8 The Geometric Asperity (GA) Model for Friction Decorate the disk s perimeter/surface with a variable number of disks of varying size. Simplest/ most-efficient modeling recipe. Effective friction is defined by maximum tangential to vertical force ratio (F t /F n ) max µ eff /(2N b (R b /R))

9 Parameters and tunability Rb/R and Nb are the only two parameters that define friction. Typically, choose Nb and then vary friction by tuning Rb/R (F t /F n ) max µ eff /(2N b (R b /R))

10 Packing Agreement between GA and CS modeling Packings have similar probabilities at all frictions, even the mobility distributions, if interpreted appropriately

11 Geometric Isostaticity at Frictional Jamming Exact microscopic overlap isostaticity when coarse-grained particle-particle contacts become hypostatic in agreement with Cundall- Strack

12 MRJ to RLP as a problem of geometric cross- sections At each contact, there can be a simple single bump-bump contact or double bump-divot contact doubles give 2 overlaps at the price of 1 contact! Low friction, their cross-section is small At high friction, single and double have equal cross-sections The competition for the ~6 isostatic contacts/particle leads to the transition from MRJ to RLP at a finite friction.

13 Signatures of the transition at the vibrational DOS Reorganization of the density of states at around the crossover friction 0.1 and loss of low-frequency gear peak Linear scale compares well with parts of old data, had missed the low-μ gear-motion divergence Log scale No plateau but instead peak! Linear scale

14 From the rigidity transition to the gear- rigidity transition 1980s 2000s Connectivity/Contact percolation Rigidity percolation/jamming No network bending that can generate contacts Gear Rigidity, Network Bending that generates new contacts

15 G = The gear- ness order parameter Z!0 0 d! P <ij> <d i(!)d j (!) N P i <d i(!) > 2 G > 0 characterizes the amount of gear-ness in vibrational spectrum G displays a transition at finite μ(in progrss)

16 Gear- Rigidity percolation Multi-asperity contacts in the GA model form spanning clusters at μ*~0.1 Same behavior for immobile (ζ<0.5) CS contacts (in progress)

17 μ$ μ * $ The gear- percolation transition Random Loose Packing Maximally Random Jammed Gear Percolation 1/φ J " 1/φ rigidity transition through low energy gear motions, facilitating low density packings Nature teaches us how to create random loose packings. Can there be smarter human protocols to generate non- MRJ packings? (Torquato et al. JAP 109, , 2011) exponents similar but different from usual 2D percolation -- preliminary estimates (μ * ~0.1): ν~2.4(4/3),β~0.6(5/36),γ~4.5(43/18),d f ~1.75 (91/48)

18 Conclusions New studies of frictional jamming, both statics and dynamics, using geometric friction models (GA) correcting old confusions in acoustic properties (arxiv: , submitted to PRL) Possible Material mechanical response by design in controlling the shape to acquire effective frictional response Novel insights for the mechanism of generating random loose packings: The gear-rigidity transition (in preparation) Thanks to C. O Hern, M. D. Shattuck, C. F. Schreck, V. K. Kumar, T. Bertrand, M. Wang, D. Kwok, S. S. Ashwin