Friction stabilizes saddle packings Prof. Corey S. O’Hern Department of Mechanical Engineering and Materials Science Department of Applied Physics Department of Physics Yale University DTRA BRBAA08-H-2-0108, Duration: 4-1-10 to 3-31-15 NSF-DMR-1006537, Duration: 9-1-10 to 8-31-13 NSF-MRSEC-1119826, Duration: 9-1-11 to 8-31-16

single grain `Material’ > 1m Material properties  grain properties Structure & interactions between grains will determine material properties

What is the characteristic μ* above which static packings transition 2d bidisperse μ* What is the characteristic μ* above which static packings transition from frictionless to frictional? Does this crossover depend on N? Is frictional jamming similar to frictionless jamming? L. Silbert, “Jamming of frictional spheres and random loose packing,” Soft Matter 6 (2010) 2918

small μ; <z>=4; ϕJ~0.84 large μ; <z>~3; ϕJ~0.76

Contact Interactions  =2 Total potential energy ij overlapped non-overlapped  =2 ij Total potential energy

Potential Energy Landscape (PEL) shrink grow Local minimum Mechanically stable packing Degenerate minima Mechanically stable packing overlapped non-overlapped

Frictionless Granular Packings Amorphous Vanishing overlaps Isostatic: Nc = dN - d + 1 Force balance Mechanically stable 1.4  Jammed packing: J, zJ, V ~ P ~ 0

{ϕ, } 20 distinct isostatic MS packings with 11 (9) contacts N/2 large, N/2 small with diameter ratio σL/σS=1.4

Classification of Packings Distance matrix particles i, j Second invariant All particle pairs, or only those in contact

Distance Matrix for Frictionless Packings 6 8

Frictionless Packing Probabilities: N=6 LS MD S. S. Ashwin, J. Blawzdziewicz, C. S. O’Hern, & M. D. Shattuck,PRE 85 (2012) 061307.

P(μ=0,z=4)=1; <z>(0)=4 P(μ=∞,z=3)=1; <z>(∞)=3 What is P(z,μ)? Describe in terms of saddles of frictionless states. P(μ=0,z=4)=1; <z>(0)=4 P(μ=∞,z=3)=1; <z>(∞)=3 C. Song, P. Wang, & H. A. Makse, Nature 453 (2008) 629

Cundall-Strack Model for Friction j

Cundall-Strack Packings for N=6 . lines

Packings of frictional particles are saddle packings of frictionless particles Saddle number Nc Nc=2N-1 . 1 1 μ . . N/2-1 N/2-1 Nc=3/2N

Probability for Nc contacts for N=6 Cundall-Strack Packings

Monte-Carlo plus Compression Grow to 1st contact Move each particle randomly Grow to 1st contact

Double-sided Spring Method isostatic Nc=11 Saddle 1 Nc=10 Ns x Nc saddle 1 Ns x Nc x (Nc-1)/2 saddle 2 double-sided . .

Which saddles are populated at a given friction coefficient μ? Determine μmin for each saddle packing

3N

2Nc variables > 3N equations 2Nc – 3N = dimension of nullspace Use eigenvectors as basis. All linear combinations of eigenvectors. Find mu_min in this space. Do not allow invalid solutions.

Geometrical Family #1 CS packings μmin L

Volume of null space for 1st order saddles Vn (μ) ~ μ6 V(μ) = # of valid solutions with μ>μmin # of valid solutions

Lr ~ μ r Length of line where mu>mu_min for each 1st order saddle.

Am/A0=5^m (C^m_{2N-1})^2.5 m+Nm-N0=1.3 m

Conclusions Understand properties of frictional packings in terms saddle packings of frictionless particles 2. Can predict μ* (e.g. max |dz/dμ|) and show that crossover is only weakly dependent on system size. 3. Analogies to calculations of vanishing diffusion coefficient for dense hard-sphere liquids as ϕ approaches ϕJ

μ=10-3 μ=1