1. 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 , Duration: to
NSF-DMR , Duration: to
NSF-MRSEC , Duration: to

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

3. 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

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

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

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

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

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

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

10. Distance Matrix for Frictionless Packings6 8

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

12. 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

13. Cundall-Strack Model for Frictionj

15. Cundall-Strack Packings for N=6.
lines

16. 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

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

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

19. 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 . .

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

23. 3N

24. 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.

25. Geometrical Family #1
CS packings
μmin L

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

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

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

31. 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

32. μ=10-3
μ=1