The influence of particle shape on jamming: From ellipsoids to - - PowerPoint PPT Presentation

• Prof. Corey S. O Hern

Department of Mechanical Engineering & Materials Science Department of Physics Yale University

The influence of particle shape on jamming: From ellipsoids to dimers to bumpy particles to friction

http://jamming.research.yale.edu/

The O Hern Group

The O'Hern group in the Fall 2011: (from left to right) Thibault Bertrand, Diego Caballero, Wendell Smith, Mate Nagy, Mark Shattuck, Alice Zhou, Jared Harwayne-Gidansky, Corey O'Hern, Georgia Lill, Maxwell Micali, Minglei Wang, Robert Hoy, Tianqi Shen, Carl Schreck, S. S. Ashwin, and Stefanos Papanikolaou

Apply driving to attain reversible set of states Different driving mechanisms lead to different sets of states! What are the microstates of granular packings and what determines their probabilities?



PRE 57 (1998) 1971 EPJE 3 (2000) 309

Statistical Mechanics of Granular Media

Attributes of Simple Granular Materials

• 1. Finite number of macroscopic spherical grains
• 2. Dissipative and repulsive contact interactions; exist at

`zero temperature unless driven by external forces

• 3. Non-spherical particle shapes
• 4. Static frictional and `history-dependent interactions

repulsive central forces, Fij~  ~ (1-rij/ij), =1 zero force, Fij = 0

mr ai  r F

ij j



 br vi

Simple Granular Model: Frictionless Disks

 Minimize energy V(r) to reach T=0 at each 

V r r

 

V r

ij

 

i j



V r

ij

  

 1 r

ij

 ij      



for overlapping particles

V( r)

 r

Mechanically stable packing Local minimum

 r

V( r)

 r

V( r)

Degenerate minima shrink grow

• verlapped

non-overlapped

Mechanically stable packing

MS Packing-Generation Algorithm

QuickTime and a TGA decompressor are needed to see this picture.

c

max 

 2 3  0.91

N=256 512 1024

c

min 

<c>

Jamming of spherical particles via isotropic compression

• P. Chaudhuri, L.Berthier, & S. Sastry, Phys. Rev. Lett, 104, 165701 (2010).
• C. F. Schreck, C. S. O Hern, & L. E. Silber, Phys. Rev. E 84, 011305 (2011).

fast quench; amorphous; isostatic Slow quench; xtalline; hyperstatic b

Jammed = mechanically stable (MS) configuration with extremely small particle overlaps; net forces (and torques) are zero on each particle; quadratically stable to small perturbations

 r

 

V  r

 

 

MS packng

Nc  Nc

iso  Nd f  d 1

z  ziso  2d f ; z  2Nc N

Isostaticity

Configuration is mechanically stable if dynamical matrix contains dfN-d eigenvalues 2 > 0 (periodic b.c.s)

M,   2V( r) r

r   r   r0

,=x, y, z, particle index

 r

0 = positions of

MS packing

Shape Matters: Packings of Frictionless Ellipsoidal Particles Are Stabilized by Quartic Modes

• C. F. Schreck, M. Mailman, B. Chakraborty, & C. S. O Hern, Constraints and vibrations in static

packings of ellipsoidal particles, submitted to PRE (2012).

• A. Donev, S. Torquato, & F. H. Stillinger, Phys. Rev. E 71 (2005) 011105.
• Z. Zeravic, N. Xu, A. J. Liu, S. R. Nagel, & W. van Saarloos, EPL 87 (2009) 26001.

QuickTime and a Photo - JPEG decompressor are needed to see this picture.

Packings of ellipse-shaped particles

compression method-fixed aspect ratio  bidisperse a2 b2 a1 b1 a1 b1  a2 b2  

a1 a2  1.4

Pairwise Repulsive Interactions: True Contact Distance

 ij

r

ij

V r

ij

  0

V r

ij

 

  1 r

ij

 ij      



r   ij r   ij       

V r

ij

  0

=2; linear springs

annealing

Average Contact Number for Ellipse Packings

compression

Naïve isostatic condition for ellipses:

Not a discontinuous jump from <z> = 4 to 6.

Missing contacts

z  ziso  2d f

z  ziso  2d f

Naïve isostatic condition for ellipsoids:

Average Contact Number for Ellipsoid Packings

Not a discontinuous jump from <z> = 6 to 10.

If z < ziso, are ellipsoid packings mechanically stable?

Density of Vibrational Modes from Dynamical Matrix

• C. F. Schreck, M. Mailman, B. Chakraborty, & C. S. O Hern, Constraints and vibrations in static

packings of ellipsoidal particles, submitted to PRE (2012). =1 1.001 1.05 2

Dynamical matrix eigenvalues 2 > 0 for all dfN - d modes

N(ziso-z) modes

Scaling of Characteristic Frequencies

3 1 2

=10-8

lowest frequency for disks lowest frequency for ellipses

Slope=2 Slope=4

Perturbations along lowest frequency eigenmodes

*

 * : 

 

12

 1

 

14

=1 =1.1

Crossover frequency scales as

concave constraint convex constraint

Ellipsoid packings are quartically stabilized at =0; i.e. For N(ziso-z) modes, V4; for Nz modes, V2

Spherical, ellipsoidal particles inaccessible inaccessible Only ellipsoidal particles

What is the difference between between a dimer and an ellipse?

a b a b  = a/b

<z> c  

dimers ellipses dimers ellipses

Dimer packings are isostatic with no quartic modes

filled=dimers

• pen=ellipses

Slope=1 Slope=0.5

Weaker linear response to shear

(-c)

Microstates of Frictional Packings: Geometrical Families

QuickTime and a Cinepak decompressor are needed to see this picture.

Frictional Geometrical Families

Frictional Geometric Families

xc  s yc  s

Plot of all centers of mass that evolve to MS packing A

A

Bumpy Particle Model for Friction

Linear repulsive spring bump-bump, bump-particle, and particle- particle interactions

max  R 2NbRb

frictional contact frictionless contact

Hertz-Mindlin Friction Model

tij

F

t   F n

Provides energy sink when contacts break

Advantages of Bumpy-Particle Model over Hertz-Mindlin

No ad hoc sliding, history dependence Forces depend only on particle positions and orientations; Use dynamical matrix to calculate vibrational response Test Hertz-Mindlin mobility distribution, P(m) m  F

t

F

n

QuickTime and a GIF decompressor are needed to see this picture.

Hertz-Mindlin Bumpy-particle model

c=0.6131, Nc=10, Nc

bb  17

`Minimum Distance from Reference MS Packing

bumpy particles Hertz-Mindlin

Comparison of Hertz-Mindlin and Bumpy-Particle Minimum-Distance Maps

family index

2N-1 3N-1 isostatic

max

Nc

bb

3N-1 2N-1

Energy Minimization Tolerance

isostatic

• L. Silbert, Soft Matter, 6 (2010) 2918.

Hertz-Mindlin Results

Conclusions

Nonspherical particle shapes changes simple `jamming scenario for spherical grains

• 1. Quartic modes lead to linear softening, perhaps nonlinear

strengthening

• 2. For bumpy particles, microstates occur as geometrical

families, instead of random points in configuration space