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

The influence of particle shape on jamming: From ellipsoids to dimers to bumpy particles to friction Prof. Corey S. O Hern Department of Mechanical Engineering & Materials Science Department of Physics Yale University

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

Statistical Mechanics of Granular Media  PRE 57 (1998) 1971 EPJE 3 (2000) 309 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?

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

Simple Granular Model: Frictionless Disks repulsive central forces, F ij ~   ~ (1-r ij /  ij )  ,  =1  zero force, F ij = 0 r m r  b r  a i  F v i Minimize energy V(r) ij to reach T=0 at each  j    V r            1  r r V r ij V r   for overlapping particles ij ij  ij   i  j

MS Packing-Generation Algorithm V (  V (  V (  r ) r ) r ) shrink   grow Local  minimum  r r  0 0 Mechanically stable 0 r Degenerate packing minima Mechanically stable overlapped non-overlapped packing

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Jamming of spherical particles via isotropic compression  max   c  0.91 Slow quench; 2 3 xtalline; hyperstatic N=256 512 1024 <  c > min  fast quench;  c amorphous; isostatic b 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).

Jammed = mechanically stable (MS) configuration with extremely small particle overlaps; net forces (and torques) are zero on each particle; quadratically stable to small perturbations Isostaticity      V r iso  Nd f  d  1 N c  N c    r z  2 N c z  z iso  2 d f ; MS packng N

Configuration is mechanically stable if dynamical matrix contains d f N-d eigenvalues  2 > 0 (periodic b.c.s) M  ,    2 V (  r )  ,  =x, y, z, particle index   r   r r 0 = positions of MS packing  r    r 0

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.

Packings of ellipse-shaped particles bidisperse a 2 b 2 a 1 QuickTime and a b 1 Photo - JPEG decompressor are needed to see this picture. a 1  a 2   b 1 b 2 a 1  1.4 a 2 compression method-fixed aspect ratio 

Pairwise Repulsive Interactions: True Contact Distance This image cannot currently be displayed. This image cannot currently be displayed. This image cannot currently be displayed.  ij r This image cannot currently be displayed. ij    0    0 V r V r ij ij      1  r   ij r   ij        =2; linear springs  ij V r    ij  r   ij 0  

Average Contact Number for Ellipse Packings annealing Missing contacts compression Naïve isostatic condition for ellipses: z  z iso  2 d f Not a discontinuous jump from <z> = 4 to 6.

Average Contact Number for Ellipsoid Packings Naïve isostatic condition for ellipsoids: z  z iso  2 d f Not a discontinuous jump from <z> = 6 to 10.

If z < z iso , are ellipsoid packings mechanically stable?

Density of Vibrational Modes from Dynamical Matrix N(z iso -z) modes  =1 1.001 1.05 2 Dynamical matrix eigenvalues  2 > 0 for all d f N - d 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).

Scaling of Characteristic Frequencies  =10 -8  3  2  1

Perturbations along lowest frequency eigenmodes Slope=2 lowest frequency  =1 for disks Slope=4 lowest frequency for ellipses  *  =1.1   12    * : Crossover frequency scales as   14   1

Ellipsoid packings are quartically stabilized at  =0; i.e. For N(z iso -z) modes,  V  4 ; for Nz modes,  V  2 inaccessible inaccessible convex concave constraint constraint Spherical, ellipsoidal Only ellipsoidal particles particles

What is the difference between between a dimer and an ellipse? b b a a  = a/b

Dimer packings are isostatic with no quartic modes <z>  c dimers dimers ellipses ellipses  

Weaker linear response to shear filled=dimers Slope=0.5 open=ellipses Slope=1 (  -  c )

Microstates of Frictional Packings: Geometrical Families

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

Frictional Geometric Families y c  s A x c  s Plot of all centers of mass that evolve to MS packing A

Bumpy Particle Model for Friction  R  max  2 N b R b Linear repulsive spring bump-bump, bump-particle, and particle- particle interactions

frictional contact frictionless contact

Hertz-Mindlin Friction Model t ij t   F 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

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bb  17  c =0.6131, N c =10, N c Hertz-Mindlin Bumpy-particle model

`Minimum Distance from Reference MS Packing

Comparison of Hertz-Mindlin and Bumpy-Particle Minimum-Distance Maps bumpy particles Hertz-Mindlin family index

isostatic 3N-1 2N-1

Energy Minimization Tolerance isostatic 3N-1 bb N c 2N-1  max

Hertz-Mindlin Results L. Silbert, Soft Matter, 6 (2010) 2918.

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