Statistical Mechanics of Jamming of frictional grains Dapeng ( Max ) - - PowerPoint PPT Presentation

Statistical Mechanics of Jamming

• f frictional grains

Bob Behringer & Jie Zhang, Duke University Dapeng (Max) Bi & Bulbul Chakraborty

Jamming

“A wide variety of systems, including granular media, colloidal suspensions

and molecular systems, exhibit non-equilibrium transitions from a fluid-like to a solid-like state characterized solely by the sudden arrest of their dynamics.” (Trappe et al, Nature, 411, 772 (2001))

Dry Granular: A special class since a solid can only be produced through

external stress

Increasing volume fraction

Jamming is the transition between them: is this a phase transition? Jammed (resists shear) Unjammed phase flows like a liquid

Granular: Shear can lead to jamming !

Experiments in 2D Packings of Grains with Friction

Quasistatic shear Solidity stems from applied stress itself

Jamming Diagram

Jammed Unjammed Jammed Unjammed

Shear Jammed Fragile

• J

S J

• a

b

Includes Anisotropic states Only Isotropically Jammed States

(Cates et al, PRL, 81, 1841 (1998))

(Old) and (New)

Force Networks

• 1

2 3 4 5 6 0.0 0.2 0.4 0.6 0.8 1.0

f ff P f

• 0.0

0.2 0.4 0.6 0.8 1.0 1.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

k fNR threshold

f > k favg Define k-networks

The fraction of non- rattlers can be used to distinguish between two types of states

(Cates et al, PRL, 81, 1841 (1998))

Phase Diagram Fragile Shear-Jammed Yield-Stress line

SJ states have finite yield stress at unjamming (first

• rder transition)

Cyclic Shear

Quasistatic cyclic strain creating jammed packings

Percolation (Cyclic Shear)

Quasistatic Steps

ǫxx

100 200 300 400 500 600 700 0.6 0.7 0.8 0.9 1.0

0.3 0.3

ξy/Ly for f > 0 cluster ξy/Ly for f > favg cluster ξx/Lx for f > favg cluster

fNR

}

SJ states

fNR = 0.83

What types of Solids are these?

Response of shear -jammed states Is Point J a special point? Fabric and Stress Relationship Statistical Framework: Force and torque balance are of paramount importance Concept of a Stress Temperature (Angoricity)

Topological Invariants of Mechanically Stable Packings Force and torque balance: Take a continuum approach

(x, 0) (x, L) (x + ∆x, 0) (x + ∆x, L)

(0, y) (0, y + ∆y) (L, y + ∆y) (L, y)

Cleanest with periodic boundary conditions

Thermal System

Thermal reservoir (E,V,N)

The distribution of energy is then given by: P(Em) = 1 Z Ω(Em, Vm, m)e−βEm

Granular materials (athermal)

Stress Ensemble

All blue lines have same

r F

y

L

l

• f ν

x (x0)

• Fx −

f ν

x (x0)

r f

x ν (x)

r F

x −

r f

x ν (x)

r F

x

All blue strips have the same “temperature” wrt to fluctuations

Need to consider a collection of 1d systems

The probability distribution of stresses in this subsystem

Distribution of Local Stresses

( ) ( )

Stress Ensemble Correctly Predicts Stress Distributions Shear-Jammed states have significant correlations These are less evident for the states above Point J

Correlation Length

• f ν

x (x0)

Force tile for a grain with 5 contacts area director Increasing density: tiles become more isotropic

Angoricity

• The stress distributions of the packings are characterized by a

temperature-like quantity, named angoricity

• Boltzman-like distribution
• Angoricity is a tensor that reflects the intrinsic 1d conservation

principle

• Correlation length appear is distribution
• Fragile/SJ: Phase Transition ? Nematic Model, Simulations

Bumpy Particles (with Corey O’Hern)

• 0.79

0.80 0.81 0.82 0.83 0.84 0.85 0.0 0.1 0.2 0.3 0.4 0.5

• 1

2 3 4 5 0.0 0.1 0.2 0.3 0.4

(N/m)

b a

• Order Parameter

Anisotropy of fabric vs Shear Stress Minimum anisotropy of SJ states

“Free energy”

F()

• −0()

0()

< J

}

Inaccessible region, no SJ states

slope=0()

a

F()

• slope=0

b

≥ J

Force Networks

Increasing non-rattler fraction

• 1

2 3 4 5 6 0.0 0.2 0.4 0.6 0.8 1.0

f ff P f

ǫmin φ

Z = 3 Z = 3.1

• 0.00

0.05 0.10 0.15 0.20 0.79 0.80 0.81 0.82 0.83 0.84 0.85

τ (N/m) ǫ

φ = 0.798

ǫmin

• 0.0

0.5 1.0 1.5 2.0 2.5

• 0.00

0.05 0.10 0.15 0.20 0.25 0.30 0.0 0.2 0.4 0.6 0.8 1.0

fNR

Fraction of non-rattlers Shear Stress Forward Shear

Pressure is not just a function of packing fraction Stresses are determined by fraction of non-rattlers

Summary so far

Anisotropic jammed states occur in frictional systems at packing fractions well below the isotropic jamming threshold The fraction of non-rattlers emerges as parameter that controls the behavior of force networks, and therefore the rigidity Two qualitatively different classes of states: For infinitely rigid grains, fragile states cannot sustain any load that is not along there force-network axis. Both are fragile jammed states. Yield stress of SJ states vanishes discontinuously (Hayakawa, simulations) The observed anisotropic jamming is a reflection of dilatancy: under constant area conditions, shearing leads to jamming