JammingPeter Olsson, Umeå University

Stephen Teitel, University of Rochester

Supported by: US Department of Energy

Swedish High Performance Computing Center North

Quantum Jamming in the ħ→ 0 limit

what is jamming?

transition from flowing to rigid in condensed matter systems

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the structural glass transition

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shear stress

solid:shear modulus

liquid:shear viscosity

liquid solid

glass

coolTm

coolTg

short rangecorrelations

long rangecorrelations

??????correlations

the structural glass transitionliquid:

shear modulus

shear viscosity

glass:shear modulus

shear viscosity

glass transitionviscosity diverges

equilibrium transition? (diverging length scale) dynamic transition? (diverging time scale)

no transition? (glass is just slow liquid)

one of the greatest unresolved problems of condensed matter physics

transition from flowing to rigid but disordered structurethermally driven

sheared foams

polydisperse densely packed gas bubbles

transition from flowing to rigid but disordered structureshear driven

thermal fluctuations negligible

critical yield stress

foam has shear flow like a liquid

foam ceases to flow and behaves like an elastic solid

granular materials

large weakly interacting grains

thermal fluctuations negligible

transition from flowing to rigid but disordered structurevolume density driven

the jamming transition

critical volume density

grains flow like a liquid

grains jam, a finite shearmodulus develops

This false color image is taken from Dan Howell's experiments. This is a 2D experiment in which a collection of disks undergoes steady shearing. The red regions mean large local force, and the blue regions mean weak local force. The stress chains show in red. The key point is that on at least the scale of this experiment, forces in granular systems are inhomogeneous and itermittent if the system is deformed. We detect the forces by means of photoelasticity: when the grains deform, they rotate the polarization of light passing through them.

Howell, Behringer, Veje, PRL 1999 and Veje, Howell, Behringer, PRE 1999

isostatic limit in d dimensions

number of contacts:

number of force balance equations: Nd(for repulsive frictionless particles)

Z is average contactsper particle

isostatic stability when these are equal

seems well obeyed at jamming c

T

Jyield stress

glas

s

conjecture by Liu and Nagel (Nature 1998)

jamming, foams, glass, all different aspects of a unified phase diagram with three axes:

temperature volume density applied shear stress (nonequilibrium axis)

“point J” is a critical point“the epitome of disorder”

here we consider the plane at T = 0 in 2D

flowing ➝ rigid but disordered

surface below which

states are jammed

jamming transition“point J”

critical scaling at point J influencesbehavior at finite T and finite .

understanding = 0 jamming at “point J” may have implications for understanding the glass transition at finite

shear stress

shear viscosity of a flowing granular material

velocity gradient

shear viscosity

if jamming is like a critical point we expect

above jamming

below jamming

⇒ shear flow in fluid state

model granular material

bidisperse mixture of soft disks in two dimensions at T = 0equal numbers of disks with diameters d1 = 1, d2 = 1.4

for N disks in area LxLy the volume density is

interaction V(r) (frictionless)

non-overlapping ⇒ non-interacting

overlapping ⇒harmonic repulsion

r

(Durian, PRL 1995 (foams);O’Hern, Silbert, Liu, Nagel, PRE 2003)

overdamped dynamics

Lx = Ly

N = 1024 for < 0.844 N = 2048 for ≥ 0.844t ~ 1/N, integrate with Heun’s methodtotal shear displacement ~ 10, ranging from 1 to 200 depending on N and

simulation parameters

finite size effects negligible(can’t get too close to c)

animation at: = 0.830 0.838 < c ≃ 0.8415 = 10-5

results for small = 10-5 (represents → 0 limit, “point J”)

as N increases, vanishes continuously at c ≃ 0.8415

smaller systems jam below c

results for finite shear stress

c

c

scaling about “point J” for finite shear stress

scaling hypothesis (2nd order phase transitions):

at a 2nd order critical point, a diverging correlation length determines all critical behavior

quantities that vanish at the critical point all scale as some power of

rescaling the correlation length, → b, corresponds to rescaling

J

c

control parameters

≡c ,

critical “point J”

,

~ b1/ , ~ b , ~ b

we thus get the scaling law

choose length rescaling factor b

crossover scaling variable

crossover scaling exponent

scaling law

crossover scaling function

scaling collapse of viscosity

point J is a true 2nd order critical point

correlation length

transverse velocity correlation function (average shear flow along x)

distance to minimum gives correlation length

regions separated by are anti-correlated

motion is by rotationof regions of size

scaling collapse of correlation length

diverges at point J

phase diagram in plane

volume density

shea

r stre

ss

jammed

flowing

“point J”

0 c

c

'

'

cz0

conclusions

• point J is a true 2nd order critical point

• critical scaling extends to non-equilibrium driven steady states at finite shear stress in agreement with proposal by Liu and Nagel

• correlation length diverges at point J

• diverging correlation length is more readily observed in driven non-equilibrium steady state than in equilibrium state

• finite temperature?