Nonequilibrium phase transitions& reversibility in non-Brownian
suspensions of spheres and rods
David Pine
Center for Soft Matter Research
Department of Physics
New York University

ISSP International Workshop on Soft Matter Physics
Tokyo
11 August 2010Collaborators
Jerry Gollub

Laurent Corté

Haverford

Mines, ParisTech

John Brady

Weining Man

CaltechSan Francisco State

Alex Leshansky

Sharon Gerbode

Technion

CornellPaul Chaikin

Sean Paradiso

NYU

UMass

Laurence Ramos

Emma Filippidi

Montpellier

NYU

Elisabeth Guazzelli

Alex Francescini

Marseille

NYUOverview
Random organization:
New kind of non-equilibrium phase transition
New process for organization of materials
Self-organized criticality in sheared sedimenting
suspensions
Random organization is suspensions of rods:
orientational order & delayed transitions

Overview
Random organization:
New kind of non-equilibrium phase transition
New process for organization of materials
Self-organized criticality in sheared sedimenting
suspensions
Random organization is suspensions of rods:
orientational order & delayed transitions

Background: Low Reynolds
number flows are time reversible

Couette cell(top view)

From ʻLow Reynolds Number Flowsʼ National Committee for Fluid Mechanics Films, Education Development Center Inc., Newton, MA

Time scales & equations
Low Reynolds number�

ma = F�∂ρ+ u · ∇ u = −∇p + η∇ 2u = 0∂tno inertia∇·u=0Navier-Stokes equationU Lρ
Re =�1ηlarge viscosity

Re ≈ 0.03• no inertia• creeping (Stokes flow)

Flow is reversible in low Reynolds number limit

What happens if you add particles?
An aside about diﬀusion

sliding plate shears fluid• big particles ~ 230 μm• viscous solvent ~ 1000×water

Stokes equations still apply2−∇p + η∇ u=0∇·u=0Particle positions are slaved to the liquid …
…but particles modify flow

Dtime to diﬀuse
particle diameter(2R)24πηR3t==DkB T> 100 years

What happens if you add particles?
∆x

sliding plate shears fluid∆yStokes equations still apply−∇p + η∇2 u
∇·u

=0=0

Particle positions are slaved to the liquid …
…but particles modify flow∆xstrain = γ =∆yWhat happens if you add particles?
∆x

sliding plate shears fluid∆yStokes equations still apply−∇p + η∇2 u∇·u=0=0Particle positions are slaved to the liquid …
…but particles modify flow

∆xstrain = γ =∆yWhat happens if you add particles?
∆x

sliding plate shears fluid∆yStokes equations still apply−∇p + η∇2 u
∇·u

=

0=0

Particle positions are slaved to the liquid …
…but particles modify flow

∆xstrain = γ =∆yWhat happens if you add particles?
∆x

sliding plate shears fluid∆yStokes equations still apply−∇p + η∇2 u
∇·u

=0=0

Particle positions are slaved to the liquid …
…but particles modify flow∆xstrain = γ =∆yWhat happens if you add particles?
∆x

sliding plate shears fluid∆yStokes equations still apply−∇p + η∇2 u∇·u=0=0

Particle positions are slaved to the liquid …
…but particles modify flow

∆xstrain = γ =∆yWhat happens if you add particles?
Particles return to their initial positions
system is reversible

sliding plate shears fluid

In fact, particle positions depend only onplate displacement throughout trajectory

Stokes equations still apply−∇p + η∇2 u∇·u=0=0Particle positions are slaved to the liquid …
…but particles modify flow

How sensitive is reversibility when there are many particles?

Particle motion in steady shear flow
experiments of Breedveld et al.
[ J. Chem. Phys. 114, 5923 (2001) ]ystrainsliding plate shears fluid∆x = γ ∆yvx
x= γ̇ ystrain rate

Experiments: particle motion is diﬀusive (& anisotropic)

�(∆x)2 � = 2Dx t�(∆y)2 � = 2Dy t�(∆z)2 � = 2Dz tParticles undergo random walk due torandom encounters with nearby particles(hydrodynamic interactions) Sierou and Brady (2004)

BUT are these trajectories ¿ reversible ? when theflow is reversed as in G.I. Taylor experiment ?

Experiment to test reversibility
SAMPLEtransparent cell
(gap = 2.5 mm)

210-250 µm PMMA spheres

dyed particles
for visibility (~1%)• density matchedliquid & spheres• index matchedliquid, spheres,and Couette cellHg

Hg

• solvent viscosity≈ 3000× water• Reynolds number≈ 0.001
particle volume fraction = 0.10 - 0.40

Re =vel × gap × density
viscosity

Experiment to test reversibility(top view)
SAMPLE
210-250 µm PMMA spheres• density matchedliquid & spheres• index matchedliquid, spheres,and Couette cellγ = γ0 sin(ωt)• solvent viscosity

max displacement
γ0 ≡gap• Reynolds number≈ 3000× water≈ 0.001
particle volume fraction = 0.10 - 0.40

Re =vel × gap × density
viscosityShear flow with tracers

2πT =≈ 30 sωγ = γ0 sin(ωt)

Movie: Periodically sampled
γ0 = 1.0

γ = γ0 sin(ωt)

volume fraction = 0.30

~ 600 frames= 600 oscillations= 5 hours

Movie: Periodically sampled
γ0 = 2.5γ = γ0 sin(ωt)

volume fraction = 0.30

~ 600 frames= 600 oscillations= 5 hours

Particle tracking

γ0 = 2.0
particle size = 9.6 pixels

Particle tracking software of David Grier, John Crocker, Eric Weeks, …

�(∆x/a)2 �mean sq. disp.

Mean square displacement�(∆x/a)2 �1�(∆z/a)2 �0.1

γ0 = 1.53
Dx = 5.87 ± 0.51 × 10−3Dz = 1.35 ± 0.65 × 10−30.01
10100

1000

dimensionless time(cumulative strain)
���2 �∆x= 2Dx γcumulativeadiﬀusion coeﬃcientDiﬀusion coeﬃcient vs. strain amplitude
diﬀusion coeﬃcientvolume fraction = 0.30
0.02

0.01

0.00
0.0

Re = 0.001
Re = 0.005

threshold

0.5

1.0

no Re
dependence

1.5

2.0

2.5

3.0

strain amplitude γ0

Nature 438, 997 (2005)

Why is there a threshold ?
A simple simulation

Paul Chaikin• Randomly place particles in a box• Subject system to sinusoidal shear strain & find particles that collide• Give a random kick to particles that come within distance d of each other• Repeat and watch what happens …
Close encounters lead toIDEA➠ small irreversible motionsab

γ0 = b/a

Why is there a threshold ?
Simulation results
γ0 = 2φ = 0.2

γ0 = 5ab

γ0 = b/a

Self organization below threshold
Simulation results
γ0 = 2

before shear

φ = 0.2

γ0 = 2after shear

Below threshold, system organizes itself sothat particles avoid close encounters
Purely dynamical process ➛ No free energy to minimize

Pair correlation functions
flow, xφ = 0.2vorticity, z32103210ngrbiAbso

fraction of active particles

fa (t)−t/τ
efa (t) = fa∞ + (f0∞ − fa∞ ) αtfa∞

Approaching steady stateetivcA

strain amplitude γ0
fa∞ = |γ0 − γ0c |βsystem stops evolving
and becomes reversible
time (# of cycles)“Absorbing” state:States check in, but they can’t check out.

Diverging time scale near transition−t/τefa (t) = fa∞ + (f0∞ − fa∞ ) αt(103 cycles)τ = |γ0 − γ0c |−ντν ≈ 1.33

strain amplitude γ0threshold strain amplitude

Phase diagram
15105

irreversible& chaotic
reversible(only after self
Self-organizingorganization)

into a reversible state

00.0

0.1

0.2

0.3

0.4

volume fraction

0.5

Does model describe experiments ?
laser sheet with fluorescent dye in liquidcamerafilterlaser sheet

can follow all particles in illuminated planevorticity direction, zOscillating shear

shear flow direction, xStrobed movies
Reversible

tracers

laser sheet
illumination

Irreversible

tracers

laser sheet
illumination

Evolution of particle movement
below threshold

φ = 0.30

γ0 = 1.8

vorticity (vertical) direction →γ0 = 1.2

above threshold← shear flow direction →Mean square displacement per cycle

fits

103

e−t/τ
∼ αt

102
101

�

∆r12 (t)�(µm2 )

104

100
0

20

40

60

80

time (# of cycles)100dynamic viscosities“Rheology” of suspensions102η�

in phase response101
out of phase response
100

η ��

apply force

10-1

σ0 cos(ωt)

10-2

γ0 cos(ωt + δ)

0.0

1.5

1.0

1.5

strain amplitude

2.0

measure response

increase occurs atreversibility threshold“Rheology” of suspensions
Measurement protocol
Shear far above threshold[randomizes particle positions]
Stop
Shear below threshold- record η ″(t)[rheological capacitance]out of phase viscosity η ��−t/τ
efa (t) = fa∞ + (f0∞ − fa∞ ) αtγ0 = 2.0

10-1

γ0 = 1.5

γ0 = 1.2
γ0 = 0.8
10-2020

4060

80

time (# of cycles)100Diverging relaxation timenear transition−t/τ
efa (t) = fa∞ + (f0∞ − fa∞ ) αt

τ = |γ0 − γ0c |−νη ��relaxation time τν ≈ 1.1

200150�(∆r1 )2 �100|γ0 − γ0c |5000123

strain amplitude γ0
Nature Physics 4, 420 (2008)

Overview
Random organization:
New kind of non-equilibrium phase transition
New process for organization of materials
Self-organized criticality in sheared sedimenting
suspensions
Random organization is suspensions of rods:
orientational order & delayed transitions

Experiments with gravitygRh( 0 )

Experiments with gravity
60
5040z/d 30
20h(γ0 )10000.2

0.4

0.6

ρ(z)

0.811.2

Experiments with gravity

0.70.6∆ρ → 00.5

�φ� 0.4
0.3
0.2
0.1
00246γ0810

Simulations with gravity
Without shear, particles all fall to the bottom of cell
With periodic shear, particles “fluﬀ up” to density ϕ}steadystatedensity

flux down due tosedimentation∆mg
Jsed = cvsed ≈ cζflux up due tocollisions∂c
Jdif f = −D(c)∂zHeight vs density profiles
Sedimentation vs diﬀusion
80

density profiles:
flat vs exponential

y/d60
40
2000.0h00.2

0.4

0.6

Height vs density profiles
Sedimentation vs diﬀusion

0.8
0.6
0.4

80

0.2
0.0
10-5

y/d60
40

10-310-2

ξ mean interparticle spacing

20
0
0.0

10-4

h0
0.2

0.4

0.6

Height vs density profiles
Sedimentation vs diﬀusion

0.8
0.6
0.4

80

0.2
0.0
10-5

y/d60
40

10-310-2

ξ mean interparticle spacing

200
0.0

10-4

ξ sedimentation
τs =time
vsh0}

0.2

0.4

0.6

dense layer at bottom of cell

Height vs density profiles
Sedimentation vs diﬀusion

y/d0.6h20 vsτD
competition betweenA==τs4Dξ diﬀusion & sedimentation

80
60

0.8
0.4
0.2
0.0
10-5

0.24

A<110-3

10-2

diﬀusion dominates

40
0.78

ξ mean interparticle spacing

A>120

ξ sedimentation
τs =time
vs

h0

2.4

00.0

10-4}

0.2

0.4

0.6

dense layer at bottom of cell

h20
time to relax dense layer
τD ==throughout sample
4D

Height vs density profiles
Sedimentation vs diﬀusiony/d3

h20 vsτD
competition betweenA==τs4Dξ sedimentation & diﬀusion

8060421010-20.24

A<1100101

diﬀusion dominates

400.78ξ mean interparticle spacing

A>120

ξ sedimentation
τs =timevsh0

2.400.010-1}

0.2

0.4

0.6

dense layer at bottom of cell

h20
time to relax dense layer
τD ==throughout sample
4D

Cluster sizes & lifetimes•New clusters nucleate at bottom wall
• monitor cluster lifetime (t = tdeath-tbirth)• monitor cluster size ( s = ∑ ai )t (number of cycles)

800

600

t

400

2000y15x105-20

020

Cluster sizes & lifetimes
100800

10-4
10-6
10-8 010102

104cluster lifetime, t106

100

P(s)

10-2t (number of cycles)

P(t)

10-2

600

t

400

200

10-4
10-6
10-8 0100102104

cluster size, sy
15106

self-organized critical state

x
10

5-20

020

strain amplitude γ0

Steady state concentrations
12108

ϕ starts high→ goes lower

64
2
0
0.0

0.1

0.2

0.3

0.4

volume fraction φConcentration evolves towards thecritical concentration for each strain

Height vs density profiles
260

0.7

50

0.6

100.54010-1 100 101 102 103

0.4
300.3
200.2

1000.10 0.2 0.4 0.6 0.8 1.0 1.2001234

56

Overview
Random organization:
New kind of non-equilibrium phase transition
New process for organization of materials
Self-organized criticality in sheared sedimenting
suspensions
Random organization is suspensions of rods:
orientational order & delayed transitions

Reversibility with rods
0.136 mm1.50 mmdensity & indexedmatched nylon rods

Roverlap
concentration~1% fluorescent rods

unpublishedRod behavior similar to spheres
Start with randomized rods, then oscillatory strain as before ...

}abovecritical}

below
criticalMore datastrain
3.80
3.70
3.50
3.30
3.15
3.00
2.80
2.78
2.58
2.55
2.30
1.90
1.40

Scaling of activity data
For data on spheres ...
−t/τ
∞
∞
∞ e
fa (t) = fa + (f0 − fa ) αtMore generally we expect ...
(at criticality)
Manna universality class (absorbing phase transitions)
Total number of particles conserved
Infinite number of absorbing states

Scaled data|γc − γ|−β (η �� /η0�� )102
abovecritical

101
100

below
critical

10-1
10-210-1100101t|γc − γ|ν

102Scaling data

Manna

More datastrain
3.80
3.70
3.50
3.30
3.15
3.00
2.80
2.78
2.58
2.55
2.30
1.90
1.40

Scaling data

Scaling with two critical strains

Evolution of orientation
0.8
0.6
γ0 = 2.80
γ0 = 2.60
γ0 = 2.01
γ0 = 1.85
γ0 = 1.60
γ0 = 0.01

0.4
0.2
0.0

0100

200 300 400
time (cycles)500

Phase diagram

activity
goes tozero

rods orient,then activity
goes to zeroactivity
remains
finiteSummary• Low Reynolds # flow with particles is not reversible ...unless theparticles self-organize• Phase transition & self organization ...
• do not result from the minimization of a free energy• do result from random displacements due to close encounters• change character if displacements are correlated• States above and below reversibility transition are very diﬀerent• below: one state is selected – no dynamics• above: system explores endless sequence of states – non-ergodic• Model is similar to directed percolation with conserved particles• Gravity → Self-organized critical state• Rods lead to a line of critical points due to orientational orderingCollaborators
Jerry Gollub

Laurent Corté

Haverford

Mines, ParisTech

John Brady

Weining ManCaltech

San Francisco State

Alex Leshansky

Sharon Gerbode

TechnionCornell

Paul Chaikin

Sean Paradiso

NYU

UMass

Laurence Ramos

Emma Filippidi

Montpellier

NYU

Elisabeth Guazzelli

Alex Francescini

Marseille

NYU

Nature 438, 997 (2005)

Nature Physics 4, 420 (2008)

Phys. Rev. Lett. 103, 248301 (2009)