THE PHYSICS OF FOAM• Boulder School for Condensed Matter and Materials Physics

July 1-26, 2002: Physics of Soft Condensed Matter

1. IntroductionFormationMicroscopics

2. StructureExperimentSimulation

3. StabilityCoarseningDrainage

4. RheologyLinear responseRearrangement & flow

Douglas J. DURIANUCLA Physics & AstronomyLos Angeles, CA 90095-1547

<durian@physics.ucla.edu>

Princen-Prud’homme model• shear a 2D honeycomb, always respecting Plateau’s rules

• deformation is not affine: vertices must rotate to maintain 120-120-120

stress, σ

strain, γ

slope

, Go

γy

shear modulus: Go = γfilm/Sqrt[3]ayield strain: γy = 1.2

etc.

γfilm

a

Yes, but…• Princen-Prud’homme is for 2D periodic static dry foam

– dimensionality?• generally expect G ~ Laplace pressure (surface tension/bubble size)

– wetness?• shear modulus must vanish for wet foams

– dissipation?• nonzero strainrate or oscillation frequency?

– disorder?• smaller rearrangement size (not system)?• smaller yield strain?

small-strain non-affine motion(DJD)

• bubble motion is up & down as well as more & less• compare bonds and displacements (normalized to affine expectation)

• trends vs polydisersity and wetness

φ=1

φ=0.

84

width=0.75 width=0.1

Frequency dependence, G*(ω)• Simplest possibility: Kelvin solid

G*(ω) = G0 + iηω i.e. G’(ω) = G0 and G”(ω) = ηω• But typical data looks much different:

– G’(ω) isn’t flat, and increases at high ω…– G”(ω) doesn’t vanish at low ω…

(A. Saint-Jalmes & DJD)

High-ω rheology• Due to non-affine motion, another term dominates:

G*(ω) = G0(1 + Sqrt[iω/ωc])**

shown by dotted curve for ε=0.08 foam:

[**seen and explained by Liu, Ramaswamy, Mason, Gang, Weitz for compressed emulsion using DWS microrheology]

(A.D. Gopal & DJD)

NB: This gives a robust wayto extract the shear modulusfrom G*(ω) vs ω data.Here Go = 2300 dyne/cm2

Short-τ DWS• long-τ: rearrangements give exponential decay• short-τ: thermal interface fluctuations (Y=<∆r2(τ)>)

– amplitude of fluctuations:– microrheology:

angstroms 313±=δ

22 dyne/cm 3001000 ±≈≈

δRTkG B

o

Unjamming vs gas fraction• Data for shear modulus:

– symbols: polydisperse foam– solid curve: monodisperse emulsion (Mason & Weitz)– dashed curve: polydisperse emulsion (Princen & Kiss)

0

400

800

1200

1600Couette cell

cone-plate

0

0.1

0.2

0.3

0.4

0.5

G (d

yn/c

m2 ) G

/ ( σ/R)

0.5 0.6 0.7 0.8 0.9 1φ

0.63

unjammed jammed

polydispersitymakes little or no

difference!

(A. Saint-Jalmes & DJD)

Behavior near the transition(DJD)• simulation of 2D bubble model

3

4

5

6

Z

1

10

10 2

0.75 0.80 0.85 0.90 0.95 1.00

τr

φ

shearmodulus

coordinationnumber

stress-relaxationtime

0.00

0.05

0.10

0.15

0.20

0.750.10

G width

20x20 polydisperse bubbles

τr appears todiverge at φrcp ~0.84

better “point J” statistics byO’Hern, Langer, Liu, Nagel

Unjamming vs time• elasticity vanishes at long times…

– stress relaxes as the bubbles coarsen– time scale is set by foam age

• ie how long for size distribution to change• not set by the time between coarsening-induced rearrangements (20s)

• Even though this is not a thermally activated mechanism like diffusion or reptation, the rheology is linear– G*(ω) and G(t) date are indeed related by Fourier transform

(A.D. Gopal & DJD)

Unjamming vs shear• make bubbles rearrange & explore packing configurations

– slow shear:• sudden avalanche-like rearrangements of a few bubbles at a time

– fast shear:• rearrangements merge together into continuous smooth flow

slow & jerky fast & smooth

Rearrangement sizes• even the largest are only a few bubbles across

– picture of a very large event– distribution of energy drops (before-after) has a cutoff

but it moves out on approach to φc “point J”

NB: shear deformation is uniform

– UCLA:• direct observation of free surface in Couette cell• viscosity and G*(ω) are indep. of sample thickness & cell geometry• DWS gives expected decay time in transmission and backscattering• viscous fingering morphology

– Hohler / Cohen-Addad lab (Marne-la-Vallee):• multiple light scattering and rheology

– Dennin lab (UC Irvine):• 2D bubble rafts and lipid monolayers

– Weitz lab (Exxon/Penn/Harvard):• Rheology of emulsions

– Computer simulations:• bubble model (DJD-Langer-Liu-Nagel)• Surface evolver (Kraynik)• 2D (Weaire)• vertex model (Kawasaki)

uniform shear-band exponential

important scalesnotation:

values for Foamy:

like-avalancheor smooth are entsrearrangemwhether :dominate entsrearrangem induced-shearor -coarseningwhether :

strain yield:eventsent rearrangem ofduration :

ntsrearrangme induced -coarseningbetween time:

d

oq

d

oq

τγγτγγ

γτ

τ

ym

yc

y

==

ss

ss

m

c

y

/5.0/003.0

05.01.0

20

d

oq

=====

γγγτ

τ

bubble motion via DWSlow shear high shear

0.01

0.1

1

0 0.005 0.01 0.015

|g1(τ

)|2

τ (sec)

0.0005 s -1

γ = 0.02 s -1.0.007 s -1

(a)

τo = time betweenrearrangements ateach scattering site

0.01

0.1

1

0 0.005 0.01 0.015

|g1(τ

)|2

τ (sec)

0.0005 s -1

γ = 0.02 s -1.0.007 s -1

(a)

0 10-10

τ2 (sec 2)

5.5 s-1

20 s-1

10 s-1

(b)

τs =30

kl * Ýγ

time for adjacentscattering sites to convect apart by λ

(A.D. Gopal & DJD)

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−≈

olLg

τττ

2

1 *6exp)(

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−≈

22

1 *6exp)(

slLg

τττ

DWS times vs strainrate(A.D. Gopal & DJD)

• behavior changes at expected strainrate scales

0.001

0.01

0.1

1

10

10 2

10 3

10 4

0.0001 0.001 0.01 0.1 1 10

1/τ0

1/τs

rate

con

stan

t (s

ec-1

)

strain rate (1/s)

γm = γ

y/τ

d.γ

c = γ

y/τ

0q.

laminar bubble motionunjammed/liquid-like

discrete rearrangementsjammed/solid-like

Rheological signature?• simplest expectation is Bingham plastic:

– but 1/γ isn’t seen in either data or bubble-model:– no real signature of γm in data

( ) ( )myydy γγσηγτγσσ 11 hence and 1 +=+=

..

(A.D. Gopal & DJD)

Superimpose step-strain!• Stress jump and relaxation time both measure elasticity

5.5

5.6

5.7

5.8

5.9

6

strain rate = 0.01/s∆γ =0.09

Stra

in

420

430

440

450

460

470

480

-10 -5 0 5 10 15 20 25

2 )

t (s)

G(0)

τRStre

ss (

dyne

/cm

2 )

S

train transient relaxation shear modulus

bubble motion vs rheology• All measures of elasticity vanish at same point where

rearrangements merge together into continuous flow• “unjamming” shear rate = yield strain / event duration

0

1000

2000

3000

4000

G(0

, γ)

[dyn

e/cm

2 ]

γm

=γy/τ

d

.

10-310-210-1110102103

10-5 10-4 10-3 10-2 10-1 1 10 102

(c)

τ R [

sec]

shear rate [1/sec]

τ0

τS -1/2

This completes the connectionbetween bubble-scale and

macroscopic foam behavior

Jamming• We’ve now seen three ways to unjam a foam

• ie for bubbles to rearrange and explore configuration space

– vs liquid fraction (gas bubble packing)– vs time (as foam coarsens)– vs shear

• Trajectories in the phase diagram?– does shear play role of temperature?

(A.J. Liu & S.R. Nagel)

Three effective temperatures

• Compressibility & pressure fluctuations

• Viscosity & shear stress fluctuations

• Heat capacity & energy fluctuations

– T’s all reduce to (dS/dU)-1 for equilibrated thermal systems– What is their value & shear rate dependence; are they equal?

• Difficult to measure, so resort to simulation…

κS−1 =

AT

p − p( )2

η =AT

dt σ xy(t)σ xy (0)0

∞

∫c

∂ U∂T

=1

T 2 U − U( )2

(I.K. Ono, C.S O’Hern, DJD, S.A. Langer, A.J. Liu, S.R. Nagel)

The three Teff’s agree!• N=400 bubbles in 2D box at φ=0.90 area fraction

– Teff approaches constant at zero strain rate– Teff increases very slowly with strain rate

γ.

TpTxyTUT

T

0.006

0.004

0.002

0

0.008

0.004

0

bidisperse

polydisperse

0.0001 0.001 0.01 0.1 1

…also agree with T=(dS/dU)-1

• Monte-Carlo results for Ω(U), the probability for a randomly constructed configuration to have energy U

• For this system, the effective temperature has all the attributes of a true statistical mechanical temperature.

-100

-80

-60

-40

-20

0

N=163664100144

log 10

(Ω)

0.001

0.01

0.1

1

0.001 0.01 0.1 1U/N

TS T

UTU =U/0.65N=U/(f/2)N

S=

TU from shearing runs

Mini-conclusions• Statistical Mechanics works for certain driven athermal

systems, unmodified but for an effective temperature– When does stat-mech succeed & what sets the value of Teff?

• Elemental fluidized bed (R.P. Ojha, DJD)

– upflow of gas: many fast degrees of freedom, constant-temperature reservoir• Uniformly sheared foam (I.K. Ono, C.S O’Hern, DJD, S.A. Langer, A.J. Liu, S.R. Nagel)

– neighboring bubbles: many configurations with the same topology• In general

– Perhaps want fluctuations to dominate the dissipation of injected energy?

– When does stat-mech fail & what to do then?• eg anisotropic velocity fluctuations in sheared sand• eg P(v)~Exp[-v3/2] in shaken sand• eg flocking in self-propelled particles

THE END.• Thank you for your interest in foam!