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

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

1. IntroductionFormation

Microscopics

2. StructureExperiment

Simulation

3. StabilityCoarsening

Drainage

4. RheologyLinear response

Rearrangement & 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,

strain,

slop

e, G

o

y

shear modulus: Go = film/Sqrt[3]ayield strain: y = 1.2

a

film

etc.

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

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

• trends vs polydisersity and wetness

(DJD)

=1

=0.

84

width=0.75 width=0.1

Frequency dependence, G*()

• Simplest possibility: Kelvin solidG*() = G0 + i {i.e. G’() = G0 and G”() = }

• But typical data looks much different:– G’() isn’t flat, and increases at high …

– G”() doesn’t vanish at low …

(A. Saint-Jalmes & DJD)

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

G*() = G0(1 + Sqrt[ic])**

shown by dotted curve for =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*() vs data.Here Go = 2300 dyne/cm2

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

– amplitude of fluctuations:

– microrheology:

angstroms 313

22

dyne/cm 3001000R

TkG 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 (

dyn

/cm

2 ) G / (

/R)

0.5 0.6 0.7 0.8 0.9 1

0.63

unjammed jammed

polydispersitymakes little or no

difference!

(A. Saint-Jalmes & DJD)

Behavior near the transition• simulation of 2D bubble model

3

4

5

6

Z

1

10

102

0.75 0.80 0.85 0.90 0.95 1.00

r

shearmodulus

coordinationnumber

stress-relaxationtime

0.00

0.05

0.10

0.15

0.20

0.750.10

G width

20x20 polydisperse bubbles

r appears todiverge at rcp ~0.84

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

(DJD)

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*() 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 c “point J”}

NB: shear deformation is uniform

– UCLA:• direct observation of free surface in Couette cell• viscosity and G*() 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

ym

yc

y

s

s

s

s

m

c

y

/5.0

/003.0

05.0

1.0

20

d

oq

bubble motion via DWSlow shear high shear

0.01

0.1

1

0 0.005 0.01 0.015

|g1(

)|2

(sec)

0.0005 s-1

= 0.02 s-1.

0.007 s-1

(a)

o = time betweenrearrangements ateach scattering site

0.01

0.1

1

0 0.005 0.01 0.015

|g1(

)|2

(sec)

0.0005 s-1

= 0.02 s-1.

0.007 s-1

(a)

0 10-10

2 (sec2)

5.5 s-1

20 s-1

10 s-1

(b)

s 30

kl * Ý

time for adjacentscattering sites to convect apart by

(A.D. Gopal & DJD)

ol

Lg

2

1 *6exp)(

22

1 *6exp)(

sl

Lg

DWS times vs strainrate• behavior changes at expected strainrate scales

discrete rearrangementsjammed/solid-like

laminar bubble motionunjammed/liquid-like

0.001

0.01

0.1

1

10

102

103

104

0.0001 0.001 0.01 0.1 1 10

1/0

1/s

rate

con

stan

t (s

ec-1

)

strain rate (1/s)

m =

y/

d

.c =

y/

0q

.

(A.D. Gopal & DJD)

Rheological signature?• simplest expectation is Bingham plastic:

– but 1/isn’t seen in either data or bubble-model:

– no real signature of m in data

myydy 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=0.09

Str

ain

420

430

440

450

460

470

480

-10 -5 0 5 10 15 20 25

Str

ess

(dy

ne/

cm

2 )

t (s)

G(0)

R

Str

ess

(dy

ne/c

m2 )

Str

ain 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

, )

[dy

ne/c

m2 ]

m

=y/

d

.

10-3

10-2

10-1

110

102103

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

(c)

R [

sec]

shear rate [1/sec]

0

S -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…

S 1

AT

p p 2

A

Tdt xy(t) xy (0)

0

c

UT

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 =0.90 area fraction

– Teff approaches constant at zero strain rate

– Teff increases very slowly with strain rate

.

Tp

Txy

TUT

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 (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 1

0(

)

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!