Are Foams Soft Glassy Materials?
Sylvie Cohen-Addad
Laboratoire de Physique des Matériaux Divisés et des Interfaces

ISSP International Workshop on Soft Matter Physics 2010

A hierarchy of length scales

100 µm1 cm10 nmgas
1 nm

1 µmliquidFoam structure
A surface minimization problem that goes back to Plateau (1873) and Kelvin (1887)100	  µm

64%90%99%
Gas content

Bubble surface energy! 1010
Thermal energy

Ageing
Drainage

D. Durian

DurianFlow through a porous medium

Coalescence

Coarsening

Avalanche-likedynamics

Statistically self-similar growthSolid-like or liquid-like mechanical behavior

50 cm

2 mm

Froghopper

Origin of foam elasticity and yielding

Surface tension T
Bubble diameter d1 mmRouyer

Shear modulus

G ≈ T/d ≈ 100 Pa

Yield strainγY ≈ 1

Yield stressσY ≈ T/d

Derjaguin, Koll. Zeitschrift 1933
Princen, JCIS 1983
Kraynik Reinelt, PRE 2000

Are generic mechanisms at the origin of the mechanical behaviorof soft disordered materials ?
Foam

50 µm

Emulsion

4 µm

Onion phase

Colloidal suspension

Granular media

20 µm

CloitreRamosWeeks

Liu Nagel, Nature 1998Viscoelastic relaxations
γ10aP(

3102G”
0.1 Hz110-3Gopal Durian PRL 2003

sulu 102do
M

G’

103Modulus (Pa)

Modulus (Pa)

Labiausse et al, JOR 2004

10σ)
Shear

10-2

10-1

Strain amplitude γo

10100

G’
1/2G”
γo= 0.001

110-410-2

100Frequency (Hz)

What is the link between foam structure and relaxations ?
What is the pertinent length scale to modelize foam viscoelasticity ?

102Nonlinear slow relaxations
Large Amplitude Oscillatory Shear

Hyun et al JNNFM 2002

! (t) = ! o cos(" t)Stress ! (t) = " o ( G'cos(# t) + G"sin(# t)) + $! (t)increasing γoσ(t)Strainσ(t)σ(t) (Pa)

γ(t)

γ(t)

γ(t)

Foam approximately behaves like an elastoplastic materialG

1G' / G- 3/2
0.1

G'' / G1

16 G $ # o 'G' !
3" &% # y )(4 G $#o'G '' !
" &% # y )(Gillette shaving foam

* 3/2

Wet φ = 92%
d = 30 µm or 40 µm
T = 30 mN/mSlowly coarsening*1

AOK / PEO / LOH foam-1

Dry φ = 97%
d = 50 µm
T = 22 mN/m
≈ No coarsening

0.1

0.1

σy1! /!
"10yRouyer et al, EPJE 2008

… but stress harmonics depend on physical chemistry! (t) = " o ( G 'cos(# t)+ G ''sin(# t)) + $! (t)Anharmonic stress residualq=< !" 2 >< "2 >

Elastoplastic model

Gillette shaving foamq

0.1

AOK / PEO / LOH foam
0.01

0.1

1! " / !y10

The SGR model does not fully capture foam relaxations
Soft Glassy Rheology model

Elastic energyof a mesoscopic regionSollich et al, PRL 1997

Bubble configuration

10.1

0.1- 3/236 !m 1 Hz
28 !m

1 Hz

SGR

0.1

0.3 Hz0.1-1SGRq

AOK Gillette

SGR

G'' / GG' / G

10.01

1 Hz
50 !m
0.3 Hz
1! /!
o

10
y

0.1

1! /!
o

10
y

0.1

1! /!
o

10y

ci 450
ts
a400l
E
350
010

Mechanical memory effectQuiescentEnergy density

Elastic modulus G’ (Pa)

su
lu
do
m500

After shear

After LAO Shear

1012t-t

10shear3(s)10104t - tshift

FoamSpin glass

Höhler et al, EPL 1999

Berthier Bouchaud, PRB 2002

Polymers Kovacs et al, JPC 1963
Microgels Cloitre et al, PRL 2000
Pastes Derec et al, PRE 2003

… but no shear rejuvenation in foams!
LaserCC

D

Time interval between rearrangements at a given
place measured using multiple light scattering
Durian Weitz Pine, Science 1991

Bubble rearrangements dynamics is the opposite of the SGR prediction

τ / τ (tp )64201000

tp
1200

1400

1600

1800

Foam age (s)Cohen-Addad et al, PRL 2001

2000

aP(Linear relaxations
103Gopal 2003

Modulus (Pa)sulu 102
do
M
10G’

1/2G”
1/τ

110-4

10-2

100

102

Frequency (Hz)Frequency(Hz)What is the coupling between ageing and linear relaxations?
Do slow relaxations correspond to glassy dynamics?

Dissipation at a mesoscopic scale

42
3

1Local stressTime

Coarsening-induced bubble rearrangements is an intrinsic source of dynamics in foams

2D dry foam as a model systemstress

Surface evolver software simulates quasistatic equilibrium and coarsening

Vincent-Bonnieu

Shear stress due to surface tension Ton an area A (Bachelor, JFM 1970)

t! XY =Tt X tY d !
$# !"#How coarsening induces creep flow
StressYield stress

σ5Compliance/γ(t) / σ 4)t"3(! 2500 bubbles

rearrangement

50 bubbles

10-5000500

Yield stress

1000

1500

2000

2500

time (a.u.)

Vincent-Bonnieu et al, EPL 2006

A mesoscopic model to link rearrangements andmacroscopic flow analytically
λT
StressKabla et al, PRL2003

Microstructurechange upon arearrangement

λT

Rearrangement = force dipole
acting on an isotropic elastic
continuum
Length of new film λSurface tension TMacroscopic flowat the rate ofrearrangements

The dipole tensor of a 2D rearrangement
Shear direction

Tλα
TConjecture
Intrinsic orientation of P:
Dipole strength ∝ λ T

Average macroscopic shear strain step due to a randomly placed
rearrangement as predicted by continuum mechanics:
G shear modulus
A sample area

$T!" #sin2%AGα

The orientation of rearrangements is biased by the applied stress
Strain δγ0.005

σ = - 0.25 σyσ=0σ = + 0.25 σy!" #0-0.005

ααα$T
sin2%AG

Distribution of orientation of new films
706050%50) 40"(! 30201000%-80-90!+ " !" = # $ρ-40-45

ρ+00

"

4045

angle α (°)ρ80 -80
90-40-45

ρ+00!

4045

angle α (°)ρ80 -80-40-45

ρ+00!

4045

angle α (°)

80

A Maxwell constitutive law links the slow relaxation
to the bubble rearrangement rate R
and to the area Ameso of a rearranged mesoscopic zone
1 d"
1
#R $T <%>! dt
G(γ /σ)(T/d)

Normalized compliance! = 0.25 !

70! = 0.15 !

60 Ny2.6

Number of rearrangements

2.7

yu50 m2.5be30 r
40

2.4

Ameso = χ T <λ>≅ (1.5 d)220 o

2.3

102.2

00204060

Time (s)80100fT
1ev
eVincent-Bonnieu et al, EPL 2006 ; Vincent-Bonnieu
et al, cond-mat/0609363
ntsCreep rheometry and in situ multiple light scattering
Laser
Yield stress

γd 2ezila 1mroN

CC

D

σCoarsening
FastAOK PEO LOH foam with N2Medium
Gillette foam
Slow
AOK PEO LOH foam with N2 /C6F14

slope G/ηγ(t) G / σNormalized compliancenistressaσrts

gas fraction = 93%0050

100time (s)Maxwell creepGητ = η / G ≅ 200 to 1000 s

Coarsening induced rearrangements are) at the origin of the slow linear relaxations( 10Creep characteristic timeη/G (s)4Coarsening rate

fastGmediumslow

/ 10

3

1!= #GR Vmeso
"1!
2

10

3

10

10

4

5

10

3
Time interval between
in a volume d3
1 /(R d rearrangements) (s)• Volume of a mesoscopic rearranged zone

Vmeso ≅ (3 d)3• In agreement with the numerical simulation and mesoscopic model
Cohen-Addad et al, PRL 2004

Shear modulus (Pa)103G*(ω) = 1/(iω L[J(t)](iω))

G’
10

2

1/2

G’’
10

1

10-4

10-310-2

10-1

10

0

10

110

2

Frequency (Hz)Do interfacial rheology and foam viscoelasticity interplay?

A generic model of viscoelasticity based on disorder
Lennard-Jones glass

Disordered foamGoldenberg et al, EPL 2007

Durian, PRE 1997

shear
Slip
ηeff gGg
fc !
!eff!
i f$G *( f ) = G # 1 +& + i 2 ' (o ff"c %Liu et al, PRL 1996

Interfacial rheology
Surface dilatation δA/ASurface stress!s = E"A1 "A+#AA "tInterfacial dilatational elasticity and viscosity• Low rigidity with synthetic surfactants SDS, TTAB…
• High rigidity with fatty acids or LOH/surfactant mixtures.

Mechanisms of viscous dissipation
Buzza Lu Cates, J Phys II 1995

Rigid interfaces: Marangoni flow in films
E dilatational interfacial elasticity
η liquid viscositydδ

Viscous resistanceDriving forcegE#3fc !
!
!
10$10Hz2"eff " d!d "!eff $+#dEg!
dMobile interfaces
Low interfacial elasticity: Flow in films junctionsJunction

T surface tension
κ surface viscosity"!eff ! ! +ddTg!
dT /dfc !
!1#10 4 Hz! +" /d)aP(Foam with rigid interfaces
Gillette shaving foamsu 700l500u
do 300m

GE = 70 mN/mG’
3Gas fraction = 92%
G(t)/G(to)21rae 100
h80
S

1
0.9
0.8
0.7
0.6
0.5

G’’

60

0.4

1/2

40

0.3
101

10100

Frequency (Hz)!
i f$
G *( f ) = G # 1 ++ i 2 ' (o f&fc %"Bubble size
29 µmd(to)/d(t)
33 µm
36 µm
42 µm
51 µm
62 µm
75 µm
t
o

100

age t (min)1000

Scaling of the characteristic frequency
300

SLES CAPB foam

E = 10 mN/mf (Hz)

200c

Rigid interfaces
Gillettte foam

100

E = 70 mN/m

0G!

100Td200

300

400

2

1

10

2

468

10

Solution viscosity (mPa s)

G (Pa)

Marangoni flow in films

E!
fc !
" d2Viscous flow in film junctions

T /dfc !
! + " /dKrishan et al, PRE 2010

3200

SLES CAPB foam
SLES CAPB LOH foam

3012! / ! 40%(! /" E) / (! /" E) 40%

Mobile interfaces)aP( 1000Physico-chemistry matters !
Rigid interfaces

Modulus (Pa)sul 100udo 10
M
10.0001 0.001

Mobile interfaces

0.01

0.1

110100Frequency (Hz)Slow relaxations

Fast relaxations

R. Höhler
A.-L. Biance

P. Sollich

F. Rouyer

S.M. Fielding

K. Krishan
Y. Khidas
A. HelalR. Lespiat
S. Vincent-Bonnieu
S. Costah'p://www2.univ-­‐mlv.fr/lpmdi/RHE/index2.php