Are Foams Soft Glassy Materials? Sylvie Cohen-Addad Laboratoire de - - PowerPoint PPT Presentation

Laboratoire de Physique des Matériaux Divisés et des Interfaces

Are Foams Soft Glassy Materials?

Sylvie Cohen-Addad

ISSP International Workshop on Soft Matter Physics 2010

A hierarchy of length scales

1 µm 1 nm 1 cm 100 µm 10 nm

liquid gas

Foam structure

A surface minimization problem that goes back to Plateau (1873) and Kelvin (1887)

100 ¡µm

Gas content

Bubblesurface energy Thermal energy !1010

64% 90% 99%

Ageing

Drainage Coalescence

Flow through a porous medium Statistically self-similar growth

Coarsening

Durian

Avalanche-like dynamics

• D. Durian

2 mm

Froghopper

50 cm

Solid-like or liquid-like mechanical behavior

Origin of foam elasticity and yielding

Rouyer

Shear modulus Yield stress G ≈ T/d ≈ 100 Pa σY ≈ T/d

Surface tension T Bubble diameter d

Derjaguin, Koll. Zeitschrift 1933 Princen, JCIS 1983 Kraynik Reinelt, PRE 2000 1 mm

Yield strain γY ≈ 1

Are generic mechanisms at the origin of the mechanical behavior

• f soft disordered materials ?

Cloitre

Foam Emulsion Colloidal suspension Granular media

Liu Nagel, Nature 1998

4 µm

Weeks

Onion phase

Ramos

20 µm 50 µm

101 102 103 10-4 10-2 100 102

M

• d

u l u s Frequency (Hz)

Viscoelastic relaxations

G” γ σ G’

What is the link between foam structure and relaxations ? What is the pertinent length scale to modelize foam viscoelasticity ? Shear Modulus (Pa)

Gopal Durian PRL 2003

101 102 103 10-3 10-2 10-1 100

Modulus (Pa) Strain amplitude

Labiausse et al, JOR 2004

G’ G”

0.1 Hz γo= 0.001

γo

1/2

Nonlinear slow relaxations

Large Amplitude Oscillatory Shear

Hyun et al JNNFM 2002

σ(t) (Pa) γ(t)

γ(t) γ(t) σ(t) σ(t)

! (t) = ! o cos(" t)

Strain Stress

!(t) = " o G'cos(# t) + G"sin(# t)

( ) + $!(t)

increasing γo

0.1 1

G' / G

• 3/2

Foam approximately behaves like an elastoplastic material

Gillette shaving foam Wet φ = 92% d = 30 µm or 40 µm T = 30 mN/m Slowly coarsening AOK / PEO / LOH foam Dry φ = 97% d = 50 µm T = 22 mN/m ≈ No coarsening

0.1 1 0.1 1 10

! " / !y G'' / G

• 1

G' ! 16 G 3" # o # y $ % & ' ( )

* 3/2

G'' ! 4 G " # o # y $ % & ' ( )

*1

Rouyer et al, EPJE 2008

G σy

… but stress harmonics depend on physical chemistry

Gillette shaving foam AOK / PEO / LOH foam

q = <!" 2 > < " 2 > !(t) = " o G'cos(#t)+G''sin(#t)

( )+ $!(t)

Anharmonic stress residual Elastoplastic model

0.01 0.1 0.1 1 10

q ! " / !y

The SGR model does not fully capture foam relaxations

Bubble configuration

Elastic energy

• f a mesoscopic region

Soft Glassy Rheology model

Sollich et al, PRL 1997

0.1 1 0.1 1 10

G'' / G

• 1

!

• / !

y

0.1 1 0.1 1 10

G' / G

• 3/2

!

• / !

y

Gillette AOK 50 !m 1 Hz 0.3 Hz 1 Hz 0.3 Hz 1 Hz 28 !m 36 !m

SGR SGR SGR

0.01 0.1 0.1 1 10

q

!

• / !

y

Mechanical memory effect

Foam

Höhler et al, EPL 1999

350 400 450 500 10 10

1

10

2

10

3

10

4

E l a s t i c m t - t

shear (s)

Quiescent After shear

Energy density t - tshift

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

E l a s t i c m

• d

u l u s G ’ ( P a )

Spin glass

Berthier Bouchaud, PRB 2002

After LAO Shear

2 4 6 1000 1200 1400 1600 1800 2000

… but no shear rejuvenation in foams!

Cohen-Addad et al, PRL 2001

τ / τ (tp )

Foam age (s)

tp

Bubble rearrangements dynamics is the opposite of the SGR prediction

Time interval between rearrangements at a given place measured using multiple light scattering

Durian Weitz Pine, Science 1991

CCD Laser

Linear relaxations

101 102 103 10-4 10-2 100 102

M

• d

u l u s

1/τ

1/2

Modulus (Pa) G” G’ What is the coupling between ageing and linear relaxations? Do slow relaxations correspond to glassy dynamics?

Frequency (Hz)

Gopal 2003

Dissipation at a mesoscopic scale

1 2 3 4

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

Time Local stress

2D dry foam as a model system

Surface evolver software simulates quasistatic equilibrium and coarsening

stress

XY X Y

T t t d !

"#

= # $

!

!

t

Shear stress due to surface tension T

• n an area A (Bachelor, JFM 1970)

Vincent-Bonnieu

How coarsening induces creep flow

Stress

Yield stress

Vincent-Bonnieu et al, EPL 2006

1 2 3 4 5

• 500

500 1000 1500 2000 2500

! ( t ) / "

time (a.u.)

50 bubbles 500 bubbles

rearrangement

γ(t) / σ

Compliance Yield stress σ

A mesoscopic model to link rearrangements and macroscopic flow analytically

Microstructure change upon a rearrangement Rearrangement = force dipole acting on an isotropic elastic continuum Length of new film λ Surface tension T Macroscopic flow at the rate of rearrangements

Kabla et al, PRL2003

Stress

λ T λ T

The dipole tensor of a 2D rearrangement

Conjecture Intrinsic orientation of P: α Dipole strength ∝ λ T α

G shear modulus A sample area

Average macroscopic shear strain step due to a randomly placed rearrangement as predicted by continuum mechanics:

Shear direction

!" # $ T A G sin2%

λ T T

The orientation of rearrangements is biased by the applied stress

σ = - 0.25 σy σ = + 0.25 σy σ = 0

10 20 30 40 50 60 70

• 80
• 40

40 80

! ( " ) "

• 80
• 40

40 80

!

• 80
• 40

40 80

!

ρ- ρ- ρ+ ρ+ ρ- ρ+

Strain δγ

0.005

• 0.005

50% 0%

• 90 -45 0 45 90

angle α (°)

• 45 0 45

angle α (°)

• 45 0 45

angle α (°) !" # $ T A G sin2% Distribution of orientation of new films

α α α

!+ " !" = # $

A Maxwell constitutive law links the slow relaxation to the bubble rearrangement rate R and to the area Ameso of a rearranged mesoscopic zone

Ameso = χ T <λ>≅ (1.5 d)2

1 ! d" dt # 1 G R $ T < % >

2.2 2.3 2.4 2.5 2.6 2.7 10 20 30 40 50 60 70 20 40 60 80 100

N u m b e r

• f

T 1 e

Time (s) ! = 0.25 !y ! = 0.15 !y

(γ /σ)(T/d) Number of rearrangements

Vincent-Bonnieu et al, EPL 2006 ; Vincent-Bonnieu et al, cond-mat/0609363

Normalized compliance

1 2 50 100

N

• r

m a l i z e d

time (s)

Creep rheometry and in situ multiple light scattering

Coarsening Fast Medium Slow slope G/η

η

Maxwell creep G

stress σ

τ = η / G ≅ 200 to 1000 s

CCD Laser

σ γ

Yield stress

Normalized compliance

AOK PEO LOH foam with N2 Gillette foam AOK PEO LOH foam with N2 /C6F14 gas fraction = 93% γ(t) G / σ

Coarsening induced rearrangements are at the origin of the slow linear relaxation

Cohen-Addad et al, PRL 2004

102 10

3

10

4

103 104 105

! / G ( s 1 /(R d3) (s)

fast slow medium

1 Coarsening rate

Time interval between rearrangements in a volume d3

! =

"

G # 1 R Vmeso

Creep characteristic time η/G (s)

• Volume of a mesoscopic rearranged zone Vmeso ≅ (3 d)3
• In agreement with the numerical simulation and mesoscopic model

101 102 103 10-4 10-3 10-2 10-1 100 101 102

Shear modulus (Pa) Frequency (Hz)

G’ G’’

G*(ω) = 1/(iω L[J(t)](iω))

Do interfacial rheology and foam viscoelasticity interplay?

1/2

G *( f ) = G 1 + i f fc ! " # $ % & + i 2' (o f

A generic model of viscoelasticity based on disorder

Liu et al, PRL 1996

Slip shear ηeff g Lennard-Jones glass Disordered foam

Goldenberg et al, EPL 2007 Durian, PRE 1997

fc ! g !eff

G

Surface dilatation δA/A Surface stress

! s = E "A A + # 1 A "A "t

• Low rigidity with synthetic surfactants SDS, TTAB…
• High rigidity with fatty acids or LOH/surfactant mixtures.

Interfacial dilatational elasticity and viscosity

Interfacial rheology

Mechanisms of viscous dissipation

Rigid interfaces: Marangoni flow in films

Viscous resistance Driving force

g ! E d

eff

d d ! " ! # $ +

Buzza Lu Cates, J Phys II 1995

E dilatational interfacial elasticity η liquid viscosity

fc ! g "eff ! E # " d 2 ! 10$103 Hz

δ d

Mobile interfaces

g ! T d !eff ! ! + " d

Low interfacial elasticity: Flow in films junctions

fc ! T / d ! +" /d !1#104 Hz

T surface tension κ surface viscosity

d

Junction

Foam with rigid interfaces

40 60 80 100 300 500 700 1 10 100

S h e a r m

• d

u l u s Frequency (Hz)

1/2 1

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3

10 100 1000

age t (min)

G(t)/G(to) to d(to)/d(t)

G’ G’’

Bubble size 29 µm 33 µm 36 µm 42 µm 51 µm 62 µm 75 µm Gas fraction = 92%

G

Gillette shaving foam E = 70 mN/m

G *( f ) = G 1 + i f fc ! " # $ % & + i 2' (o f

100 200 300 100 200 300 400

f

c (Hz)

G (Pa)

Scaling of the characteristic frequency

Mobile interfaces

SLES CAPB foam

E = 10 mN/m

Rigid interfaces

Gillettte foam

E = 70 mN/m

Marangoni flow in films Viscous flow in film junctions

fc ! T / d ! + " / d

fc ! E ! " d 2

G ! T d

Krishan et al, PRE 2010

1 2 3 1 2 3 2 4 6 8 10 12

! / ! (! /" E) / (! /" E) Solution viscosity (mPa s) SLES CAPB foam SLES CAPB LOH foam

40% 40%

1 10 100 1000 0.0001 0.001 0.01 0.1 1 10 100

M

• d

u l u s ( P Frequency (Hz)

Physico-chemistry matters !

Rigid interfaces Mobile interfaces Modulus (Pa)

Fast relaxations Slow relaxations

• R. Höhler

A.-L. Biance

• F. Rouyer
• K. Krishan
• Y. Khidas
• A. Helal
• R. Lespiat
• S. Vincent-Bonnieu
• S. Costa

h'p://www2.univ-­‑mlv.fr/lpmdi/RHE/index2.php

• P. Sollich

S.M. Fielding