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

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 µ m 1 cm gas 10 nm 1 nm 1 µ m liquid

Foam structure A surface minimization problem that goes back to Plateau (1873) and Kelvin (1887) 100 ¡µm 64% 90% 99% Gas content Bubblesurface energy ! 10 10 Thermal energy

Ageing Drainage Coalescence Coarsening Avalanche-like Statistically self-similar growth dynamics Durian D. Durian Flow through a porous medium

Solid-like or liquid-like mechanical behavior 50 cm 2 mm Froghopper

Origin of foam elasticity and yielding Surface tension T Bubble diameter d 1 mm Rouyer G ≈ T/d ≈ 100 Pa Shear modulus Derjaguin, Koll. Zeitschrift 1933 γ Y ≈ 1 Yield strain Princen, JCIS 1983 Kraynik Reinelt, PRE 2000 σ Y ≈ T/d Yield stress

Are generic mechanisms at the origin of the mechanical behavior of soft disordered materials ? Foam Emulsion Colloidal suspension Granular media Onion phase 4 µ m 20 µ m 50 µ m Cloitre Ramos Weeks Liu Nagel, Nature 1998

Viscoelastic relaxations σ Shear γ 10 3 10 3 Labiausse et al, JOR 2004 Gopal Durian PRL 2003 s Modulus (Pa) Modulus (Pa) u G’ G’ l u 10 2 10 2 d 1/2 o G” G” M γ o = 0.001 0.1 Hz 10 1 10 1 10 -4 10 -2 10 0 10 2 10 -3 10 -2 10 -1 10 0 γ o Frequency (Hz) Strain amplitude What is the link between foam structure and relaxations ? What is the pertinent length scale to modelize foam viscoelasticity ?

Nonlinear slow relaxations Large Amplitude Oscillatory Shear Hyun et al JNNFM 2002 ! ( t ) = ! o cos( " t ) Strain ( ) + $ ! ( t ) ! ( t ) = " o G'cos ( # t ) + G"sin ( # t ) Stress σ (t) increasing γ o γ (t) σ (t) (Pa) σ (t) γ (t) γ (t)

Foam approximately behaves like an elastoplastic material σ y G 1 - 3/2 G' / G Gillette shaving foam * 3/2 $ ' # o 0.1 G ' ! 16 G Wet φ = 92% & ) 3 " # y % ( d = 30 µ m or 40 µ m T = 30 mN/m Slowly coarsening * 1 $ ' # o 1 G '' ! 4 G & ) AOK / PEO / LOH foam " # y % ( - 1 G'' / G Dry φ = 97% d = 50 µ m 0.1 T = 22 mN/m ≈ No coarsening 0.1 1 10 ! " / ! y Rouyer et al, EPJE 2008

… but stress harmonics depend on physical chemistry ( ) + $ ! ( t ) ! ( t ) = " o G 'cos( # t ) + G ''sin( # t ) < ! " 2 > q = Anharmonic stress residual < " 2 > Elastoplastic model Gillette shaving foam 0.1 q AOK / PEO / LOH foam 0.01 0.1 1 10 ! " / ! y

The SGR model does not fully capture foam relaxations Soft Glassy Rheology model Elastic energy Sollich et al, PRL 1997 of a mesoscopic region Bubble configuration 1 1 0.1 G'' / G - 1 G' / G SGR SGR SGR - 3/2 36 ! m 1 Hz q Gillette 1 Hz 28 ! m 0.1 0.1 0.3 Hz 0.01 50 ! m 1 Hz AOK 0.3 Hz 0.1 1 10 0.1 1 10 0.1 1 10 ! o / ! ! o / ! ! o / ! y y y

Mechanical memory effect ) a m 500 P ( Quiescent ’ G c Energy density s i 450 u l t u d After shear s After LAO Shear o m a 400 c l i t s E a l 350 E 0 1 2 3 4 10 10 10 10 10 t - t shift t - t shear (s) Foam Spin 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! Laser CCD 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 6 4 τ / τ (t p ) 2 t p 0 1000 1200 1400 1600 1800 2000 Foam age (s) Cohen-Addad et al, PRL 2001

Linear relaxations 10 3 Gopal 2003 s u Modulus (Pa) G’ l u 10 2 d 1/2 o G” M 1/ τ 10 1 10 -4 10 -2 10 0 10 2 Frequency (Hz) What is the coupling between ageing and linear relaxations? Do slow relaxations correspond to glassy dynamics?

Dissipation at a mesoscopic scale 4 2 1 3 Local stress Time Coarsening-induced bubble rearrangements is an intrinsic source of dynamics in foams

2D dry foam as a model system Surface evolver software simulates quasistatic equilibrium and coarsening stress Vincent-Bonnieu T t = # $ ! Shear stress due to surface tension T t t d ! XY X Y on an area A (Bachelor, JFM 1970) "# !

How coarsening induces creep flow Stress Yield stress σ 5 Compliance / 500 bubbles γ (t) / σ rearrangement 4 ) 50 bubbles " t 3 ( ! 2 1 0 -500 0 500 1000 1500 2000 2500 Yield stress time (a.u.) Vincent-Bonnieu et al, EPL 2006

A mesoscopic model to link rearrangements and macroscopic flow analytically λ T Stress Kabla et al, PRL2003 λ T Rearrangement = force dipole Macroscopic flow Microstructure at the rate of change upon a acting on an isotropic elastic continuum rearrangements rearrangement Length of new film λ Surface tension T

The dipole tensor of a 2D rearrangement Shear direction T Conjecture λ Intrinsic orientation of P : α α Dipole strength ∝ λ T T Average macroscopic shear strain step due to a randomly placed rearrangement as predicted by continuum mechanics: !" # $ T G shear modulus A G sin 2 % A sample area

The orientation of rearrangements is biased by the applied stress Strain δγ σ = - 0.25 σ y σ = 0 σ = + 0.25 σ y 0.005 !" # $ T A G sin 2 % 0 -0.005 α α α Distribution of orientation of new films 70 60 50% 50 ) ! + " ! " = # $ " 40 ( ! 30 20 ρ - ρ + ρ - ρ + ρ - ρ + 10 0% 0 -45 0 45 -45 0 45 -80 -40 0 40 80 -80 -40 0 40 80 -80 -40 0 40 80 -90 -45 0 45 90 ! " ! angle α (°) angle α (°) angle α (°)

A Maxwell constitutive law links the slow relaxation to the bubble rearrangement rate R and to the area A meso of a rearranged mesoscopic zone d " 1 dt # 1 G R $ T < % > ! 2.7 Number of rearrangements Normalized compliance ! = 0.25 ! y 70 ! = 0.15 ! y N 2.6 60 u ( γ / σ ) (T/d) 50 m 2.5 b A meso = χ T < λ > ≅ (1.5 d) 2 40 e r 2.4 30 20 o 2.3 f 10 T 2.2 0 1 0 20 40 60 80 100 Time (s) e Vincent-Bonnieu et al, EPL 2006 ; Vincent-Bonnieu et al, cond-mat/0609363

Creep rheometry and in situ multiple light scattering Laser CCD stress σ Yield stress σ γ d 2 Coarsening Normalized compliance e Fast AOK PEO LOH foam with N 2 z slope G/ η i γ (t) G / σ Medium Gillette foam l Slow AOK PEO LOH foam with N 2 /C 6 F 14 a 1 m gas fraction = 93% r o N 0 0 50 100 time (s) τ = η / G ≅ 200 to 1000 s Maxwell creep η G

Coarsening induced rearrangements are at the origin of the slow linear relaxation s ( 4 10 Coarsening rate Creep characteristic time fast G medium 1 " η /G (s) G # ! = slow / 3 10 1 R V meso ! 10 2 10 3 10 4 10 5 1 /(R d 3 ) (s) Time interval between rearrangements in a volume d 3 • Volume of a mesoscopic rearranged zone V meso ≅ (3 d) 3 • In agreement with the numerical simulation and mesoscopic model Cohen-Addad et al, PRL 2004

10 3 G*( ω ) = 1/( i ω L[J(t)]( i ω )) Shear modulus (Pa) G’ 10 2 1/2 G’’ 10 1 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 Frequency (Hz) Do interfacial rheology and foam viscoelasticity interplay?

A generic model of viscoelasticity based on disorder Lennard-Jones glass Disordered foam Durian, PRE 1997 Goldenberg et al, EPL 2007 f c ! g shear ! eff Slip ! $ η eff g i f G *( f ) = G 1 + & + i 2 ' ( o f # " % f c G Liu et al, PRL 1996

Interfacial rheology Surface dilatation δ A/A Surface stress ! s = E " A " A A + # 1 " t A Interfacial 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 viscosity d ! " d ! $ + Viscous resistance δ eff # d g ! E Driving force d ! E # f c ! g " d 2 ! 10 $ 10 3 Hz " eff

Mobile interfaces Low interfacial elasticity: Flow in films junctions Junction T surface tension κ surface viscosity ! eff ! ! + " d d g ! T d T / d ! + " / d ! 1 # 10 4 Hz f c !

Foam with rigid interfaces Gillette shaving foam E = 70 mN/m s 700 u G l G’ Gas fraction = 92% 500 u G(t)/G(t o ) d 3 o 300 Bubble size m 2 d(t o )/d(t) 29 µ m 1 r 33 µ m a 36 µ m 1 0.9 e 42 µ m 100 0.8 h 0.7 51 µ m 80 G’’ 0.6 S 62 µ m 0.5 60 75 µ m 0.4 t o 1/2 40 0.3 10 100 1000 age t (min) 1 10 100 Frequency (Hz) ! $ i f G *( f ) = G 1 + & + i 2 ' ( o f # " % f c

Scaling of the characteristic frequency 300 Mobile interfaces 40% SLES CAPB foam SLES CAPB foam 3 3 SLES CAPB LOH foam E = 10 mN/m ( ! / " E) / ( ! / " E) 200 c (Hz) 2 2 ! / ! Rigid interfaces 40% Gillettte foam f 1 1 100 E = 70 mN/m 0 0 0 2 4 6 8 10 12 Solution viscosity (mPa s) 0 0 100 200 300 400 G (Pa) G ! T d f c ! E ! Marangoni flow in films " d 2 T / d f c ! Viscous flow in film junctions ! + " / d Krishan et al, PRE 2010