www.imperial.ac.uk/events Sp Spre reading, re retrac action an and sustained oscillations of surfa factant-laden lenses G. Karapetsas1, R. V. Craster2 & O. K. Matar1 1Department of Chemical Engineering 2Department of Mathematics Imperial College London Workshop on Surfactant Driven Thin Films Flow Fields Institute, Toronto, 24 February, 2012
Motivation Van Nierop et al. PoF 2006 Daniels et al. 2007 Stocker & Bush JFM 2007
Formulation I ε = < < 2 V L 1 2 = S i empty space at interface i Surfactant transport and chemical kinetics + ↔ ↔ * * + ↔ * * S c c * * Nc m S c c 23 2 23 2 2 12 2 12 + ↔ + + ↔ + + ↔ + * * * * * * S c S c S c S c S c S c 13 12 12 13 13 23 23 13 23 12 12 23 Approximations Lubrication theory Rapid vertical diffusion
Formulation II Governing Equations h ∫ = − 12 h u dz 12 , t 1 0 x h h h ∫ ∫ ∫ ∫ = − 12 + 23 = − 13 h u dz u dz = σ h u dz where u dz f ( h , ) 23 , t 1 2 13 , t 1 i i i 0 h 0 12 x x [ ] ( ) − h h c c β β h ∫ + = 23 12 2 , x − − − 2 , x 23 c u dz x c 2 c 12 J c 2 c 23 J J ( ) 2 , t − 2 − − c 2 c 12 − c 2 c 23 2 h h h h Pe h h h h h 12 23 12 23 12 c 2 23 12 23 12 [ ] ( ) − h h m m h ∫ + = 23 12 2 , x + = 2 , x 23 m u dz x J J sorption fluxes ( ) i 2 , t − 2 − 2 h h h h Pe h 12 23 12 23 12 m 2
Formulation III c ( ) + = 12 , xx + c u c J 12 , t s , 12 12 c 2 c 12 x Pe 12 c ( ) + = 13 , xx + c u c J 13 , t s , 13 13 ev 13 x Pe 13 c ( ) = J sorption fluxes + = 23 , xx + + c u c J J i 23 , t s , 23 23 c 2 c 23 ev 23 x Pe 23 Equation of state ( ( ) ) ( ) ( ) − 3 1 / 3 σ = + Σ + + Σ − 1 1 1 c 1 1 i i i i Sheludko 1967 ( ) = i 12 , 13 , 23 Σ = σ − σ σ δ = σ σ * * * * * i io im im i im 23 m
Formulation IV Boundary Conditions x = x Contact line ( ) c = = h h h 12 13 23 = + σ = + σ h h f ( ) h h g ( ) Force balance 12 , x 23 , x i 13 , x 23 , x i σ = F ( h , ) 0 Continuity of pressure i , xx i ( ) x x x ∫ ∫ ∫ ∞ c + = − = c 2 h dx 2 h dx V 2 h h dx V Mass conservation 12 13 1 23 12 2 0 x 0 c
Formulation V Boundary Conditions x = x Contact line ( ) c c = β + β 23 , x J J c 12 c 23 c 12 c 23 c 13 c 23 c 13 c 23 Pe 23 = x x c c c 13 , x = + J J 12 , x = β + J J c 13 c 12 c 13 c 23 c 13 c 12 c 13 c 12 c 12 c 23 Pe Pe 13 = x x 12 = x x c c
= σ − σ − S 1 S : spreading parameter Results I 13 12 Clean fluid σ 12 =1, ρ=1, μ=1 σ − σ * * σ = i im i σ − σ * * io im ρ * ρ = 2 ρ * 1 µ * µ = 2 µ * 1 Joanny 1987 1 / 7 x c ~ t Fraaije and Cazabat 1989
( ) ( ) M = * * * Σ = σ * − σ * σ * δ = σ * σ * M V c 2 cmc i io im im i im 23 m Results II Surfactant-laden drop M=8, δ 23 =1.9, δ 12 =1, Σ i =0.1, ρ=μ=1
( ) ( ) M = * * * Σ = σ * − σ * σ * δ = σ * σ * M V c 2 cmc i io im im i im 23 m Results IIΙ Effect of M δ 23 =1.9, δ 12 =1, Σ i =0.1 Long time drop shapes, t=10 5
[ ] ( ) ( ) = − − − J k R c 1 c c 1 c c 13 c 12 c 13 c 12 c 13 c 12 13 12 12 13 = x x Results IV c Adsorption at the contact line M=8, δ 23 =1.9, δ 12 =1, Σ i =0.1 + ↔ + * * S c S c 13 12 12 13
[ ] ( ) ( ) = − − − J k R c 1 c c 1 c c 13 c 12 c 13 c 12 c 13 c 12 13 12 12 13 = x x Results IV c Adsorption at the contact line M=8, δ 23 =1.9, δ 12 =1, Σ i =0.1 + ↔ + * * S c S c 13 12 12 13
Results V Oil water interface: Oleic acid + NaOH Na-oleate Van Nierop et al. PoF 2006
Results VI Stocker & Bush JFM 2007
Results VI Stocker & Bush JFM 2007
Results VI Stocker & Bush JFM 2007
13 = k kinetic parameter for evaporatio n Results VI ev Stocker & Bush JFM 2007
( ) ( ) M = * * * Σ = σ * − σ * σ * δ = σ * σ * M V c 2 cmc i io im im i im 23 m Results VII Effect of density ratio, ρ M=8, δ 23 =1.9, δ 12 =1, Σ i =0.1 ρ * ρ = 2 ρ * 1 Long time drop shapes, t=10 5
Conclusions We have studied the spreading of surfactant-laden drops on thin layers of another liquid. The presence of Marangoni stresses gives rise to very rich dynamics which may include: Spreading until the drop reaches equilibrium (S < 0). Continuous spreading (S > 0) Spreading followed by retraction. Self-sustained oscillations.
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