Surfactants and their hydrodynamic consequences: focus on surface tension effects on drops

1. Surfactants and their hydrodynamic consequences: focus on surface tension effects on drops Kathleen StebeChemical and Biomolecular EngineeringUniversity of Pennsylvania

2. Aims: • Stresses at fluid interfaces • Compositional dependence of surface tension: surfactants • How surfactants can dictate interfacial stresses of moving drops • How surfactants can create interesting drop break up modes

3. Common bc: • No slip at solids • Velocities continuous at fluid interfaces • Viscous stresses continuous at fluid interfaces

4. U Flow around spherical drop • Hadamard-Rybsinski • ss, small Re, linear eom • Continuous stresses, velocities at surface; far field U • Integrate drag, equate to body force Uterminal: Not obeyed for small drops in early exp. Why do we care? UHR=3/2UStokesf(l) f(l) approached 2/3 for l big • Motivation: • Retarded drop migration • Remobilization • Tip streaming Vs

6. Adsorption-desorption fluxes (t-)d1/2] + (D/b)[ Ct - () d] (1) adsorption flux desorption flux At equilibrium: Frumkin isotherm Equation of state By Gibbs adsorption equation entropic enthalpic Isotherm and eqn of state related by Gibbs adsorption equation

7. Common bc: No slip at solids Velocities continuous at fluid interfaces Viscous stresses continuous at fluid interfaces

8. Surfactants, surface tension and their hydrodynamic consequences Isothermal system: g (G) • 1. Dynamic surface tension studies: TRANSPORT COEFFICIENTS • Soluble surfactants exchange between interface and solution: • adsorption/desorption; diffusion (convection) • monomers, micelles • 2. Instantaneous surface stresses on moving drops/bubbles: • g (G(t,s)) • G: memory! prevailing flow and flow history • highly coupled hydrodynamics/mass transfer • Very sensitive to hindered diffusion fluxes, desorption fluxes

10. Dynamic surface tension

11. Dynamic surface tension (t-)d1/2] + (D/b)[ Ct - () d] (1) or

12. C12E6 CH3(CH2)11-(OCH2CH2)6-OH Fit equilibrium and compression data to obtain constants

13. Dynamics: obtain transport kinetic constants C12E6 α=1.4x10-4s-1 b=4.0x10-4m3(mol-s) -1 D=6x10-10m2/s Pan et al 1998.

15. Microtensiometry: fast diffusion flux -reveal kinetic barriers? Jin et al, J. Adhesion 2004 b<<h b RD-K~10s of microns

16. Microtensiometry Static bubble/drop tensiometer: Focuses on RDK C12E8 Langmuir, Anna et al 2010 CMU

17. at CMC- ads-des times range from .12-63 s diffusion times from 10-2 to greater than 102 s desorption kinetic constant: ao=10-2 s-1-10-6s-1

18. above CMC: micellar transport Bhole, et al, 2010

19. U Flow around spherical drop • Hadamard-Rybsinski • ss, small Re, linear eom • Continuous stresses, velocities at surface; far field U • Integrate drag, equate to body force Uterminal: UHR=3/2UStokesf(l) f(l) approached 2/3 for l big Vs Not obeyed for small drops in early exp. Why do we care?

20. Common bc: No slip at solids Velocities continuous at fluid interfaces Viscous stresses continuous at fluid interfaces

21. U U D ghigh adsorb Vs Vs glow desorb Surfactants on an undeforming drop • Balance of surface convection and mass transfer: determines G • Marangoni stresses resist motion, terminal velocity reduced; U Ustokes Shift Hill’s vortices: Slow mass transfer, reaction in drops Increase drag D

22. U Surface stress balance and surfaceequation of state determinesg(G(q)) ghigh Vs glow

23. U Surface stress balance ; Marangoni number ghigh Vs glow

24. 0.25 G 0.2 0.15 Surface Concentration, 0.1 0.05 U 0 0 40 80 120 160 0 Position Along the Bubble Surface, q 0.7 0.6 s 0.5 0.4 Surface Velocity, V 0.3 0.2 0.1 0 0 40 80 120 160 0 Position Along the Bubble, q ‘Insoluble' surfactants: stagnant caps Sadhal and Johnson 1983 Davis and Acrivos 1966 Savic 1953 q ghigh Vs glow Palaparthi, Papageorgiou, Maldarelli, JFM 2006

25. U Adsorption-desorption controlled surfactants: Bi ghigh Vs glow • Chen and Stebe 1996 ; Levan and Holbrook 1983

26. h Diffusion controlled surfactants:

27. Diffusion controlled surfactants: Wang, Papageorgiou, Maldarelli JFM 1999, 2002 Palaparthi, Papageorgiou, Maldarelli, JFM 2006 Liao et al. JCIS 2004 in capillary flows: Stebe, Lin, Maldarelli, Phys. Fluids 1991 In exp: needed to go above CMC

28. Pressure drops conc Triton X 100 soln BSA conc BSA soln

29. Summary: surfactants on undeforming drops • Surfactants collect near points of converging surface flow: • Depending on rates of mass transfer, surfactants can slow surface velocity, retard terminal velocity. • Effect ranges from immobile interface (slow surfactant mass transfer) to completely mobile interface (rapid surfactant mass transfer) • Implications for mass transfer/reaction in drops • Ads/des and diffusion mixed control the norm; micelles-micelle free zones, micellar adsorption What happens on deforming drops?

30. Surfactants and the Taylor extensional flow l=1 l<1 g high g (G) g low Leal and collab: clean drop break up modes; surfactant effects Stebe and collab: surfactant effects Vlahovska; others

31. Surfactants and Deformation Mechanisms gtipDf <g>Df x 1 x<<1

32. Mass Balance: ‘Insoluble’ Surfactant o vs o Ds Fix material parameters: Vary: E=0.2 L=103

33. Time Marching with Boundary Integral Method initial conditions: sphere,G(s)=g(s)=1 Stokes Eqs. mapped to s ovelocity odeformation omass conservation: update G Continue until: stable shape (vn zero); iterate Ca -or- no stable shape (vn grows): study shapes oupdate g, stress condition

34. Dilute initial coverage x<0.5 Initially weak Marangoni stresses Eventual accumulation near tip Ma Stress Drop deformation x x x Ca=0.04 x tip equator Eggleton, Stebe

35. Stagnant Cap Limit for x<<1 Mass balance at steady state low initial coverage stagnant mobile Surface concentration Tangential velocity x x

36. Drop deformation x Stagnant interface at steady state High Initial Coverage x 1 x

37. Ma stresses for x 1 Ma stress Marangoni stresses regulate surface velocity so that  remains nearly uniform

38. Ma large in these studies- hence surface in incompressible limit. Can reformulate to have surface pressure (Marangoni stresses) become Lagrange multiplier that enforces incompressibility constraint. This limit is commonly invoked in surface viscosity experiments based on Brownian migration of particles/proteins in which surface flow is small enough that Marangoni number is large.

40. ordaughter : rparent ~ 0.01 ogdaughter<<gparent Implications for deformation:tip streaming for small x: large G gradients Taylor 1934 Rumscheidt and Mason 1961 Janssen et al. 1997 (tip streaming –tip dropping) deBruijn 1993 (Shear flow)* Hu, Pine and Leal 2000 Michael Seigel JFM Anna Bontoux Stone PRL l<1 *Tip streaming: dilute surfactants Eggleton et al PRL 2001

41. 1.0 0.0 -1.0 x=0.1 Low coverage, large gradients l=0.05 Steady state profiles 0 2 4 C 0.1

42. First Daughter Drop G∞ G 0 geq g 0

43. 1.0 0.0 -1.0 0 2 4 0 4 8 Comparison of time evolving drop shapes: time marching BIM x=0.1 Cacrit=0.065 x=0.75 Cacrit=0.055

45. (a) (b) (c) (d) (e) FROM SHELLY ANNA , CMU Flow focusing microfluidic device for producing a range of sizes of liquid drops by varying the applied flow rates. Outer (oil) phase contains Span 80 surfactant.

46. Tip streaming; tail streaming Balaji Gopalan and Joseph Katz Ali Borhan, tail streaming simulations Fang Jin (Jin and Stebe) Tails as drops detach

47. Drop detachment, necks and surfactantsSurfactants collect at sites of strong surface contraction if mass transfer is slower than the contraction rateThis can alter drop detachmentWe investigate this in the adsorption/desorption controlled limit in the diffusion controlled limit in the mixed-controlled limit (unpublished)

48. U b A viscous drop injected into a viscous fluid: a numerical study Axisymmetric Bouyant drop Fix: Front tracking scheme: marker & cell with continuum surface force

49. Surfactant free drop evolution Primary Neck Secondary Neck Thins at primary neck

50. Surfactant free drop evolution Necking dynamics: local balance of viscous stresses and capillary forces Self similar dynamics Brenner, Lister and Stone; Zhang and Lister; Cohen and Nagle