1. Denny Vitasari, Paul Grassia, Peter Martin Foam and Minimal Surface, 24 – 28 February 2014 1 Surface viscous effect on surfactant transport onto a foam lamella

2. Background – Foam fractionation 2 Foam fractionation: Separation of surface active material using rising column of foam. Foam fractionation column with reflux: Some of the top product is returned to the column Transport of surfactant onto the film interface determines the efficiency of a foam fractionation column.

3. Foam structure – Dry foam 3 Lamella : thin film separating the air bubbles within foam. Plateau border : three lamellae meet at 120  to form an edge

4. 2D illustration of a foam lamella 4 Due to reflux, the surface tension at the Plateau border is lower than that at the lamella  transport of surfactant from the surface of Plateau border to the surface of film  Marangoni effect. Pressure in the Plateau border is lower due to curvature (Young-Laplace law)  suction of liquid to the Plateau border  film drainage. Surface viscous effect takes place and opposes surface motion.

5. Aim Modelling the surface velocity profile and the surfactant transport onto a foam lamella in the presence of surface viscous stress. 5

6. Surface velocity profile Surface velocity film drainage Marangoni effect surface viscous effect

7. Case without film drainage 7 Simplification as benchmark for the real system Dimensionless surface velocity: Marangoni effect surface viscous effect

8. Illustration of a lamella and Plateau border 8 Boundary condition:

9. Parameters for simulation 9 ParameterSymbolValueUnit Characteristic `Marangoni’ time scale L 2  /(G  0 )3.125  10 -2 s Initial half lamella thickness 00 20  10 -6 m Half lamella lengthL 5  10 -3 m Liquid viscosity  7  10 -3 Pa s Surface viscosity ss (31±12)  10 -3 Pa m s Curvature radius of the Plateau bordera 5  10 -4 m Surfactant surface concentration at PB  Pb 2  10 -6 mol m -2 Initial surface concentration at film  F0 1  10 -6 mol m -2 Foam film made from solution of Bovine serum albumin (BSA) with cosurfactant propylene glycol alginate (PGA) (Durand and Stone, 2006)

10. Key parameters 10 ParameterRange of values

11. 11 Effect of surface viscosity only at the boundary layer near Plateau border Surface movement slows down due to surface viscosity.

12. 12 Solution using Green’s function: Largest magnitude of surface velocity at the jump point. Surface viscous effect reduces peak surface velocity. Flux near Plateau border in the absence of local Marangoni force there.

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16. 16 Differential equation

17. Surface velocity profile 17 The turn around of surface velocity is less sharp at a later time due to surfactant surface concentration gradients being spread over larger distances, implying also a weaker Marangoni effect. Weaker Marangoni effect results in lower surface velocity.

18. Surfactant transport via material point method 18

19. Evolution of surfactant surface concentration 19 Without surface viscosity

20. Conclusions 20

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