Denny Vitasari, Paul Grassia , Peter Martin Foam and Minimal Surface, 24 – 28 February 2014

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Denny Vitasari, Paul Grassia, Peter MartinFoam and Minimal Surface, 24 – 28 February 2014

Surface viscous effect on surfactant transport onto a foam

lamella

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Background – Foam fractionation

• 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.

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Foam structure – Dry foam

Lamella: thin film separating the air bubbles within foam.

Plateau border: three lamellae meet at 120 to form an edge

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2D illustration of a foam lamella

• 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.

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Aim

Modelling the surface velocity profile and the surfactant transport onto a foam

lamella in the presence of surface viscous stress.

Surface velocity profile

Surface velocity

film drainage

Marangoni effect

surface viscous effect

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Case without film drainageSimplification as benchmark for the real system

Dimensionless surface velocity:

Marangoni effect

surface viscous effect

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Illustration of a lamella and Plateau border

𝐿′+𝐿′𝑃𝑏=1+𝑎′𝜋 /6

Boundary condition:

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Parameters for simulation

Parameter Symbol Value UnitCharacteristic `Marangoni’ time scale L2/(G0) 3.12510-2 s

Initial half lamella thickness 0 2010-6 m

Half lamella length L 510-3 mLiquid viscosity 710-3 Pa sSurface viscosity s (31±12)10-3 Pa m s

Curvature radius of the Plateau border a 510-4 mSurfactant surface concentration at PB Pb 210-6 mol m-2

Initial surface concentration at film F0 110-6 mol m-2

Foam film made from solution of Bovine serum albumin (BSA) with cosurfactant propylene glycol alginate (PGA) (Durand and Stone, 2006)

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Key parameters

• (dimensionless surface viscosity parameter) is a key parameter to determine the effect of surface viscosity upon the system.

• (dimensionless radius of curvature of Plateau border relative to film length) determines magnitude of surface velocity near the Plateau border, hence the rate of surfactant transport onto the film.

Parameter Range of values

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Surface velocity: very small s

Special case where

Effect of surface viscosity only at the boundary layer near Plateau border

Surface movement slows down due to surface viscosity.

δ0s

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Surface velocity:very small s

Jump of at to at With very small δ0s the surface velocity profile represents a Dirac delta function at

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.

𝑥0′ =0.5

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Jump of at to at

Value of relative to determines the perturbation of surface velocity near Plateau border:

• upward perturbation from

• downward perturbation from

Surface velocity:very small s

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Jump of at to at

is shifted along the lamella length.

The location of the largest magnitude of surface velocity shifts as the jump point shifts.

Surface velocity:very small s

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Jump of at to at

Surface velocity:very small s

Value of relative to determines the perturbation of surface velocity near Plateau border:

• upward perturbation from

• downward perturbation from

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General solution for arbitrary

Integration of the Green’s function The Green’s function is easier to obtain when is small but more complicated to obtain when is larger.

Finite difference approximation Applicable for arbitrary s

Differential equation

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Surface velocity profile

• 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.

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Surfactant transport via material point method• Surface velocity () applies on every

material point.

• Surface excess () averaged between two material points.

• Plot vs position

• Surfactant is conserved between material points rectangle area preserved.

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Evolution of surfactant surface concentration

With surface viscosity Without surface viscosity

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Conclusions In the absence of film drainage, surface viscous effect balances the

Marangoni force and slows down the surface movement. Jump of surfactant surface concentration:

• Largest magnitude of surface velocity at the jump point.

• Surface movement delocalized away from Marangoni surface tension gradients due to surface viscous effect (delocalisation distance )

Critical radius of curvature of Plateau border: • ~ : magnitude of surface velocity greatly reduced by

requirement to satisfy symmetry condition on midpoint of Plateau border face.

In the presence of surface viscosity, the surface concentration of surfactant at a given time is lower than that without surface viscosity.

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Thank you