Surface viscous effect
on surfactant transport onto a foam lamella
Denny Vitasari, Paul Grassia, Peter Martin
Foam and Minimal Surface, 24 – 28 February 2014
1
Background – Foam fractionation
• Foam fractionation:
Separation of surface
active material using
rising column of
foam.
Transport of surfactant
onto the film interface
determines the
efficiency of a foam
fractionation column.
• Foam fractionation
column with reflux:
Some of the top
product is returned to
the column
2
Foam structure – Dry foam
Lamella:
thin film
separating the
air bubbles
within foam.
Plateau border:
three lamellae meet
at 120 to form an
edge
3
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.
4
Aim
Modelling the surface velocity profile and
the surfactant transport onto a foam
lamella in the presence of surface viscous
stress.
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Surface velocity profile
Surface velocity
film
drainage
Marangoni
effect
surface
viscous effect
Case without film drainage
Simplification as benchmark for the real system
Dimensionless surface velocity:
Marangoni
effect
surface
viscous effect
7
Illustration of a lamella
and Plateau border
𝐿′ + 𝐿′ 𝑃𝑏 = 1 + 𝑎′ 𝜋 6
Boundary condition:
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Parameters for simulation
Foam film made from solution of Bovine serum albumin (BSA) with cosurfactant
propylene glycol alginate (PGA) (Durand and Stone, 2006)
Parameter
Symbol
Value
Unit
Characteristic `Marangoni’ time scale
L2/(G0)
3.12510-2
s
Initial half lamella thickness
0
2010-6
m
Half lamella length
L
510-3
m
Liquid viscosity

710-3
Pa s
Surface viscosity
(31±12)10-3
Pa m s
Curvature radius of the Plateau border
s
a
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
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Key parameters
• 𝛿0′ 𝜇𝑠 (dimensionless surface viscosity parameter) is a
•
key parameter to determine the effect of surface
viscosity upon the system.
𝑎′ (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
𝛿0′ 𝜇𝑠
𝑎′
2.7 × 10−4 − 5.4 × 10−2
0.1 − 0.3
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Surface velocity: very small
′
δ0 𝜇 s
Effect of surface viscosity only at the boundary layer near Plateau border
Special case where 𝜕lnΓ′ 𝜕𝑥 ′ = 𝛽𝑥′
δ0𝜇s
Surface movement
slows down due to
surface viscosity.
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Surface velocity:
very small δ′0 𝜇s
Jump of Γ ′ = Γ0′ at 𝑥 ′ ≤ 𝑥0′ to Γ ′ = 1 at 𝑥 ′ ≥ 𝑥0′
With very small δ0𝜇s the surface velocity profile represents a Dirac delta function at 𝑥 ′ = 𝑥0′
Solution using Green’s function:
𝑥0′ = 0.5
• 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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Surface velocity:
′
very small δ0 𝜇s
Jump of Γ ′ = Γ0′ at 𝑥 ′ ≤ 𝑥0′ to Γ ′ = 1 at 𝑥 ′ ≥ 𝑥0′
Value of 𝑎′ relative to 𝑎′𝑐𝑟𝑖𝑡
determines the perturbation of
surface velocity near Plateau
border:
• 𝑎′ < 𝑎′𝑐𝑟𝑖𝑡  upward
perturbation from 𝑎′ = 𝑎′𝑐𝑟𝑖𝑡
• 𝑎′ > 𝑎′𝑐𝑟𝑖𝑡  downward
perturbation from 𝑎′ = 𝑎′𝑐𝑟𝑖𝑡
13
Surface velocity:
very small δ′0 𝜇s
Jump of Γ ′ = Γ0′ at 𝑥 ′ ≤ 𝑥0′ to Γ ′ = 1 at 𝑥 ′ ≥ 𝑥0′
𝑥0′ is shifted along the
lamella length.
The location of the
largest magnitude of
surface velocity shifts as
the jump point shifts.
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Surface velocity:
′
very small δ0 𝜇s
Jump of Γ ′ = Γ0′ at 𝑥 ′ ≤ 1 to Γ ′ = 1 at 𝑥 ′ ≥ 1
Value of 𝑎′ relative to 𝑎′𝑐𝑟𝑖𝑡
determines the perturbation of
surface velocity near Plateau
border:
• 𝑎′ < 𝑎′𝑐𝑟𝑖𝑡  upward
perturbation from 𝑎′ = 𝑎′𝑐𝑟𝑖𝑡
• 𝑎′ > 𝑎′𝑐𝑟𝑖𝑡  downward
perturbation from 𝑎′ = 𝑎′𝑐𝑟𝑖𝑡
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General solution for arbitrary 𝜕lnΓ′ 𝜕𝑥′
Differential equation
Integration of the Green’s function
 The Green’s function is easier to obtain when δ′0 𝜇𝑠 is small but more
complicated to obtain when δ′0 𝜇𝑠 is larger.
Finite difference approximation
 Applicable for arbitrary δ′0 𝜇s
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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 (𝑢𝑠 ) applies on every
material point.
• Surface excess () averaged between
two material points.
• Plot  vs position 𝑥
• Surfactant is conserved between
material points  rectangle area
preserved.
18
Evolution of surfactant surface concentration
With surface viscosity δ′0 𝜇𝑠 = 0.0266
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 ~ δ′0 𝜇𝑠 )
 Critical radius of curvature of Plateau border:
• 𝑎′ ≪ 𝑎′𝑐𝑟𝑖𝑡 ~ δ′0 𝜇𝑠 : 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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