Material point method simulation Surfactant transport onto a foam lamella Material point method simulation Denny Vitasari, Paul Grassia, Peter Martin Annual Manchester SIAM Student Chapter Conference 2013 20 May 2013

Background – 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.

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

2D illustration of a foam lamella Due to reflux, the surface tension at the Plateau border (Pb) is lower than that at the lamella (F)  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)  liquid is sucked to the Plateau border  film drainage.

Assumptions The lamella is always flat and has a uniform thickness along the length. At initial time the surface concentration (F0) of surfactant along the film is uniform. The surface concentration (Pb) of surfactant at the Plateau border interface is fixed.

Gibbs-Marangoni parameter Film velocity profile viscosity The equation for velocity profile of liquid on the lamella surface: Gibbs-Marangoni parameter Film drainage Marangoni effect Surfactant mass balance:

Rate of film drainage Mobile interface (Breward-Howell, 20021) Rigid interface (Reynolds, 18862) 1. Breward, CJW and Howell, PD, Journal of Fluid Mechanics, 458:379-406, 2002 2. Reynolds, O, Philosophical Transaction of the Royal Society of London, 177:157-234, 1886

Analytical solution Case no film drainage Solution of surfactant mass balance equation: Complementary error function Shift one boundary condition to - and solve the equation analytically to result in a complementary error function. Reflection method to correct the boundary condition. Violation of boundary condition due to reflection method. Improving accuracy using additional reflections. Surfactant mass balance: Boundary conditions: Complementary error function (1 reflection):

Analytical solution Case no film drainage Solution of surfactant mass balance equation: Fourier series Fourier series obtained from method of separation variable: Surfactant mass balance: Boundary conditions: Fourier series:

Numerical simulation of surfactant concentration () Material point method3 The surface velocity (us) applies on every material point  material point change its position. Surface excess () averages between two material points. Surfactant is conserved  same area of the rectangle. 3. Embley, B and Grassia, P, Colloids and Surfaces A, 382: 8-17, 2011

Bookkeeping operation Every time step: material points change their positions  uneven spatial interval over time. The spatial interval (Δx) is restored and the value of  is corrected. Take a weighted average of  in the restored interval as the new value. both sides move to the right left side moves to the right right side moves to the left both sides move to the left

Analytical solution Case with film drainage (Quasi) steady state (rigid interface): us = 0 (no surfactant flux on the surface) Solution: Asymptotic solution (mobile interface): Uniform inner solution inner pulls boundary layer near the Plateau border Solution:

Parameters for simulation Symbol Value Unit Characteristic `Marangoni’ time scale L2/(G0) 3.12510-2 s Characteristic thinning time scale (mobile) 0/(d/dt)0 1.4810-3 Characteristic thinning time scale (rigid) 2.08 Initial half lamella thickness 0 2010-6 m Half lamella length L 510-3 Liquid viscosity  110-3 Pa s Curvature radius of the Plateau border a 510-4 Surfactant surface concentration at PB Pb 210-6 mol m-2 Initial surface concentration at film film 110-6 Surface tension of solution at PB Pb 4510-3 N m-1 Gibbs-Marangoni parameter G 4010-3

Results dimensionless form

Surface excess profile ( vs x) no film drainage t Surface excess of surfactant () increases with time

Verification of numerical result Case no film drainage Complementary error function t’ = 0.005 Simulation at early time  fewer reflection terms needed. Numerical simulation fits well with the analytical solution.

Verification of numerical result Case no film drainage Fourier series t’ = 2 Simulation at later time  fewer Fourier terms needed. Numerical simulation fits well with the analytical solution. Accuracy of the numerical result increases with more grid elements.

Surface excess profile ( vs x) film drainage: rigid interface Surface excess of surfactant () increases with time but slightly more slowly than in case with no film drainage.

Surface excess profile ( vs x) film drainage: mobile interface Surface excess of surfactant () decreases with time: surfactant washed off film by film drainage.

Spatially-averaged surface excess  Surface excess  increases with time for the case of no drainage and draining film with a rigid interface. Surface excess  decreases with time for the case of draining film with a mobile interface. Surface excess  of draining film with a rigid interface is slightly lower than that of no drainage.

(Quasi) steady state solution Film drainage: rigid interface The (quasi) static solution has weak spatial variation in . Quasi static solution does not significantly change with time. Agreement between numerical and (quasi) static solution only possible at reasonably long time.

Asymptotic boundary-layer solution Film drainage: mobile interface Inner region (near the film centre) and outer region (near the Plateau border). Inner region: Marangoni effect is negligible. Outer region: Marangoni effect is retained. Agreement between numerical and asymptotic analytical results.

Conclusions The equations of surfactant transport onto a foam lamella can be solved numerically using a material point method followed by a bookkeeping operation. The numerical simulation is validated by analytical solution obtained from complementary error function and Fourier series in the case of no film drainage. In a foam fractionation column with reflux, when Marangoni flow dominates the film drainage, the surface concentration of surfactant increases with time. When film drainage dominates the Marangoni effect such as in a film with mobile interface, surfactant is washed away to Plateau border and its concentration decreases with time. Quasi steady state solution agreed with the numerical simulation for the case of a film with a rigid interface, in the limit of long times. Asymptotic boundary-layer solution agreed with the numerical simulation for the case of a film with a mobile interface.

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