1. A phase field model for binary fluid-surfactant system
Kuan-Yu Chen (陳冠羽)
Advisor: Ming-Chih Lai (賴明治)
Department of Applied Mathematics, National Chiao Tung University, Taiwan

2. East Asian Postgraduate Workshop on Soft Matter
Outline of this talk
Introduction
Mathematical formulations
- model for binary fluid system
- model for surfactant
Numerical schemes
Validations and Results
Conclusion and future works
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Introduction
1D / 2D Problem
surfactant (surface active agent)
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Example
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Model formulation
Cahn-Hilliard surface free energy *
φ : phase function (order parameter), 0<=φ<=1
ε : interface width scale
f(φ) is the bulk energy density
J. W. Cahn and J. E. Hilliard, “Free energy of a nonuniform system. I. Interfacial energy,” J. Chem. Phys 28, 258 (1958).
J. Kim, “Numerical simulations of phase separation dynamics in a water-oil-surfactant system,” J. Colloid & Int. Sci. 303, 272 (2006)
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Cahn-Hilliard equation
Mφ is the mobility
Mass conservation
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Property of Cahn-Hilliard energy
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Coupling binary-fluid & surfactant energy
α ~ O(ε2), potential term coefficient
θ ~ O(ε2), entropy term coefficient
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Coupling binary-fluid & surfactant system
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Surfactant equation
MΓ is the mobility
Mass conservation
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Simplified surfactant equation
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Property of Coupled energy
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Numerical scheme
Phase field Equation
Let L be Standard Laplacian discretization :
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Neumann Boundary Condition => cosine transform
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Surfactant Equation
Using similar manner in phase field solver
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Validations & Results
Convergence Test (1D)
domain: 0<= x <= 2π
initial: φ(x)= *cos(6x), Γ(x)= *exp(-(x-π)2/0.52)
parameters: ε2=α=θ=0.0001
test on T=1, dt~O(dx2)
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mesh size
phase function (φ)
L2-error order
surfactant (Γ)
L2-error order
128
2.4225x10-3
1.5623x10-3
256
5.5523x10-4
2.1253
3.4942x10-4
2.1606
512
1.0695x10-4
2.3760
7.3528x10-5
2.2485
comparison based on finest mesh (1024)
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Time evolution
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Mass & Energy
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Sample Test (2D)
domain: 0<= x <= 2π, 0<= y <= 2π
initial: φ(x,y)= *cos(6x)*cos(6y), Γ(x,y)= *exp(-((x-π)2 +(y-π)2 )/0.52)
parameters: ε=α=θ=0.04
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Time evolution
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Mass & Energy
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23. Conclusion and future works
We develop a phase field model for binary fluid-surfactant system.
We propose a simple numerical scheme for our model, which keeping the mass conservation and energy decreasing properties.
Challenge : Coupled with fluid dynamics (i.e. Navier-Stokes systems)
Other possible formulations for binary fluid-surfactant system ?
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Undergoing Work
Incompressible Navier-Stokes Equation with binary fluid-surfactant system.
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Binary-fluid & surfactant system under flow field
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Thanks for your attention
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