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A phase field model for binary fluid-surfactant system

This talk presents a phase field model for binary fluid-surfactant systems, discussing mathematical formulations and numerical schemes for the model. It covers validations, results, conclusions, and future works. Challenges include coupling with fluid dynamics. Ongoing work involves Incompressible Navier-Stokes Equation with binary fluid-surfactant systems under flow fields.

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A phase field model for binary fluid-surfactant system

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  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. Outline of this talk • Introduction • Mathematical formulations - model for binary fluid system - model for surfactant • Numerical schemes • Validations and Results • Conclusion and future works East Asian Postgraduate Workshop on Soft Matter

  3. Introduction • 1D / 2D Problem • surfactant (surface active agent) East Asian Postgraduate Workshop on Soft Matter

  4. Example East Asian Postgraduate Workshop on Soft Matter

  5. 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) East Asian Postgraduate Workshop on Soft Matter

  6. Cahn-Hilliard equation • Mφ is the mobility • Mass conservation East Asian Postgraduate Workshop on Soft Matter

  7. Property of Cahn-Hilliard energy East Asian Postgraduate Workshop on Soft Matter

  8. Coupling binary-fluid & surfactant energy • α ~ O(ε2), potential term coefficient • θ ~ O(ε2), entropy term coefficient East Asian Postgraduate Workshop on Soft Matter

  9. Coupling binary-fluid & surfactant system East Asian Postgraduate Workshop on Soft Matter

  10. Surfactant equation • MΓ is the mobility • Mass conservation East Asian Postgraduate Workshop on Soft Matter

  11. Simplified surfactant equation East Asian Postgraduate Workshop on Soft Matter

  12. Property of Coupled energy East Asian Postgraduate Workshop on Soft Matter

  13. Numerical scheme • Phase field Equation • Let L be Standard Laplacian discretization : East Asian Postgraduate Workshop on Soft Matter

  14. Neumann Boundary Condition => cosine transform East Asian Postgraduate Workshop on Soft Matter

  15. Surfactant Equation • Using similar manner in phase field solver East Asian Postgraduate Workshop on Soft Matter

  16. Validations & Results • Convergence Test (1D) domain: 0<= x <= 2π initial: φ(x)=0.3 + 0.01*cos(6x), Γ(x)=0.1 + 0.03*exp(-(x-π)2/0.52) parameters: ε2=α=θ=0.0001 test on T=1, dt~O(dx2) East Asian Postgraduate Workshop on Soft Matter

  17. East Asian Postgraduate Workshop on Soft Matter

  18. Time evolution East Asian Postgraduate Workshop on Soft Matter

  19. Mass & Energy East Asian Postgraduate Workshop on Soft Matter

  20. Sample Test (2D) domain: 0<= x <= 2π, 0<= y <= 2π initial: φ(x,y)=0.3 + 0.01*cos(6x)*cos(6y), Γ(x,y)=0.1 + 0.03*exp(-((x-π)2 +(y-π)2 )/0.52) parameters: ε=α=θ=0.04 East Asian Postgraduate Workshop on Soft Matter

  21. Time evolution East Asian Postgraduate Workshop on Soft Matter

  22. Mass & Energy East Asian Postgraduate Workshop on Soft Matter

  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 ? East Asian Postgraduate Workshop on Soft Matter

  24. Undergoing Work • Incompressible Navier-Stokes Equation with binary fluid-surfactant system. East Asian Postgraduate Workshop on Soft Matter

  25. Binary-fluid & surfactant system under flow field East Asian Postgraduate Workshop on Soft Matter

  26. Thanks for your attention East Asian Postgraduate Workshop on Soft Matter

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