Archive for Rational Mechanics and Analysis

, Volume 183, Issue 3, pp 411–456 | Cite as

Surfactants in Foam Stability: A Phase-Field Model

  • Irene Fonseca
  • Massimiliano Morini
  • Valeriy Slastikov
Article

Abstract

The role of surfactants in stabilizing the formation of bubbles in foams is studied using a phase-field model. The analysis is centered on a van der Walls–Cahn– Hilliard-type energy with an added term which accounts for the interplay between the presence of a surfactant density and the creation of interfaces. In particular, it is concluded that the surfactant segregates to the interfaces, and that the prescription of the distribution of surfactant will dictate the locus of interfaces, which is in agreement with the experimental results.

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References

  1. 1.
    Ambrosio L., Fusco N., Pallara D. (2000) Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, New YorkMATHGoogle Scholar
  2. 2.
    Ambrosio L., De Lellis C., Mantegazza C. (1999) Line energies for gradient vector fields in the plane. Calc. Var. Partial Differential Equations 9, 327–355CrossRefMathSciNetGoogle Scholar
  3. 3.
    Aviles P., Giga Y. (1999) On lower semicontinuity of a defect energy obtained by a singular limit of the Ginzburg-Landau type energy for gradient fields. Proc. Roy. Soc. Edinburgh Sect. A 129, 1–17MathSciNetGoogle Scholar
  4. 4.
    Baldo S. (1990) Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids. Ann. Inst. H. Poincaré Anal. Non Linéaire 7, 67–90MathSciNetGoogle Scholar
  5. 5.
    Barroso A.C., Fonseca I. (1994) Anisotropic singular perturbations–the vectorial case. Proc. Roy. Soc. Edinburgh Sect. A 124, 527–571MathSciNetGoogle Scholar
  6. 6.
    Bouchittè G. (1990) Singular perturbations of variational problems arising from a two-phase transition model. Appl. Math. Optim. 21, 289–314CrossRefMathSciNetGoogle Scholar
  7. 7.
    Bronsard L., Kohn R.V. (1990) On the slowness of phase boundary motion in one space dimension. Commun. Pure Appl. Math. 43, 983–997MathSciNetGoogle Scholar
  8. 8.
    Carr J., Pego R. (1989) Metastable patterns in solutions of u t = ε2 u xxf(u). Commun. Pure Appl. Math. 42, 523–576MathSciNetGoogle Scholar
  9. 9.
    Conti S., Fonseca I., Leoni G. (2002) A Γ-convergence result for the two-gradient theory of phase transitions. Commun. Pure Appl. Math. 55, 857–936CrossRefMathSciNetGoogle Scholar
  10. 10.
    Conti S., Schweizer B. (2006) A sharp-interface limit for the geometrically linear two-well problem in two dimensions. Arch. Ration. Mech. Anal. 179, 413–452CrossRefMathSciNetGoogle Scholar
  11. 11.
    Dal Maso G. (1993) An Introduction to Γ-Convergence. Progress in Nonlinear Differential Equations and their Applications, 8. Birkhäuser BostonGoogle Scholar
  12. 12.
    Fonseca I., Mantegazza C. (2000) Second order singular perturbation models for phase transitions. SIAM J. Math. Anal. 31, 1121–1143CrossRefMathSciNetGoogle Scholar
  13. 13.
    Fonseca I., Tartar L. (1989) The gradient theory of phase transitions for systems with two potential wells. Proc. Roy. Soc. Edinburgh Sect. A 111, 89–102MathSciNetGoogle Scholar
  14. 14.
    Grant C.P. (1995) Slow motion in one-dimensional Cahn-Morral systems. SIAM J. Math. Anal. 26, 21–34CrossRefMathSciNetGoogle Scholar
  15. 15.
    Gurtin M.E. (1987) Some results and conjectures in the gradient theory of phase transitions. Metastability and Incompletely Posed Problems. IMA Volumes in Mathematics and Its Applications, 3, pp. 135–146. Springer, New YorkGoogle Scholar
  16. 16.
    Gurtin M.E., Matano H. (1988) On the structure of equilibrium phase transitions within the gradient theory of fluids. Quart. Appl. Math. 46, 301–317MathSciNetGoogle Scholar
  17. 17.
    Jin W., Kohn R.V. (2000) Singular perturbation and the energy of folds. J. Nonlinear Sci. 10, 355–390CrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Kraynik A.M. (2003) Foam structure: From soap froth to solid foams. MRS Bulletin 28, 275–278Google Scholar
  19. 19.
    Kohn R.V., Sternberg P. (1989) Local minimisers and singular perturbations. Proc. Roy. Soc. Edinburgh Sect. A 111, 69–84MathSciNetGoogle Scholar
  20. 20.
    Modica L., Mortola S. Un esempio di Γ-convergenza. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 14, 285–299 (1977)Google Scholar
  21. 21.
    Modica L. (1987) The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 98, 123–142CrossRefMathSciNetGoogle Scholar
  22. 22.
    Owen N.C., Sternberg P. (1991) Nonconvex variational problems with anisotropic perturbations. Nonlinear Anal. 16, 705–719CrossRefMathSciNetGoogle Scholar
  23. 23.
    Perkins, R., Sekerka, R., Warren, J., Langer, S. Private communicationGoogle Scholar
  24. 24.
    Rockafellar R.T. (1970) Convex Analysis. Princeton Mathematical Series, no. 28, Princeton University Press, PrincetonMATHGoogle Scholar
  25. 25.
    Sternberg P. (1988) The effect of a singular perturbation on nonconvex variational problems. Arch. Ration. Mech. Anal. 101, 209–260CrossRefMathSciNetGoogle Scholar
  26. 26.
    Sternberg P. Vector-valued local minimizers of nonconvex variational problems. Current directions in nonlinear partial differential equations. Rocky Mountain J. Math. 21, 799–807Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Irene Fonseca
    • 1
  • Massimiliano Morini
    • 2
  • Valeriy Slastikov
    • 3
  1. 1.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA
  2. 2.Classe di MatematicaSISSAMiramare Grignano (Trieste)Italy
  3. 3.Mathematics InstituteUniversity of WarwickCoventryUK

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