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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Ambrosio L., Fusco N., Pallara D. (2000) Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, New York
Ambrosio L., De Lellis C., Mantegazza C. (1999) Line energies for gradient vector fields in the plane. Calc. Var. Partial Differential Equations 9, 327–355
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, 1–17
Baldo S. (1990) Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids. Ann. Inst. H. Poincaré Anal. Non Linéaire 7, 67–90
Barroso A.C., Fonseca I. (1994) Anisotropic singular perturbations–the vectorial case. Proc. Roy. Soc. Edinburgh Sect. A 124, 527–571
Bouchittè G. (1990) Singular perturbations of variational problems arising from a two-phase transition model. Appl. Math. Optim. 21, 289–314
Bronsard L., Kohn R.V. (1990) On the slowness of phase boundary motion in one space dimension. Commun. Pure Appl. Math. 43, 983–997
Carr J., Pego R. (1989) Metastable patterns in solutions of u t = ε2 u xx − f(u). Commun. Pure Appl. Math. 42, 523–576
Conti S., Fonseca I., Leoni G. (2002) A Γ-convergence result for the two-gradient theory of phase transitions. Commun. Pure Appl. Math. 55, 857–936
Conti S., Schweizer B. (2006) A sharp-interface limit for the geometrically linear two-well problem in two dimensions. Arch. Ration. Mech. Anal. 179, 413–452
Dal Maso G. (1993) An Introduction to Γ-Convergence. Progress in Nonlinear Differential Equations and their Applications, 8. Birkhäuser Boston
Fonseca I., Mantegazza C. (2000) Second order singular perturbation models for phase transitions. SIAM J. Math. Anal. 31, 1121–1143
Fonseca I., Tartar L. (1989) The gradient theory of phase transitions for systems with two potential wells. Proc. Roy. Soc. Edinburgh Sect. A 111, 89–102
Grant C.P. (1995) Slow motion in one-dimensional Cahn-Morral systems. SIAM J. Math. Anal. 26, 21–34
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 York
Gurtin M.E., Matano H. (1988) On the structure of equilibrium phase transitions within the gradient theory of fluids. Quart. Appl. Math. 46, 301–317
Jin W., Kohn R.V. (2000) Singular perturbation and the energy of folds. J. Nonlinear Sci. 10, 355–390
Kraynik A.M. (2003) Foam structure: From soap froth to solid foams. MRS Bulletin 28, 275–278
Kohn R.V., Sternberg P. (1989) Local minimisers and singular perturbations. Proc. Roy. Soc. Edinburgh Sect. A 111, 69–84
Modica L., Mortola S. Un esempio di Γ−-convergenza. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 14, 285–299 (1977)
Modica L. (1987) The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 98, 123–142
Owen N.C., Sternberg P. (1991) Nonconvex variational problems with anisotropic perturbations. Nonlinear Anal. 16, 705–719
Perkins, R., Sekerka, R., Warren, J., Langer, S. Private communication
Rockafellar R.T. (1970) Convex Analysis. Princeton Mathematical Series, no. 28, Princeton University Press, Princeton
Sternberg P. (1988) The effect of a singular perturbation on nonconvex variational problems. Arch. Ration. Mech. Anal. 101, 209–260
Sternberg P. Vector-valued local minimizers of nonconvex variational problems. Current directions in nonlinear partial differential equations. Rocky Mountain J. Math. 21, 799–807
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Fonseca, I., Morini, M. & Slastikov, V. Surfactants in Foam Stability: A Phase-Field Model. Arch Rational Mech Anal 183, 411–456 (2007). https://doi.org/10.1007/s00205-006-0012-x
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DOI: https://doi.org/10.1007/s00205-006-0012-x