Two-dimensional steady states in off-critical mixtures with high interface tension

  • Fathi A. M. Bribesh
  • Santiago Madruga
  • Uwe Thiele
Regular Article


We present 2D steady concentration profiles of confined layers of off-critical polymer blends. The layer rests on a solid substrate and has a flat free surface due to very high surface tension. The profiles correspond to non-linear steady solutions of the Cahn-Hilliard equation in a rectangular domain. The free polymer-gas interface is considered to be sharp, while the internal interfaces are diffuse. We explore the rich solution structure (including laterally structured layers, stratified layers, checkerboard structures, oblique states and droplets) as a function of mean concentration.


Free Surface European Physical Journal Special Topic Bifurcation Diagram Homogeneous State Rectangular Domain 
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  1. 1.
    M. Geoghegan, G. Krausch, Prog. Polym. Sci. 28, 261 (2003)CrossRefGoogle Scholar
  2. 2.
    J.W. Cahn, J.E. Hilliard, J. Chem. Phys. 28, 258 (1958)ADSCrossRefGoogle Scholar
  3. 3.
    J.W. Cahn, J. Chem. Phys. 42, 93 (1965)ADSCrossRefGoogle Scholar
  4. 4.
    D.M. Anderson, G.B. McFadden, A.A. Wheeler, Ann. Rev. Fluid Mech. 30, 139 (1998)ADSCrossRefMathSciNetGoogle Scholar
  5. 5.
    H.P. Fischer, P. Maass, W. Dieterich, Phys. Rev. Lett. 79, 893 (1997)ADSCrossRefGoogle Scholar
  6. 6.
    R. Kenzler, F. Eurich, P. Maass, B. Rinn, J. Schropp, E. Bohl, W. Dieterich, Comp. Phys. Comm. 133, 139 (2001)ADSCrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    O.A. Frolovskaya, A.A. Nepomnyashchy, A. Oron, A.A. Golovin, Phys. Fluids 20, 112105 (2008)ADSCrossRefGoogle Scholar
  8. 8.
    U. Thiele, S. Madruga, L. Frastia, Phys. Fluids 19, 122106 (2007)ADSCrossRefGoogle Scholar
  9. 9.
    S. Madruga, U. Thiele, Phys. Fluids 21, 062104 (2009)ADSCrossRefGoogle Scholar
  10. 10.
    F.A.M. Bribesh, L. Frastia, U. Thiele, Phys. Fluids 24, 062109 (2012)ADSCrossRefGoogle Scholar
  11. 11.
    A. Novick-Cohen, L.A. Segel, Physica D 10, 277 (1984)ADSCrossRefMathSciNetGoogle Scholar
  12. 12.
    A. Novick-Cohen, J. Stat. Phys. 38, 707 (1985)ADSCrossRefGoogle Scholar
  13. 13.
    P.C. Fife, B. Kielhofer, S. Maier-Paape, T. Wanner, Physica D 100, 257 (1997)ADSCrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    H. Kielhofer, Proc. Royal Soc. Edinburgh Section A-Mathematics 127, 1219 (1997)CrossRefMathSciNetGoogle Scholar
  15. 15.
    H. Kielhofer, Arch. Rational Mech. Anal. 155, 261 (2000)ADSCrossRefMathSciNetGoogle Scholar
  16. 16.
    T.J. Healey, H. Kielhöfer, Siam J. Math. Anal. 31, 1307 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    S. Maier-Paape, K. Mischaikow, T. Wanner, Inter. J. Bifur. Chaos 17, 1221 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    S. Maier-Paape, U. Miller, K. Mischaikow, T. Wanner, Rev. Mat. Comput. 21, 351 (2008)MathSciNetzbMATHGoogle Scholar
  19. 19.
    L. Frastia, U. Thiele, L.M. Pismen, Math. Model. Nat. Phenom. 6, 62 (2011)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© EDP Sciences and Springer 2013

Authors and Affiliations

  • Fathi A. M. Bribesh
    • 1
    • 2
  • Santiago Madruga
    • 3
  • Uwe Thiele
    • 1
  1. 1.Department of Mathematical SciencesLoughborough UniversityLoughborough, LeicestershireUK
  2. 2.Department of MathematicsZawia UniversityZawiaLibya
  3. 3.ETSI AeronáuticosUniversidad Politécnica de MadridMadridSpain

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