Skip to main content
SpringerLink
Log in

The Spatial-Temporal Lamellar Structures in the Confined Ideal Polymer Blends

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We consider the formation of self-organized spatial-temporal oscillating structures in symmetric binary polymer blends confined by two flat walls. An influence of these walls on the formation of the oscillating volume structures is studied. This phenomenon is simulated by an initial boundary-value problem for the conserved order parameter (or the concentration of one of the components in a binary mixture). Under a special choice the dynamical Puri-Binder’s boundary conditions these structures look like the lamellar structures. The behavior of the order parameter is described by the modified Cahn-Hilliard equation which models so-called the non-Fickian diffusion in the symmetric binary polymer blends. The nonlinear dynamical boundary conditions correspond to the process of adsorption-desorption on the walls. As a result, these nonlinear surface processes induce into the volume the spatial-temporal asymptotically periodic structures of relaxation, pre-turbulent or turbulent type with finite, countable or non-countable points of discontinuities on the period correspondingly. The frequency of oscillations on the period follows a power-law for the relaxation type and increases exponentially in the other cases.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Albano, E.V., Müller, M., Binder, K.: Physica A 279, 188 (2000)

    ADS  Google Scholar 

  2. Bartels, C.R., Crist, B., Fetters, L.J., Graeslly, W.W.: Macromolecules 19, 785 (1986)

    Article  ADS  Google Scholar 

  3. Binder, K.: In: Advances in Polymer Science, vol. 112, pp. 181–299. Springer, Berlin (2002)

    Google Scholar 

  4. Binder, K., Puri, S.: Z. Phys. B, Condens. Matter 86, 263 (1992)

    Article  ADS  Google Scholar 

  5. Binder, K., Puri, S.: J. Stat. Phys. 77(1/2), 145 (1994)

    ADS  Google Scholar 

  6. Binder, K., Puri, S.: Phys. Rev. E 49, 5359 (1994)

    Article  ADS  Google Scholar 

  7. Binder, K., Frisch, H.L., Stepanov, S.: J. Phys. II France, 1353–1378 (1997)

  8. Binder, K., Frisch, H.L.: Z. Phys. B 84, 403 (1991)

    Article  ADS  Google Scholar 

  9. Binder, K.: Acta Polym. 46, 204 (1995)

    Article  Google Scholar 

  10. Brandenburg, A., Käpylä, P.J., Mohammed, A.: Phys. Fluids 16(4), 1020–1027 (2004)

    Article  ADS  Google Scholar 

  11. Cahn, J.W.: Acta Metall. 9, 795 (1961)

    Article  Google Scholar 

  12. Cahn, J.W., Hilliard, J.E.: J. Chem. Phys. 28, 258 (1958)

    Article  ADS  Google Scholar 

  13. Doi, M., Edwards, S.F.: The Theory of Polymer Dynamics. Clarendon, Oxford (1986)

    Google Scholar 

  14. Diehl, H.W., Janssen, H.K.: Phys. Rev. A 45, 7145 (1992)

    Article  ADS  Google Scholar 

  15. Feudel, U., Kuznetsov, S., Pikovsky, A.: Strange Nonchaotic Attractors Dynamics Between Order and Chaos in Quasiperiodically Forced Systems. Word Scientific, Singapore (2006)

    Book  Google Scholar 

  16. Frisch, H.L., Binder, K.: Z. Phys. B, Condens. Matter 84, 403 (1991)

    Article  ADS  Google Scholar 

  17. Frisch, H.L., Binder, K., Puri, S.: Faraday Discuss. 112, 103 (1999)

    Article  ADS  Google Scholar 

  18. Galenko, P., Kharchenko, D., Lysenko, I.: Physica A 389, 3443–3455 (2010)

    Article  ADS  Google Scholar 

  19. Galenko, P.K., Lebedev, V.G.: JETP Lett. 86(7), 458–461 (2007)

    Article  ADS  Google Scholar 

  20. Galenko, P., Jou, D.: Phys. Rev. E. 71(4), 046125 (2005)

    Article  ADS  Google Scholar 

  21. Galenko, P.: Phys. Lett. A 287, 190 (2001)

    Article  ADS  Google Scholar 

  22. Galenko, P., Lebedev, V.: Philos. Mag. Lett. 87(11), 821 (2007)

    Article  ADS  Google Scholar 

  23. de Gennes, P.-G.: Rep. Prog. Phys. 187 (1969)

  24. de Gennes, P.-G.: Scaling Concepts in Polymer Physics. Cornell University, Ithaca (1979)

    Google Scholar 

  25. Graeslly, W.W.: J. Polym. Sci. Polym. Phys. Ed. 18(1), 27–34 (1980)

    Article  ADS  Google Scholar 

  26. Jou, D., Casas-Vazquez, J., Lebon, G.: Extended Irreversible Thermodynamics, 2nd edn. Springer, Berlin (1996)

    Book  MATH  Google Scholar 

  27. Julia, G.: Mémoire sur l’itération des fonctions rationnelles. J. Math. Pures Appl. 8, 47–245 (1918)

    Google Scholar 

  28. Krasnyuk, I.B., Taranets, R.M.: The spatiotemporal oscillations of order parameter for isothermal model of the surface-directed spinodal decomposition in bounded binary mixtures. Res. Lett. Phys. 2009, 4 (2009). Article ID 250203

    Google Scholar 

  29. Kuznetsov, S.P.: JETP Lett. 39(3), 133–136 (1984)

    ADS  Google Scholar 

  30. Leibler, L., Rubinshtein, M., Colby, P.H.: J. Phys. II France 3, 1581–1590 (1993)

    Article  Google Scholar 

  31. Lecoq, N., Zapolsky, H., Galenko, P.: Eur. Phys. J. Spec. Top. 117, 165–175 (2009)

    Article  Google Scholar 

  32. Puri, S.: J. Phys., Condens. Matter 17, R101 (2005)

    Article  ADS  Google Scholar 

  33. Romanenko, E.: Comput. Math. Appl. 36(10–12), 377–390 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  34. Schmidt, I., Binder, K.: Z. Phys. B 67, 369 (1987)

    Article  ADS  Google Scholar 

  35. Schmidt, I., Binder, K.: J. Phys. France 46, 1631–1644 (1985)

    Article  Google Scholar 

  36. Sharkovsky, A.N.: Ukr. Math. J. 1, 61 (1964)

    Google Scholar 

  37. Sharkovsky, A.N., Maistrenko, Yu.L., Romanenko, E.Yu.: Difference Equations and Their Applications. Mathematics and Its Applications, vol. 250, Kluwer Academic, Dordrecht (1993)

    Chapter  Google Scholar 

  38. Sharkovsky, A.N., Krasnuyk, I.B., Maistrenko, Yu.L., Dokl. Akad. Nauk USSR, Ser. A 12, 27 (1984)

    Google Scholar 

  39. Sharkovsky, A.N., Kolyada, S.F., Sivak, A.G., Fedorenko, V.V.: Dynamics of One Dimensional Maps. Mathematics and Its Applications, vol. 407, Kluwer Academic, Dordrecht (1997)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Donetsk Institute for Physics and Engineering named after O.O. Galkin of the NASU, 72, R. Luxemburg Str., 83114, Donetsk, Ukraine

    Igor B. Krasnyuk

  2. School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK

    Roman M. Taranets

Authors
  1. Igor B. Krasnyuk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Roman M. Taranets
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roman M. Taranets.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Krasnyuk, I.B., Taranets, R.M. The Spatial-Temporal Lamellar Structures in the Confined Ideal Polymer Blends. J Stat Phys 145, 1485–1498 (2011). https://doi.org/10.1007/s10955-011-0378-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-011-0378-5

Keywords

Search

Navigation