Evolution of the structure factor in a hyperbolic model of spinodal decomposition

  • N. Lecoq
  • H. Zapolsky
  • P. GalenkoEmail author


We consider the modification of the Cahn-Hilliard equation when a time delay process through a memory function is taken into account. The memory effects are seen to affect the dynamics of phase transition at short times. The process of fast spinodal decomposition associated with a conserved order parameter - concentration is studied numerically. Details of a semi-implicit numerical scheme used to simulate the kinetics of spinodal decomposition and evolution of the structure factor are discussed. Analysis of the modeled structure factor predicted by a hyperbolic model of spinodal decomposition is presented in comparison with the parabolic model of Cahn and Hilliard. It is shown that during initial periods of decomposition the structure factor exhibits wave behavior. Analytical treatments explain such behavior by existence of damped oscillations in structure factor at earliest stages of phase separation and at large values of the wave-number. These oscillations disappear gradually in time and the hyperbolic evolution approaches the pure dissipative parabolic evolution of spinodal decomposition.


Structure Factor European Physical Journal Special Topic Spinodal Decomposition Free Energy Density Hyperbolic Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    J.W. Cahn, J.E. Hilliard, J. Chem. Phys. 28, 258 (1958)CrossRefADSGoogle Scholar
  2. 2.
    H.E. Cook, Acta Metall. 18, 297 (1970)CrossRefGoogle Scholar
  3. 3.
    J.S. Langer, M. Bar-on, H.D. Miller, Phys. Rev. A 11, 1417 (1975)CrossRefADSGoogle Scholar
  4. 4.
    N.S. Andreev, G.G. Boiko, N.A. Bokov, J. Non-Cryst. Solids 5, 41 (1970)CrossRefADSGoogle Scholar
  5. 5.
    P. Guyot, J.P. Simon, in: Proceedings of the international conference on solid-solid phase transformations, edited by H.I. Aaronson, D.E. Laughlin, R.F. Sekerka, C.M. Wayman (TMS, Warrendale, 1983), p. 325Google Scholar
  6. 6.
    T. Ujihara, K. Osamura, Acta Metall. 48, 1629 (2000)Google Scholar
  7. 7.
    R. Wagner, R. Kaupmann, P.W. Voorhees, in: Phase transformations in materials, edited by G. Kostorz (Wiley, Weinheim, 2001), p. 309CrossRefGoogle Scholar
  8. 8.
    K. Binder, P. Fratzl, in: Phase transformations in materials, edited by G. Kostorz (Wiley, Weinheim, 2001), p. 409CrossRefGoogle Scholar
  9. 9.
    V.P. Skripov, A.V. Skripov, Usp. Fiz. Nauk 128, 193 (1979) [English Translation: Sov. Phys. Usp. 22, 389 (1979)]Google Scholar
  10. 10.
    A.E. Bailey, Early stage spinodal decomposition, PhD thesis (University of California, Santa Barbara, California, 1993)Google Scholar
  11. 11.
    J.P. Donley, Adiabatic spinodal decomposition, Ph.D. thesis (University of California, Santa Barbara, California, 1991)Google Scholar
  12. 12.
    J.P. Donley, J.S. Langer, Phys. Rev. Lett. 72, 1573 (1993)CrossRefADSGoogle Scholar
  13. 13.
    P. Galenko, Phys. Lett. A 287, 190 (2001)CrossRefADSGoogle Scholar
  14. 14.
    P. Galenko, V. Lebedev, Phil. Mag. Lett. 87, 821 (2007)CrossRefADSGoogle Scholar
  15. 15.
    P. Galenko, V. Lebedev, Phys. Lett. A 372, 985 (2008)CrossRefADSGoogle Scholar
  16. 16.
    P. Galenko, D. Jou, Phys. Rev. E 71, 046125 (2005)CrossRefADSGoogle Scholar
  17. 17.
    J.W. Cahn, Acta Metall. 9, 795 (1961)CrossRefGoogle Scholar
  18. 18.
    A.G. Khachaturyan, Theory of structural transformations in solids (Wiley, New York, 1983)Google Scholar
  19. 19.
    R.L.H. Essery, R.C. Ball, Europhys. Lett. 16, 379 (1991)CrossRefADSGoogle Scholar
  20. 20.
    D. Jou, J. Casas-Vazquez, M. Criado-Sancho, Thermodynamics of fluids under flow (Springer, Berlin, 2000)Google Scholar
  21. 21.
    L.Q. Chen, J. Shen, Comput. Phys. Comm. 108, 147 (1998)zbMATHCrossRefADSGoogle Scholar
  22. 22.
    See, for details, the web-site: www.crihan.frGoogle Scholar
  23. 23.
    D. Beysens, in: Materials Sciences in Space, edited by B. Feuerbacher, H. Hamacher, R.J. Naumann (Springer, Berlin, 1986), p. 191Google Scholar
  24. 24.
    D. Jou, P. Galenko, Physica A 366, 149 (2006)CrossRefADSGoogle Scholar
  25. 25.
    P. Galenko, D. Jou, Physica A 388, 3113 (2009)CrossRefMathSciNetADSGoogle Scholar

Copyright information

© EDP Sciences and Springer 2009

Authors and Affiliations

  1. 1.Groupe de Physique des Matériaux, Université de RouenSaint-Étienne du RouvrayFrance
  2. 2.Institut für Materialphysik im WeltraumDeutsches Zentrum für Luft- und Raumfahrt (DLR)KölnGermany

Personalised recommendations