Theoretical and Mathematical Physics

, Volume 99, Issue 3, pp 681–686 | Cite as

Nonlinear evolution of directional solidification patterns

  • K. Kassner
  • C. Misbah
  • A. Valance
  • H. Müller-Krumbhaar
Article
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Abstract

We have studied directional solidification theoretically for high velocities, where a time-dependent description is essential. The model equations can be simplified into an asymptotically valid strongly nonlinear equation. This quasilocal approximation preserves the essential dynamic features of the system.

Analyzing the dynamic evolution of an isotrope-nematic interface, we find, besides the usual parity-breaking solution (PB), avacillating-breathing (VB) mode, which is associated with spatial period-doubling. Both the PB and VB modes can be understood analytically, the former as the consequence of aq-2q interaction, the latter on the basis of a perturbative approach. As the relevant system parameter, a renormalized thermal gradient, is decreased, the VB mode becomes unstable with respect to parity breaking and acquires a lateral drift velocity. The interface motion is thenquasiperiodic. Further decrease of the thermal gradient drives the interface into achaotic state.

We suggest that the quasiperiodicity scenario is generic for systems in which both an oscillatory and a parity-breaking instability exist. This expectation is supported by a study of amplitude equations for the same system, in which only the two most important modes have been retained.

Keywords

High Velocity System Parameter Nonlinear Equation Dynamic Feature Thermal Gradient 

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References

  1. [1]
    A. J. Simon, J. Bechhoefer, and A. Libchaber, Phys. Rev. Lett.,63, 2574 (1988).Google Scholar
  2. [2]
    A. J. Simon and A. Libchaber, Phys. Rev. A,41, 7090 (1989).Google Scholar
  3. [3]
    J.-M. Flesselles, A. J. Simon, and A. J. Libchaber, Adv. Phys.,40 1 (1991).Google Scholar
  4. [4]
    K. Kassner, Pattern Formation in Diffusion-Limited Crystal Growth. World Scientific, Singapore, to appear in summer 1994.Google Scholar
  5. [5]
    J. Stefan, Ann. Phys. Chem.,42 269 (1891); see alsoAnn. Phys.,278.Google Scholar
  6. [6]
    L. Rubinstein, The Stefan Problem. American Mathematical Society, Providence, R. I., 1971. Translated by A. D. Solomon.Google Scholar
  7. [7]
    M. Zimmermann, A. Karma, and M. Carrard, Phys. Rev. B,42 833 (1990).Google Scholar
  8. [8]
    J. S. Langer, Lectures in the theory of pattern formation. In J. Souletie, J. Vannimenus, and R. Stora, editors, Chance and Matter, page 629, Amsterdam, 1987. Elsevier.Google Scholar
  9. [9]
    G. I. Sivashinsky, Acta Astronautica,4 1177 (1977).Google Scholar
  10. [10]
    K. Kassner, C. Misbah, and H. Müller-Krumbhaar, Phys. Rev. Lett.,67 1551 (1991).Google Scholar
  11. [11]
    K. Kassner, C. Misbah, H. Müller-Krumbhaar, and A. Valance,Directional solidification at high speed. I.Secondary instabilities. Submitted to Phys. Rev. E.Google Scholar
  12. [12]
    H. Jamgotchian, R. Trivedi, and B. Billia, Phys. Rev. E,47, 4313 (1993).Google Scholar
  13. [13]
    A. Valance, K. Kassner, and C. Misbah, Phys. Rev. Lett.,69 1544 (1992).Google Scholar
  14. [14]
    K. Kassner, C. Misbah, H. Müller-Krumbhaar, and A. Valance,Directional solidification at high speed. II.Transition to chaos. Submitted to Phys. Rev. E.Google Scholar

Copyright information

© Plenum Publishing Corporation 1994

Authors and Affiliations

  • K. Kassner
  • C. Misbah
  • A. Valance
  • H. Müller-Krumbhaar

There are no affiliations available

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