Spinodal decomposition

A phase transformation in which both phases have equivalent symmetry but differ only in composition is well-known as spinodal decomposition (see Fig. 1). This transformation has been theoretically described by Cahn and Hilliard [1, 2].

Figure 1. Three-dimensional evolution of spinodally decomposed liquid. The modelling is provided using a "hyperbolic" model for phase separation [3, 4]. Snapshots for the decomposed (blue) phase are shown for various computational times.

Few advancements were made for strongly non-equilibrium phase separation. Binder et al. [5] generalized the linearized Cahn-Hilliard theory to the case of the existence of a slowly relaxing variable. Their calculations show that the instability of the system is not of the standard diffusive type, but rather it is controlled by the relaxation of the slow structural variable.

Recently, Cahn-Hillard theory has been modified by taking into account the relaxation of diffusion flux to its local steady state [3, 4]. The flux is considered as an independent thermodynamic variable in consistency with the extended irreversible thermodynamics. As a result, a partial differential equation of a hyperbolic type for phase separation with diffusion has been derived that can be called "a hyperbolic model for spinodal decomposition". Theoretically, this model can predict spinodal decomposition for short periods of time, large characteristic velocities of the process, large concentration gradients, or deep supercoolings at the earliest stages of decomposition. A comparative analysis for both Cahn-Hilliard's parabolic model and the hyperbolic model (modified Cahn-Hilliard) of spinodal decomposition are given in Refs. [6-9]. As a test for the hyperbolic model, its predictions are compared with experimental data in Fig. 2.

Figure 2. Dependence of the amplification rate ω/k2 upon square of the wave number k2 (solid line) given by the hyperbolic model [8] in comparison with scattering data of visible light (points) as described by Ref. [10]. Experimental points were obtained on phase-separated SiO2-12 wt.%Na2O glass at the temperature 803 K.

  1. J.W. Cahn and J.E. Hilliard, J. Chem. Phys. 28, 258 (1958).
  2. J.W. Cahn, Acta Metall. 9, 795 (1961).
  3. P. Galenko, Phys. Lett. A 287, 190 (2001).
  4. P. Galenko and D. Jou, Phys. Rev. E 71, 046125-1-13 (2005).
  5. K. Binder, H.L. Frisch and J. J├Ąckle, J. Chem. Phys. 85, 1505 (1986).
  6. P. Galenko and V. Lebedev, Analysis of dispersion relation in spinodal decomposition of a binary system Philosophical Magazine Letters 87(11) 821-827 (2007).
  7. P. Galenko and V. Lebedev, Experimental test for hyperbolic model of spinodal decomposition in a binary system Letters to Journal of Experimental and Theoretical Physics 86(7) 527-529 (2007).
  8. P. Galenko and V. Lebedev, Non-equilibrium effects in in spinodal decomposition of a binary system Physics Letters A 327(7) 985-989 (2008).
  9. P. Galenko and V. Lebedev, Local nonequilibrium effect on spinodal decomposition in a binary system The International Journal of Thermodynamics (2008) 11(1) 21-28 (2008).
  10. N.S. Andreev, G.G. Boiko and N.A. Bokov, J. Non-Cryst. Solids 5, 41 (1970).
  11. D. Kharchenko, I. Lysenko and P. K. Galenko, Fluctuation effects on pattern selection in the hyperbolic model of phase decomposition. In: Stochastic Differential Equations. Editor: N. Halidias (Nova Science, New York, 2011), pp. 97-127.
  12. P.K. Galenko, D. Kharchenko, I. Lysenko, Stochastic generalization for a hyperbolic model of spinodal decomposition. Physica A, 389(17) 3443-3455 (2009).
  13. P. Galenko and D. Jou, Kinetic contribution to the fast spinodal decomposition controlled by diffusion. Physica A, 388(15-16) 3113-3123 (2009).
  14. N. Lecoq, H. Zapolsky, P. Galenko, Evolution of the structure factor in a hyperbolic model of spinodal decomposition. The European Physical Journal - Special Topics, 177 165-175 (2009).
  15. D. Kharchenko, P. Galenko, and V. Lebedev, Deterministic and stochastic phenomenological models in spinodal decomposition of a binary system. Progress in Physics of Metals [Uspekhi Fiziki Metallov], 10 27-102 (2009).

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