Issues »  Volume 10, Issue 1

Deterministic and Stochastic Dynamics in Spinodal Decomposition of a Binary System

D. O. Kharchenko$^{1}$, P. K. Galenko$^{2,3}$, V. G. Lebedev$^{4}$

$^1$Institute of Applied Physics, NAS of Ukraine, 58 Petropavlivska Str., 40000 Sumy, Ukraine
$^2$Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luftund Raumfahrt (DLR), 51170 Köln, Germany
$^3$Institut für Festkörperphysik, Ruhr-Universität Bochum, 44780 Bochum, Germany
$^4$Udmurt State University, 1 Universitetskaya Str., 426034 Izhevsk, Russia

Received: 06.03.2009; final version - 26.03.2009. Download: PDF

A model for diffusion and phase separation, which takes into account hyperbolic relaxation of the solute diffusion flux, is developed. Such a ‘hyperbolic model’ provides analysis of ‘hyperbolic evolution’ of patterns in spinodal decomposition in systems supercooled below critical temperature. Analytical results for the hyperbolic model of spinodal decomposition are summarized in comparison with outcomes of classic Cahn−Hilliard theory. Numeric modelling shows that the hyperbolic evolution leads to sharper boundary between two structures of a decomposed system in comparison with prediction of parabolic equation given by the theory of Cahn and Hilliard. Considering phase separation processes in stochastic systems with a field-dependent mobility and an internal multiplicative noise, we study dynamics of spinodal decomposition for parabolic and hyperbolic models separately. It is that the domain growth law is generalized when internal fluctuations are introduced into the model. A mean field approach is carried out in order to obtain the stationary probability, bifurcation and phase diagrams displaying re-entrant phase transitions. We relate our approach to entropy-driven phase-transitions theory.

Keywords: spinodal, diffusion, relaxation, model, liquid, structure factor, stochastic systems.

PACS: 05.40.-a, 05.45.-a, 05.70.Fh, 05.70.Ln, 64.60.-i, 64.75.Nx, 81.30.-t

DOI: https://doi.org/10.15407/ufm.10.01.027

Citation: D. O. Kharchenko, P. K. Galenko, and V. G. Lebedev, Deterministic and Stochastic Dynamics in Spinodal Decomposition of a Binary System, Usp. Fiz. Met., 10, No. 1: 27—102 (2009), doi: 10.15407/ufm.10.01.027

References (92)  
  1. H. Jinnai, T. Hashimoto, D. Lee, and S.-H. Chen, Macromolecules, 30: 130 (1997). Crossref
  2. F. Mallamace, N. Micali, and S. Trusso, J. Phys.: Condens. Matter, 8: A81 (1996). Crossref
  3. N. F. Bunkin, A. V. Lobeev, and G. A. Lyakhov, Usp. Fiz. Nauk, 167: 1070 (1997) (Physics–Uspekhi, 40: 1090 (1997)); Yu. I. Ustinovshchikov and B. E. Pushkarev, Usp. Fiz. Nauk, 176: 611 (2006) (Physics–Uspekhi, 49: 593 (2006)). Crossref
  4. D. Beysens, Materials Sciences in Space (Eds. B. Feuerbacher, H. Hamacher, and R. J. Naumann) (Berlin: Springer: 1986), p. 191. Crossref
  5. J. Mainville, Y. S. Yang, K. R. Elder, M. Sutton, K. F. Ludwig, and G. B. Sephenson, Phys. Rev. Lett., 78: 258 (1997). Crossref
  6. Yu. I. Ustinovshchikov and B. E. Pushkarev, Usp. Fiz. Nauk, 176: 611 (2006) (Physics–Uspekhi, 49: 593 (2006)). Crossref
  7. J. S. Langer and M. Baron, Annals of Physics, 78: 421 (1973); J. P. Donley and J. S. Langer, Phys. Rev. Lett., 71: 1573 (1993). Crossref
  8. G. F. Mazenko, Phys. Rev. B, 43: 5747 (1991). Crossref
  9. K. Binder, C. Billotet, and P. Mirold, Zeitschrift für Physik B–Condensed Matter, 30: 183 (1978); K. Binder, J. of Chemical Physics, 79: 6387 (1983). Crossref
  10. K. Binder and P. Fratzl, Phase Transformations in Materials (Ed. G. Kostorz) (Weinheim: Wiley: 2001), p. 409.
  11. V. L. Ginzburg and L. D. Landau, JETP, 20: 1064 (1950).
  12. J. W. Cahn and J. E. Hilliard, J. Chem. Phys., 28: 258 (1958). Crossref
  13. J. W. Cahn, Acta Metall., 9: 795 (1961). Crossref
  14. J. W. Cahn, Trans. Metall. Soc. AIME, 242: 166 (1968).
  15. V. P. Skripov and A. V. Skripov, Usp. Fiz. Nauk, 128: 193 (1979) (Sov. Phys. Usp., 22: 389 (1979)). Crossref
  16. S. Bastea and J. L. Lebovitz, Phys. Rev. E, 52: 3821, (1995). Crossref
  17. B. Liu, H. Zhang, and Y. Yang, J. Chem. Phys., 113: 719, (2000). Crossref
  18. K. Binder, H. L. Frish, and J. Jäckle, J. Chem Phys., 85: 1505 (1986). Crossref
  19. P. Galenko, Phys. Lett. A, 287: 190 (2001). Crossref
  20. P. Galenko and D. Jou, Phys. Rev. E, 71: 046125 (2005). Crossref
  21. D. Jou, J. Casas-Vazquez, and G. Lebon, Extended Irreversible Thermodynamics (Berlin: Springer: 2001). Crossref
  22. M. Grasselli and V. Pata, J. Evolution Equations, 4: 27 (2004). Crossref
  23. S. Gatti, M. Grasselli, A. Miranville, and V. Pata, J. Math. Anal. Appl., 312: 230 (2005). Crossref
  24. M. Grasselli, G. Schimperna, and S. Zelik, arXiv:0804.0988v1 [math.AP].
  25. A. Segatti, Math. Models Methods Appl. Sci., 17, No. 3: 411 (2007). Crossref
  26. P. Galenko and S. Sobolev, Phys. Rev. E, 55: 343 (1997). Crossref
  27. P. K. Galenko and D. A. Danilov, Phys. Lett. A, 235: 271 (1997). Crossref
  28. L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Addison Wesley: 1958).
  29. D. Jou and J. M. Rubi, Phys. Lett. A, 72: 78 (1979). Crossref
  30. D. Jou and C. Perez-Garsia, Physica A, 104: 3320 (1980). Crossref
  31. D. Jou, J. E. Llebot, and J. Casas-Vazquez, Physica A, 109: 208 (1981). Crossref
  32. H. Kleinert and S. V. Shabanov, Phys. Lett. A, 235: 105 (1997). Crossref
  33. M. Ibanes, J. Garcia-Ojalvo, R. Toral, and J. M. Sancho, Lecture Notes in Physics (Berlin: Springer-Verlag: 2000), vol. 557, p. 247.
  34. J. M. Sancho, J. Garcia-Ojalvo, and H. Guo, Physica D, 113: 331 (1998). Crossref
  35. C. Van der Broeck, Phys. Rev. Lett., 73: 3395 (1994). Crossref
  36. C. Van der Broeck, J. M. R. Parrondo, R. Toral, and R. Kawai, Phys. Rev. E, 55: 4084 (1997). Crossref
  37. M. Ibanes, J. Garcia-Ojalvo, R. Toral, and J. M. Sancho, Phys. Rev. E, 60: 3597 (1999). Crossref
  38. J. Buceta, M. Ibanes, J. M. Sancho, and K. Lindenberg, arXiv:cond-mat/0211181.
  39. P. K. Galenko and M. D. Krivilyov, Modelling Simul. Mater. Sci. Eng., 8: 67 (2000). Crossref
  40. D. Jou and P. Galenko, Physica A, 399: 149 (2006). Crossref
  41. P. Galenko, Phys. Rev. B, 65: 144103 (2002). Crossref
  42. J. P. Boon and S. Yip, Molecular Hydrodynamics (New York: McGraw Hill: 1970).
  43. H. B. Callen, Thermodynamics (New York: Wiley: 1960).
  44. R. Luzzi, A. R. Vasconcellos, J. Casas-Vazquez, and D. Jou, Physica A, 248: 111 (1998). Crossref
  45. P. Galenko and D. Jou, Proceedings of Joint European Thermodynamics Conference IX (Eds. B. Guy and D. Tondeur) (Saint-Etienne: Ecole Nationale Superieure des Mines: 2007), p. 75.
  46. J. Casas-Vazquez and D. Jou, Rep. Prog. Phys., 66: 1937 (2003). Crossref
  47. P. Galenko and V. Lebedev, Phil. Mag. Lett., 87: 821 (2007). Crossref
  48. P. Galenko and V. Lebedev, Phys. Lett. A, 372: 985 (2008). Crossref
  49. P. Galenko and D. Jou, Physica A, 388: 3113 (2009). Crossref
  50. A. Miranville and S. Zelik, Math. Methods Appl. Sci., 28: 709 (2005). Crossref
  51. C. Gal, Dyn. Partial Differ. Equ., 5: 39 (2008). Crossref
  52. P. Galenko and V. Lebedev, The International Journal of Thermodynamics, 11: 21 (2008).
  53. P. Galenko, Phys. Rev. B, 65: 144103 (2002). Crossref
  54. M. Hillert, Acta Metall., 9: 525 (1961). Crossref
  55. P. Galenko and V. Lebedev, Pis'ma Zh. Eksp. Teor. Phys., 86: 525 (1990) (JETP Lett., 86: 458 (2007)). Crossref
  56. N. S. Andreev, G. G. Boiko, and N. A. Bokov, J. Non-Cryst. Solids, 5: 41 (1970). Crossref
  57. N. S. Andreev and E. A. Porai-Koshits, Discuss. Faraday Soc., 50: 135 (1970). Crossref
  58. E. P. Feldman and L. I. Stefanovich, Zh. Eksp. Teor. Fiz., 98: 1695 (1990) (Sov. Phys. JETP, 71: 951 (1990)).
  59. H. E. Cook, J. Chem. Solids, 30: 2427 (1969); H. E. Cook, Acta. Metall., 18: 297 (1970). Crossref
  60. W. Horsthemke and R. Lefever, Noise-Induced Transitions (Berlin: Springer-Verlag: 1984).
  61. H. Risken, The Fokker–Planck Equation (Berlin: Springer-Verlag: 1989).
  62. C. W. Gradiner, Handbook of Stochastic Methods (Berlin: Springer-Verlag: 1985).
  63. A. I. Olemskoi and D. O. Kharchenko, Physica A, 293: 178 (2001). Crossref
  64. D. O. Kharchenko and S. V. Kohan, Eur. Phys. J. B, 29: 97 (2002). Crossref
  65. K. Wiesenfeld and F. Moss, Nature, 373: 33 (1995). Crossref
  66. L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys., 70: 223 (1998). Crossref
  67. J. Garcia-Ojalvo and J. M. Sancho, Noise in Spatially Extended Systems (New York: Springer-Verlag: 1999). Crossref
  68. D. O. Kharchenko, Physica A, 308: 113 (2002). Crossref
  69. D. O. Kharchenko and I. A. Knyaz', Eur. Phys. J. B, 32: 375 (2003). Crossref
  70. A. I. Olemskoi, D. O. Kharchenko, and I. A. Knyaz', Phys. Rev. E, 71: 041101 (2005). Crossref
  71. J. M. R. Parrondo, C. Van der Broeck, J. Buceta, and F. J. de la Rubia, Physica A, 224: 153 (1996). Crossref
  72. F. Sagués, J. M. Sancho, and J. Garcia-Ojalvo, Rev. Mod. Phys., 79: 829 (2007). Crossref
  73. B. Linder, J. Garcia-Ojalvo, A. Neiman, and L. Shimansky-Geier, Phys. Rep., 392: 321 (2004). Crossref
  74. Y. Tu, G. Grinstein, and M. A. Munoz, Phys. Rev. Lett., 78: 224 (1997). Crossref
  75. W. Genovese, M. A. Munoz, and J. M. Sancho, Phys. Rev. E, 57: R2495 (1998). Crossref
  76. M. Ibanes, J. Garcia-Ojalvo, R. Toral, and J. M. Sancho, Phys. Rev. Lett., 87: 020601 (2001). Crossref
  77. B. von Haeften, G. Izús, S. Mangioni, A. D. Sánchez, and H. S. Wio, Phys. Rev. E, 69: 021107 (2004). Crossref
  78. D. O. Kharchenko, Usp. Fiz. Met., 6: 1 (2005). Crossref
  79. E. A. Novikov, Sov. Phys. JETP, 20: 1290 (1965).
  80. J. Buceta and K. Lindenberg, Phys. Rev. E, 69: 011102 (2004). Crossref
  81. D. O. Kharchenko and A. V. Dvornichenko, Eur. Phys. J. B, 61: 95 (2008). Crossref
  82. D. O. Kharchenko and A. V. Dvornichenko, Physica A, 387: 5342 (2008). Crossref
  83. S. van Gemmert, G. T. Barkema, and S. Puri, arXiv:cond-mat/0501196.
  84. J. Garcia-Ojalvo, A. M. Lacasta, J. M. Sancho, and R. Toral, Europhys. Lett., 42: 125 (1998). Crossref
  85. L. Ramirez-Piscina, J. M. Sancho, and A. Hernandez-Machado, Phys. Rev. B, 48: 125 (1993). Crossref
  86. I. M. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids, 19: 35 (1961). Crossref
  87. C. L. Emmott and A. J. Bray, Phys. Rev. E, 59: 213 (1999). Crossref
  88. J. M. Sancho, A. Hernandez-Machado, L. Ramirez-Piscina, and A. M. Lacasta, Acta Phys. Polonica B, 24: 733 (1993).
  89. J. D. Gunton, M. San Miguel, and P. S. Sahni, Phase Transtions and Critical Phenomena (New York: Academic Press: 1983), vol. 8.
  90. D. O. Kharchenko, A. V. Dvornichenko, and I. A. Lysenko, Ukr. Jour. of Phys., 53, No. 9: 917 (2008).
  91. S. De Groot and P. Mazur, Non-Equilibrium Thermodynamics (Amsterdam: 1962).
  92. N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (Amsterdam: 1992).
Cited By (2)
  1. E. Davidoff, P.K. Galenko, D.M. Herlach, M. Kolbe et al., Acta Materialia 61, 1078 (2013).
  2. D. O. Kharchenko, I. O. Lysenko and V. O. Kharchenko, Usp. Fiz. Met. 13, 101 (2012).

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