Topological Defects and Phase Ordering Dynamics

  • A. J. Bray
Part of the NATO ASI Series book series (NSSB, volume 349)

Abstract

The theory of phase ordering dynamics — the growth of order through domain coarsening when a system is quenched from the homogeneous phase into a broken-symmetry phase — will be reviewed. Interest will focus on the scaling regime that develops at long times after the quench. The growth laws that describe the time-dependence of characteristic length scales will be determined, and the forms of the associated scaling functions discussed. Particular attention will be paid to systems described by more complicated order parameters than the simple scalars usually considered, e.g. vector and tensor fields. The latter are needed, for example, to describe phase ordering in nematic liquid crystals, on which there have been a number of recent experiments. The study of topological defects (domain walls, strings and monopoles) provides a unifying framework for discussing coarsening in these different systems.

Keywords

Vortex Manifold Autocorrelation Topo 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. J. Bray, Advances in Physics, to appear.Google Scholar
  2. [2]
    A. J. Bray in Phase Transitions in Systems with Competing Energy Scales, edited by T. Riste and D. Sherrington (Kluwer Academic, 1993).Google Scholar
  3. [3]
    J. S. Langer, in Solids Far From Equilibrium, ed. C. Godrèche (Cambridge, Cambridge, 1992).Google Scholar
  4. [4]
    P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49 435 (1977).ADSCrossRefGoogle Scholar
  5. [5]
    A. J. Bray, Phys. Rev. Lett. 62 2841 (1989).ADSCrossRefGoogle Scholar
  6. [6]
    A. J. Bray, Phys. Rev. B 41 6724 (1990).ADSCrossRefGoogle Scholar
  7. [7]
    K. Binder and D. Stauffer, Phys. Rev. Lett. 33 1006 (1974).ADSCrossRefGoogle Scholar
  8. [8]
    J. Marro, J. L. Lebowitz and M. H. Kalos, Phys. Rev. Lett. 43 282 (1979).ADSCrossRefGoogle Scholar
  9. [9]
    H. Furukawa, Prog. Theor. Phys. 59 1072 (1978).ADSCrossRefGoogle Scholar
  10. [10]
    H. Furukawa, Phys. Rev. Lett. 43 136 (1979).ADSCrossRefGoogle Scholar
  11. [11]
    H. Furukawa, J. Phys. Soc. Jpn. 58 216 (1989).ADSCrossRefGoogle Scholar
  12. [12]
    H. Furukawa, Phys. Rev. B 40 2341 (1989).ADSGoogle Scholar
  13. [13]
    G. Porod, Kolloid Z. bf 124, 83 (1951); 125 51 (1952).Google Scholar
  14. [14]
    P. Debye, H. R. Anderson and H. Brumberger, J. Appl. Phys. 28 679 (1957); G. Porod, in Small-Angle X-Ray Scattering, edited by O. Glatter and O. Kratky (Academic, New York, 1982).Google Scholar
  15. [15]
    S. M. Allen and J. W. Cahn, Acta. Metall. 27 1085 (1979).CrossRefGoogle Scholar
  16. [16]
    For a general discussion of topological defects, see e.g. M. Kléman, Points,Lines and Walls, in Liquid Crystals, Magnetic Systems, and Various Ordered Media (Wiley, New York, 1983).Google Scholar
  17. [17]
    See, for example, S. Ostlund, Phys. Rev. B 24 485 (1981).MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    A. D. Rutenberg and A. J. Bray, submitted to Phys. Rev. E.Google Scholar
  19. [19]
    Strictly speaking, the large-distance cut-off for the energy dissipation is controlled by the defect velocity (see [18]), but in practice the same results are obtained by using the defect separation.Google Scholar
  20. [20]
    A. N. Pargellis, P. Finn, J. W. Goodby, P. Pannizza, B. Yurke and P. E. Cladis, Phys. Rev. A 46, 7765 (1992).ADSCrossRefGoogle Scholar
  21. [21]
    Yurke, Pargellis, Kovac and Huse, Phys. Rev. E 47, 1525 (1993).ADSCrossRefGoogle Scholar
  22. [22]
    I am grateful to N. Turok for a useful discussion of this approach.Google Scholar
  23. [23]
    A. J. Bray, Phys. Rev. E 47, 228 (1993).ADSCrossRefGoogle Scholar
  24. [24]
    A. J. Bray and K. Humayun, Phys. Rev. E 47, R9, (1993).ADSCrossRefGoogle Scholar
  25. [25]
    A. J. Bray and S. Puri, Phys. Rev. Lett. 67, 2670 (1991).ADSCrossRefGoogle Scholar
  26. [26]
    H. Toyoki, Phys. Rev. B 45 1965 (1992).ADSGoogle Scholar
  27. [27]
    Fong Liu and G. F. Mazenko, Phys. Rev. B 45 6989 (1992).ADSCrossRefGoogle Scholar
  28. [28]
    A. J. Bray and K. Humayun, J. Phys. A 25 2191 (1992).ADSCrossRefGoogle Scholar
  29. [29]
    R. E. Blundell and A. J. Bray, Phys. Rev. E 49 4925 (1994).ADSMATHCrossRefGoogle Scholar
  30. [30]
    N. Mason, A. N. Pargellis, and B. Yurke, Phys. Rev. Lett. 70 190 (1993); for earlier work on twisted nematics see H. Orihara and Y. Ishibashi, J. Phys. Soc. Jpn. 55 2151 (1986); T. Nagaya, H. Orihara and Y. Ishibashi, ibid. 56 1898 (1987); 56 3086 (1987); 59 377 (1990).Google Scholar
  31. [31]
    I. Chuang, R. Durrer, N. Turok and B. Yurke, Science 251 1336 (1991); I. Chuang, N. Turok and B. Yurke, Phys. Rev. Lett. 66 2472 (1991); B. Yurke, A. N. Pargellis, I. Chuang and N. Turok, Physica 178B 56 (1992).Google Scholar
  32. [32]
    A. P. Y. Wong, P. Wiltzius and B. Yurke, Phys. Rev. Lett. 68 3583 (1992).ADSCrossRefGoogle Scholar
  33. [33]
    A. P. Y. Wong, P. Wiltzius, R. G. Larson and B. Yurke, Phys. Rev. E 47, 2683 (1993).ADSCrossRefGoogle Scholar
  34. [34]
    See, e.g., P. G. de Gennes, The Physics of Liquid Crystals (Clarendon, Oxford, 1974).Google Scholar
  35. [35]
    S. Green, A. N. Pargellis and B. Yurke, unpublished.Google Scholar
  36. [36]
    A. J. Bray, S. Puri, R. E. Blundell and A. M. Somoza, Phys. Rev. E 47 2261 (1993).ADSCrossRefGoogle Scholar
  37. [37]
    R. E. Blundell and A. J. Bray, Phys. Rev. A 46 R6154 (1992).ADSCrossRefGoogle Scholar
  38. [38]
    T. Ohta, D. Jasnow and K. Kawasaki, Phys. Rev. Lett. 49 1223 (1982).ADSCrossRefGoogle Scholar
  39. [39]
    K. Kawasaki, M. C. Yalabik and J. D. Gunton, Phys. Rev. A 17, 455 (1978).ADSCrossRefGoogle Scholar
  40. [40]
    G. F. Mazenko, Phys. Rev. Lett. 63 1605 (1989).ADSCrossRefGoogle Scholar
  41. [41]
    G. F. Mazenko, Phys. Rev. B 42 4487 (1990).ADSCrossRefGoogle Scholar
  42. [42]
    G. F. Mazenko, Phys. Rev. B 43 5747 (1991).ADSCrossRefGoogle Scholar
  43. [43]
    A. J. Bray and K. Humayun, Phys. Rev. E 48 1609 (1993).ADSCrossRefGoogle Scholar
  44. [44]
    Y. Oono and S. Puri, Mod. Phys. Lett. B 2 861 (1988).ADSCrossRefGoogle Scholar
  45. [45]
    M. Suzuki, Prog. Theor. Phys. 56, 77 (1976); 56, 477 (1976).Google Scholar
  46. [46]
    S. Puri and C. Roland, Phys. Lett. A 151 500 (1990).ADSCrossRefGoogle Scholar
  47. [47]
    A. J. Bray, K. Humayun and T. J. Newman, Phys. Rev. B 43 3699 (1991).ADSCrossRefGoogle Scholar
  48. [48]
    K. Humayun and A. J. Bray, Phys. Rev. B 46 10594 (1992).ADSCrossRefGoogle Scholar
  49. [49]
    F. Liu and G. F. Mazenko, Phys. Rev. B 44 9185 (1991).ADSCrossRefGoogle Scholar
  50. [50]
    C. Yeung, A. Shinozaki and Y. Oono, preprint.Google Scholar
  51. [51]
    G. F. Mazenko, preprint.Google Scholar
  52. [52]
    Eq. (68) only fixes V(ͬ) for ͬ 2 ≤ 1. Note that, for T=0, ͬ 2(x,0) ≤ 1 everywhere implies ͬ 2(x,t) ≤ 1 everywhere, so ͬ(x, t) does not depend on the form of V(ͬ) for ͬ 2 > 1. Of course, for stability against thermal fluctuations the points ͬ = ±1 must be global minima of V(ͬ). Google Scholar
  53. [53]
    T. J. Newman and A. J. Bray, J. Phys. A 23 4491 (1990).ADSCrossRefGoogle Scholar
  54. [54]
    J. G. Kissner and A. J. Bray, J. Phys. A 26 1571 (1993). Note that this paper corrects an error in reference [53].Google Scholar
  55. [55]
    R. E. Blundell, A. J. Bray and S. Sattler, Phys. Rev. E 48 2476 (1993).ADSMATHCrossRefGoogle Scholar
  56. [56]
    Y. Oono and S. Puri, Phys. Rev. Lett. 58 836 (1987); Phys. Rev. A 38 434 (1988).Google Scholar
  57. [57]
    A. J. Bray and A. D. Rutenberg, Phys. Rev. E 49 R27 (1994).ADSMATHCrossRefGoogle Scholar
  58. [58]
    M. Mondello and N. Goldenfeld, Phys. Rev. A 45 657 (1992).ADSCrossRefGoogle Scholar
  59. [59]
    H. Toyoki, J. Phys. Soc. Jpn. 60 1433 (1991).CrossRefGoogle Scholar
  60. [60]
    A. J. Bray, Phys. Rev. E 47, 3191 (1993).ADSCrossRefGoogle Scholar
  61. [61]
    A. Onuki, Prog. Theor. Phys. 74, 1155 (1985).MathSciNetADSCrossRefGoogle Scholar
  62. [62]
    M. Siegert and M. Rao, Phys. Rev. Lett. 70 1956 (1993).ADSCrossRefGoogle Scholar
  63. [63]
    M. Siegert, private communication.Google Scholar
  64. [64]
    S. Puri and A. J. Bray, unpublished.Google Scholar
  65. [65]
    A. D. Rutenberg and A. J. Bray, submitted to Phys. Rev. Lett.Google Scholar
  66. [66]
    A. D. Rutenberg and A. J. Bray, Phys. Rev. E, in press.Google Scholar
  67. [67]
    Of course, this means that the present approach will not address systems with a potential-dependent growth law, e.g. d = n for n < 2.Google Scholar
  68. [68]
    C. Roland and M. Grant, Phys. Rev. B 41 4663 (1990).ADSCrossRefGoogle Scholar
  69. [69]
    C. Jeppesen and O. G. Mouritsen, Phys. Rev. B 47, 14724 (1993).ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1995

Authors and Affiliations

  • A. J. Bray
    • 1
  1. 1.Theoretical Physics Group, Department of Physics and AstronomyThe UniversityManchesterUK

Personalised recommendations