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Phys. Rev. A 43, 1727–1743 (1991)

[Issue 4 – 15 February 1991 ]

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Kinetic roughening of interfaces in driven systems

Bruno Grossmann, Hong Guo, and Martin Grant

Centre for the Physics of Materials and Department of Physics, McGill University, Rutherford Building, 3600 University Street, Montréal, Québec, Canada H3A 2T8

We study the dynamics of an interface driven far from equilibrium in three dimensions. First we derive the Kardar-Parisi-Zhang equation from the Langevin equation for a system with a nonconserved scalar order parameter, for the cases where an external field is present, and where an asymmetric coupling to a conserved variable exists. The relationship of the phenomena to self-organized critical phenomena is discussed. Numerical results are then obtained for three models that simulate the growth of an interface: the Kardar-Parisi-Zhang equation, a discrete version of that model, and a solid-on-solid model with asymmetric rates of evaporation and condensation. We first make a study of crossover effects. In particular, we propose a crossover scaling ansatz and verify it numerically. We then estimate the dynamical scaling exponents. Within the precision of our study, the Kardar-Parisi-Zhang equation and the solid-on-solid model have the same asymptotic behavior, indicating that the models share a dynamical universality class. Furthermore, the discrete models exhibit a kinetic roughening transition. We study this by monitoring the surface step energy, which shows a dramatic jump at a finite temperature for a given driving force. At the same temperature, a finite-size-scaling analysis of the bond-energy fluctuation shows a diverging peak.

©1991 The American Physical Society

URL: http://link.aps.org/abstract/PRA/v43/p1727
DOI: 10.1103/PhysRevA.43.1727
PACS: 05.70.Ln, 05.40.+j, 64.60.Ht, 68.35.Fx

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References

1. W. K. Burton, N. Cabrera and F. C. Frank, Philos Trans. R. Soc. London, Ser. A 243, 299 (1951).
2. For a recent review, see H. van Beijeren and I. Nolden, in Structure and Dynamics of Surfaces, edited by W. Schommers and P. von Blackenhagen, Vol. 43 of Topics in Current Physics (Springer, Berlin, 1987).
3. J. D. Weeks, in Ordering in Strongly Fluctuating Condensed Matter Systems, edited by T. Riste (Plenum, New York, 1980); J. D. Weeks and G. H. Gilmer, Adv. Chem. Phys. 40, 157 (1979).
4. G. H. Gilmer, in Chemistry and Physics of Solid Surfaces, edited by R. Vanselow and R. Howe, Vol. 35 of Springer Series in Chemical Physics (Springer Verlag, Heidelberg, 1982).
5. R. K. P. Zia, in Statistical and Particle Physics: Common Problems and Techniques, edited by K. C. Bowler and A. J. McKane (SUSSP, Edinburgh, 1984).
6. D. Jasnow, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic, London, 1986), Vol. 10.
7. S. T. Chui and J. D. Weeks, Phys. Rev. B 14, 4978 (1976).
8. J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181 (1973) [ INSPEC][SPIRES]; J. M. Kosterlitz, 7, 1046 (1974) [SPIRES].
9. P. E. Wolf, F. Gallet, S. Balibar, E. Rolley and P. Nozières, J. Phys. (Paris) 46, 1987 (1985) [ INSPEC]; C. Rottman, M. Wortis, J.C. Heyraud and J.J. Métois, Phys. Rev. Lett. 52, 1009 (1984).
10. R. H. Swendsen, Phys. Rev. B 17, 3710 (1978); 15, 5421 (1977).
11. K. K. Mon, S. Wansleben, D. P. Landau and K. Binder, Phys. Rev. Lett. 60, 708 (1988).
12. C. Rottman and M. Wortis, Phys. Rev. B 29, 328 (1984).
13. it Molecular Beam Epitaxy and Heterostructures, edited by L. L. Chang and K. Ploog (Nijhoff, Dordrecht, 1985); A. W. Vere, Crystal Growth (Plenum, New York, 1987).
14. it Thin Film Processes, edited by J. L. Vossen and W. Kern (Academic, New York, 1978); A. Mazor, D. J. Srolovitz, P. S. Hagan and B. G. Bukiet, Phys. Rev. Lett. 60, 424 (1988).
15. A. Dougherty and J. P. Gollub, Phys. Rev. A 38, 3043 (1988); J. P. Franck and J. Jung, Physica D 23, 259 (1986); J. Maurer, P. Bouissou, B. Perrin and P. Tabeling, Europhys. Lett. 8, 67 (1989) [ INSPEC]; J. H. Bilgram, M. Firmann and E. Hürlimann, J. Cryst. Growth 96, 175 (1989) [ INSPEC]; F. Gallet, S. Balibar and E. Rolley, J. Phys. (Paris) 48, 369 (1987), [ INSPEC]; and references therein.
16. See, for example, the article by H. Müller Krumbhaar in Monte Carlo Methods in Statistical Physics (Springer Verlag, New York, 1979), p. 261.
17. M. Kardar, G. Parisi and Y. C. Zhang, Phys. Rev. Lett. 56, 889 (1986) [SPIRES]; see also, E. Medina, T. Hwa, M. Kardar and Y. C. Zhang, Phys. Rev. A 39, 3053 (1989).
18. F. Family and T. Vicsek, J. Phys. A 18, L75 (1985).
19. J. M. Kim and J. M. Kosterlitz, Phys. Rev. Lett. 62, 2289 (1989).
20. M. Plischke and Z. Rácz, Phys. Rev. A 32, 3825 (1985); R. Jullien and R. Botet, J. Phys. A 18, 2279 (1985) [ INSPEC]; P. Meakin, R. Jullien and R. Botet, Europhys. Lett. 1, 609 (1986) [ INSPEC]; J.G. Zabolitzky and D. Stauffer, Phys. Rev. A 34, 1523 (1986); D.E. Wolf and J. Kertész, J. Phys. A 20, L257 (1987) [ INSPEC]; P. Meakin,, L1113 (1987) [ INSPEC].
21. P. Meakin, P. Ramanlal, L.M. Sander and R.C. Ball, Phys. Rev. A 34, 5091 (1986).
22. M. Plischke, Z. Rácz and D. Liu, Phys. Rev. A 35, 3485 (1987).
23. D. A. Huse and C. L. Henley, Phys. Rev. Lett. 54, 2708 (1985); M. Kardar, 55, 2923 (1985); A. Bovier, J. Fröhlich and U. Galus, Phys. Rev. B 34, 6409 (1986); T. Nattermann and W. Renz, 38, 5184 (1988).
24. If beta equiv chi/z, with chi+z=2, the conjectures for dimension d are beta = case 1/3, due to Kardar, Parisi, and Zhang (Ref. 17); beta =1/(2d-1), due to D. E. Wolf and J. Kertesz (Ref. 29); and beta =1/(d+1), due to J. M. Kim and J. M. Kosterlitz (Ref. 19).
25. P. Bak, C. Tang and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987); P. Bak and C. Tang, Phys. Rev. A 38, 364 (1988); and, 60, 2347 (1988); and, J. Stat. Phys. 51, 797 (1988) [ INSPEC].
26. C.H. Bennett and G. Grinstein, Phys. Rev. Lett. 55, 657 (1985).
27. G. Grinstein, D. H. Lee and S. Sachdev, Phys. Rev. Lett. 64, 1927 (1990).
28. B. Grossmann, H. Guo and M. Grant, Phys. Rev. A 41, 4195 (1990).
29. R. Jullien and R. Botet, in Ref. 20; D.E. Wolf and J. Kertész, Europhys. Lett. 4, 651 (1987) [ INSPEC]; P. Meakin, in Ref. 20; D. Devillard and H.E. Stanley, Physica A 160, 298 (1989) [ INSPEC].
30. P. Meakin, in Ref. 20; R. Baiod, D. Kessler, P. Ramanlal, L.M. Sander and R. Savit, Phys. Rev. A 38, 3672 (1988); F. Family (unpublished).
31. D. Liu and M. Plischke, Phys. Rev. B 38, 4781 (1988); B.M. Forrest and L.H. Tang, Phys. Rev. Lett. 64, 1405 (1990).
32. M. Kardar and Y.C. Zhang, Phys. Rev. Lett. 58, 2087 (1987).
33. Some of these results have been presented previously in two short papers (Refs. 34 and 35).
34. H. Guo, B. Grossmann and M. Grant, Phys. Rev. Lett. 64, 1262 (1990).
35. H. Guo, B. Grossmann and M. Grant, Phys. Rev. A 41, 7082 (1990).
36. A. Chakrabarti and R. Toral, Phys. Rev. B 40, 11419 (1989).
37. Y. Saito, in Ordering in Strongly Fluctuating Condensed Matter Systems, edited by T. Riste (Plenum, New York, 1980); Z. Phys. B 32, 75 (1978).
38. P. Nozières and F. Gallet, J. Phys. (Paris) 48, 353 (1987) [ INSPEC].
39. S. Katz, J. L. Lebowitz and H. Spohn, Phys. Rev. B 28, 1655 (1983); K. T. Leung, K. K. Mon, J. L. Vallés, and R. K. P. Zia, Phys. Rev. Lett. 61, 1744 (1988).
40. H. J. Leamy and G. H. Gilmer, J. Cryst. Growth 24/25, 499 (1974).
41. J. M. Burgers, The Nonlinear Diffusion Equation (Riedel, Boston, 1974).
42. S. T. Chui and J. D. Weeks, Phys. Rev. B 40, 733 (1978).
43. S. F. Edwards and D. R. Wilkinson, Proc. R. Soc. London Ser. A 381, 17 (1982) [ INSPEC].
44. M. Grant, Phys. Rev. B 37, 5705 (1988).
45. D. Forster, D. R. Nelson and M. J. Stephen, Phys. Rev. A 16, 732 (1977).
46. P.C. Hohenberg and B.I. Halperin, Rev. Mod. Phys. 49, 435 (1977) [SPIRES].
47. J. D. Gunton, M. San Miguel, and P. S. Sahni, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic, London 1983), Vol. 8.
48. J. D. Gunton and M. Droz, in Introduction to the Theory of Metastable and Unstable States, Vol. 183 of Lecture Notes in Physics (Springer, Berlin, 1983).
49. J. Collins and H. Levine, Phys. Rev. A 31, 6119 (1985).
50. L. Jörgenson, R. Harris and M. Grant, Phys. Rev. Lett. 63, 1693 (1989).
51. R. Harris and M. Grant, J. Phys. A 23, L567 (1990) [ INSPEC].
52. S. M. Allen and J. W. Cahn, Acta Metall. 27, 1085 (1979) [ INSPEC].
53. R. Bausch, V. Dohm, H. K. Janssen, and R. K. P. Zia, Phys. Rev. Lett. 47, 1837 (1981).
54. K. Kawasaki and T. Ohta, Prog. Theor. Phys. 67, 147 (1982) [ INSPEC]; and, 68, 129 (1982) [ INSPEC].
55. M. Grant and J. D. Gunton, Phys. Rev. B 28, 5496 (1983).
56. G. Caginalp, IMA J. Appl. Math. 44, 77 (1990) [ INSPEC]; Phys. Rev. A. 39, 5887 (1989).
57. D. Boyanovsky and J. Cardy, Phys. Rev. B 27, 5557 (1983).
58. M. Grant and J. D. Gunton, Phys. Rev. B 29, 6266 (1984).
59. We thank R. K. P. Zia for suggesting this to us.
60. J. Krug and H. Spohn (unpublished).
61. By direct simulation, we have checked that the two dimensional Ising model in a field has the same exponents as the KPZ equation, for small fields, for times before droplet nucleation is important. Further results are planned to be presented in a future paper by L. Jörgenson, H. Guo, B. Grossmann, R. Harris, and M. Grant (unpublished).
62. R. Becker and W. Döring, Ann. Phys. 24, 719 (1935).
63. We have also checked other discretization schemes, such as including next nearest neighbors in the finite difference approximation for the derivatives. A negligible difference was found in the results.
64. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W.T. Vetterling, Numerical Recipes (Cambridge University Press, New York, 1986).
65. Recently, R. Fox (private communication) has derived an exact sum rule for the cumulants of the height variable for the KPZ equation. We have tested our numerical algorithm against this exact result and found agreement within less than $0.5.
66. E. Brezin, J.C. LeGuillou and J. Zinn Justin, Phys. Rev. B 10, 892 (1974).
67. L. H. Tang, T. Nattermann, and B. M. Forrest, Phys. Rev. Lett. 65, 2422 (1990).
68. See, however, J. G. Amar and F. Family, Phys. Rev. A 41, 3399 (1989),; where values of the growth exponents different from ours and those of Ref. 36 are obtained.
69. W. J. Shugard, J.D. Weeks and G.H. Gilmer, Phys. Rev. Lett. 31, 549 (1978).
70. T. Sun, H. Guo and M. Grant, Phys. Rev. A 40, 6763 (1989).
71. H. Yan, D. Kessler and L. M. Sander, Phys. Rev. Lett. 64, 926 (1990).

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