Continuum model for the growth of interfaces

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 53, Issue 1, 1996, Pages 759-778

  Keblinski, Pawel ^a,b,c,d,e   Maritan, Amos ^a,b,c,d,e   Toigo, Flavio ^a,b,c,d,e   Messier, Russell ^a,b,c,d,e   Banavar, Jayanth R ^a,b,c,d,e  

^a PENNSYLVANIA STATE UNIVERSITY   (United States)

^b SISSA   (Italy)

^c SEZIONE DI TRIESTE   (Italy)

^d INFM   (Italy)

^e The Pennsylvania State University   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000274043     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.53.759     Document Type: Article

References (68)

1. Dynamics of Fractal Surfaces, edited by F. Family and T. Vicsek (World Scientific, Singapore, 1991), and references therein;
2. L. A. Barabasi and H. E. Stanley, Fractal Concepts in Interface Growth (Cambridge University Press, Cambridge, 1995) and references therein;
3. T. Halpin Healy and Y. C. Zhang, Phys. Rep. 254, 415 (1995).
4. For a review of growth involving nonlocal effects, see G. S. Bales, R. Bruinsma, E. A. Eklund, R. P. U. Karunasiri, J. Rudnick and A. Zangwill, Science 249, 264 (1990), and references therein.
5. B. A. Movchan and A. V. Demchishin, Fiz. Met. Metall. 28, 653 (1969) [Phys Met. Metall. (USSR) 28, 83 (1969)].
6. L. R. Gilbert, R. Messier and R. Roy, Thin Solid Films 54, 149 (1978);
7. R. Messier, S. V. Krishnaswamy, L. R. Gilbert and P. Swab, J. Appl. Phys. 51, 1611 (1980).
8. R. Messier and J. E. Yehoda, J. Appl. Phys. 58, 3739 (1985).
9. R. Messier, J. Vac. Sci. Technol. A 4, 490 (1986).
10. K. H. Muller, Surf. Sci. Lett. 184, L375 (1987).
11. J. Bartholomeusz, K. H. Muller and M. R. Jacobson, Proc. SPIE 821, 2 (1987).
12. J. E. Yehoda, B. Yang, K. Vedam and R. Messier, J. Vac. Sci. Technol. A 6, 1631 (1988).
13. T. A. Witten and L. M. Sander, Phys. Rev. Lett. 47, 1400 (1981).
14. For a review see, e.g., Fractals in Physics, edited by L. Pietronero and E. Tosatti (North-Holland, Amsterdam, 1985).
15. L. M. Sander, Nature 332, 789 (1986).
16. J. Nittamann and H. E. Stanley, Nature 321, 663 (1986).
17. L. Paterson, Phys. Rev. Lett. 52, 1621 (1984).
18. J. Nittamann, G. Daccord and H. E. Stanley, Nature 314, 141 (1985);
19. T. Maxworthy, Phys. Rev. A 39, 5863 (1989).
20. E. Ben-Jacob , Phys. Rev. Lett. 55, 1315 (1985).
21. L. Neimeyer, L. Pietronero and H. Weisman, Phys. Rev. Lett. 52, 1033 (1984).
22. G. L. M. K. S. Kahanda, X. Zou, R. Farrel and P. Wong, Phys. Rev. Lett. 68, 3741 (1992).
23. P. Meakin, CRC Crit. Rev. Solid State Sci. 13, 143 (1987), and references therein.
24. See, e.g., G. H. Gilmer and M. Grabow, Proc. SPIE 821, 56 (1987).
25. P. Keblinski, A. Maritan, F. Toigo, J. Koplik and J. R. Banavar, Phys. Rev. E 49, R937 (1994).
26. P. Keblinski, A. Maritan, F. Toigo and J. R. Banavar, Phys. Rev. E 49, R4795 (1994).
27. P. Keblinski, A. Maritan, F. Toigo and J. R. Banavar, Phys. Rev. Lett. 40, 1783 (1995).
28. M. Kardar, G. Parisi and Y. Zhang, Phys. Rev. Lett. 56, 889 (1986).
29. L. Golubovic and R. P. U. Karunasiri, Phys. Rev. Lett. 66, 3156 (1991).
30. P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49, 435 (1977).
31. T. M. Rogers, K. R. Elder and R. C. Desai, Phys. Rev. B 37, 9638 (1988).
32. J. M. Kim and J. M. Kosterlitz, Phys. Rev. Lett. 62, 2289 (1989);
33. J. M. Kim, J. M. Kosterlitz and T. Ala-Nissila, J. Phys. A 24, 5569 (1991);
34. K. Moser, J. Kertesz and D. E. Wolf, Physica A 178, 215 (1991), obtain β = 0.24 +- 0.005 in d = 2 + 1 by a direct numerical integration of the KPZ equation.
35. D. Y. K. Ko and F. Seno, Phys. Rev. E 50, R1741 (1994).
36. A. Mazor, D. J. Srolovitz, P. S. Hagan and B. G. Bukiet, Phys. Rev. Lett. 60, 424 (1988);
37. D. J. Srolovitz, A. Mazor and B. G. Bukiet, J. Vac. Sci. Technol. A 6, 2371 (1988).
38. D. J. Eaglesham, H. J. Gossman and M. Cerullo, Phys. Rev. Lett. 65, 1227 (1990).
39. R. S. Williams, R. Bruinsma and J. Rudnik, in Surface Disordering Growth, Roughening and Phase Transitions, edited by R. Jullien, J. Kertesz, P. Meakin, and D. E. Wolf (Nova, New York, 1993).
40. R. Messier and A. Roy, J. Vac. Sci. Technol. 13, 1060 (1976);
41. J. R. Blanco, R. Messier, K. Vedam and P. J. McMarr, in Plasma Synthesis and Etching of Electronic Materials, edited by R. P. H. Chang and P. Abeles, MRS Symposia Proceedings No. 38 (Materials Research Society, Pittsburgh, 1985), p. 301.
42. C. Tang and S. Liang, Phys. Rev. Lett. 71, 2769 (1993);
43. S. Redner (unpublished)has carried out lattice simulations analogous to our two-rain case at grazing incidence, and has found patterns of independent growing trees that do not show any coalescence. These results suggest that there may be essential differences between the lattice and continuum models in this limit. We are grateful to Sid Redner for bringing his work to our attention.
44. Y. Kuramoto, Suppl. Prog. Theor. Phys. 64, 346 (1978);
45. G. Sivashinsky, Acta Austronautica 4, 1177 (1977).
46. M. Marsili, A. Maritan, F. Toigo, and J. R. Banavar (unpublished).
47. T. Ihle and H. Muller-Krumbhaar, Phys. Rev. Lett. 70, 3083 (1993).
48. The interaction term is similar to that studied in annihilation chemical reactions;
49. see, e.g., K. Kang and S. Redner, Phys. Rev. A 32, 435 (1985), and references therein.
50. J. S. Langer, in Directions in Condensed Matter Physics, edited by G. Grinstein and G. Mazenko (World Scientific, Singapore, 1986), pp. 164–186;
51. J. B. Collins and H. Levine, Phys. Rev. B 31, 6119 (1985);
52. G. Caginalp, Phys. Rev. A 39, 887 (1989).
53. For numerical work on the phase field model, see A. A. Wheeler, B. T. Murray and R. J. Schaefer, Physica D 66, 143 (1993);
54. R. Kobayashi, Physica D 63, 410 (1993).
55. M. Nauenberg, R. Richter and L. M. Sander, Phys. Rev. B 28, R1649 (1983);
56. G. Parisi and Y. C. Zhang, J. Stat. Phys. 41, 1 (1985);
57. D. Elderfield, J. Phys. A 18, L773 (1985);
58. D. Elderfield Y. Shapir J. Phys. A 18, L970 (1985)
59. H. Levine and Y. Tu, Phys. Rev. E 48, R4207 (1993).
60. L. M. Sander, in Fractals In Physics (Ref. 11), especially Fig. 2 on p. 245.
61. W. W. Mullins and R. F. Sekerka, J. Appl. Phys. 35, 444 (1964);
62. J. S. Langer, Rev. Mod. Phys. 52, 1 (1980).
63. P. Z. Wong, Phys. Today 41, (12), 24 (1988);
64. D. Bensimon, L. P. Kadanoff, S. Liang, B. I. Shraiman and C. Tang, Rev. Mod. Phys. 58, 997 (1986).
65. E. Brener, H. Muller-Krumbhaar and D. Temkin, Europhys. Lett. 17, 535 (1992).
66. O. Shochet, K. Kasser, E. Ben-Jacob, S. G. Lipson and H. Muller-Krumbhaar, Physica A 181, 136 (1992).
67. R. Bausch, V. Dohm, H. K. Janssen, and R. K. P. Zia, Phys. Rev. Lett. 47, 1837 (1981).
68. A. Maritan, F. Toigo, J. Koplik and J. R. Banavar, Phys. Rev. Lett. 69, 3193 (1992).