Advertisement

The Growth of Fractal Aggregates

  • Paul Meakin
Part of the NATO ASI Series book series (volume 167)

Abstract

A large variety of structures with more or less well defined fractal geometries have now been recognized.1 Many of these structures have come about as a result of nonequilibrium growth or aggregation processes. At present much of what we know about these growth and aggregation processes has come from computer simulations. Almost 20 years ago a simple ballistic aggregation model in which clusters combined with other clusters to generate successively larger and larger clusters was developed by Sutherland.2,3 This model led to the formation of tenuous structures which exhibited mass-length scaling relationships which would now be recognized as being characteristic of fractal geometry. The mass-length scaling exponents obtained from three dimensional simulations correspond to a fractal dimensionality (D) of about 1.85. More regent larger scale simulations led to values in the range 1.90 to 1.95.4,5

Keywords

Random Walker Fractal Dimensionality Smoluchowski Equation Cluster Size Distribution Diffusion Limited Aggregation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    B. B. Mandelbrot, “The Fractal Geometry of Nature”, W. H. Freeman and Company, New York (1982).MATHGoogle Scholar
  2. 2.
    D. N. Sutherland, J. Colloid and Interace Sci. 25: 373 (1967).CrossRefGoogle Scholar
  3. 3.
    D. N. Sutherland and I. Goodarz-Nia, Chem. Eng. Sci. 26: 2071 (1971).CrossRefGoogle Scholar
  4. 4.
    P. Meakin, J. Colloid and Interface Sci. 102:505 (1984); P. Meakin and B. Donn, unpublished.Google Scholar
  5. 5.
    R. Jullien, M. Kolb and R. Botet, J. Physique Lett. (Paris) 45:L211 (1984); R. Ball and R. Jullien, J. Physique Lett. (Paris) 45: L1031 (1984).CrossRefGoogle Scholar
  6. 6.
    S. R. Forrest and T. A. Witten, J. Phys. Al2: 109 (1979).Google Scholar
  7. 7.
    T. A. Witten and L. M. Sander, Phys. Rev. Lett. 47: 1400 (1981).ADSCrossRefGoogle Scholar
  8. 8.
    B. B. Mandelbrot, “Fractals in Physics”, Proceedings of the Sixth International Symposium on Fractals in Physics, ICTP, Trieste, Italy, L. Pietronero and E. Tosatti, eds., North Holland Amsterdam (1986).Google Scholar
  9. 9.
    P. Meakin, J. Phys. A18: L661 (1985).ADSGoogle Scholar
  10. 10.
    L. Patterson, Phys. Rev. Lett. 52: 1621 (1984).CrossRefGoogle Scholar
  11. 11.
    K. J. Maloy, J. Feder and T. Jossang, Phys. Rev. Lett. 55: 2688 (1985).ADSCrossRefGoogle Scholar
  12. 12.
    J. Nittman, G. Daccord and H. E. Stanley, Nature 314: 141 (1985).ADSCrossRefGoogle Scholar
  13. 13.
    L. P. Kadanoff, J. Stat. Phys. 39: 267 (1985).MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    H. Van Damme, F. Obrecht, P. Levitz, L. Gatineau and C. Laroche, Nature, 320: 731 (1986).ADSCrossRefGoogle Scholar
  15. 15.
    L. Niemeyer, L. Pietronero and A. J. Wiesmann, Phys. Rev. Lett. 52: 1033 (1984).MathSciNetADSCrossRefGoogle Scholar
  16. 16.
    R. M. Brady and R. C. Ball, Nature 309: 225 (1984).ADSCrossRefGoogle Scholar
  17. 17.
    H. Honjo, S. Ohta and M. Matsushita, J. Phys. Soc. Japan 55: 2487 (1986).ADSCrossRefGoogle Scholar
  18. 18.
    W. T. Elam, S. A. Wolf, J. Srague, D. V. Gubser, D. Van Vechten, G. L. Barz, Jr. and P. Meakin, Phys. Rev. Lett. 54: 701 (1985).ADSCrossRefGoogle Scholar
  19. 19.
    Gy Radnoczi, T. Vicsek, L. M. Sander and D. Grier, preprint (1986).Google Scholar
  20. 20.
    G. Daccord and R. Lenormand, preprint (1986).Google Scholar
  21. 21.
    P. Meakin, J. Theor. Biol. 118: 101 (1986).MathSciNetCrossRefGoogle Scholar
  22. 22.
    A. A. Tsonis and P. A. Tsonis, Persp. Biol. Med. xx:xxxx (1987).Google Scholar
  23. 23.
    P. Meakin, Phys. Rev. Lett. 51: 1119 (1983).CrossRefGoogle Scholar
  24. 24.
    M. Kolb, R. Botet and R. Jullien, Phys. Rev. Lett. 51: 1123 (1983).ADSCrossRefGoogle Scholar
  25. 25.
    P. Meakin, Phys. Lett. 107A: 269 (1985).CrossRefGoogle Scholar
  26. 26.
    R. Jullien, M. Kolb and R. Botet, J. Physique Lett. 45: L211 (1984).CrossRefGoogle Scholar
  27. 27.
    D. A. Weitz and M. Oliveria, Phys. Rev. Lett. 52: 1433 (1984).ADSCrossRefGoogle Scholar
  28. 28.
    C. Aubert and D. Cannell, Phys. Rev. Lett. 56: 738 (1986).ADSCrossRefGoogle Scholar
  29. 29.
    J. E. Martin, D. W. Schaefer and A. J. Hurd, Phys. Rev. A33: 3540 (1986).ADSCrossRefGoogle Scholar
  30. 30.
    G. Bolle, C. Cametti, P. Codastefano and P. Tartaglia, Phys. Rev. A35: 837 (1987).ADSCrossRefGoogle Scholar
  31. 31.
    R. Jullien and R. Botet, J. Phys. A17: L639 (1984).MathSciNetADSGoogle Scholar
  32. 32.
    F. Leyvraz, preprint (1986).Google Scholar
  33. 33.
    W. D. Brown and R. C. Ball, J. Phys. A18: L517 (1985).ADSGoogle Scholar
  34. 34.
    P. Meakin and F. Family, unpublished.Google Scholar
  35. 35.
    D. W. Schaefer, J. E. Martin, P. Wiltzius and D. S. Cannell, Phys. Rev. Lett. 52: 2371 (1984).ADSCrossRefGoogle Scholar
  36. 36.
    D. Johnston and G. Benedek, “Kinetics of Aggregation and Gelation”, p. 181, F. Family and D. P. Landau, eds., North Holland Amsterdam (1984).Google Scholar
  37. 37.
    D. A. Weitz, J. S. Huang, M. Y. Lin and J. Sung, Phys. Rev. Lett 54: 1416 (1985).ADSCrossRefGoogle Scholar
  38. 38.
    P. Meakin, A. Coniglio, H. E. Stanley and T. A. Witten, Phys. Rev. A34: 3325 (1986).MathSciNetADSCrossRefGoogle Scholar
  39. 39.
    M. Plischke and Z. Racz, Phys. Rev. Lett. 53: 415 (1984).ADSCrossRefGoogle Scholar
  40. 40.
    T. C. Halsey, P. Meakin and I. Procaccia, Phys. Rev. Lett. 56: 854 (1986).ADSCrossRefGoogle Scholar
  41. 41.
    P. Meakin, H. E. Stanley, A. Coniglio and T. A. Witten, Phys. Rev. A32: 2364 (1985).ADSCrossRefGoogle Scholar
  42. 42.
    C. Amitrano, A. Coniglio and F. diLiberto, Phys. Rev. Lett. 57: 1016 (1986).ADSCrossRefGoogle Scholar
  43. 43.
    L. Pietronero, C. Evertsz and A. P. Siebesma “Stoichastic Processes in Physics and Engineeering”. S. Albeverio, Ph. Blanchard, L. Streit and M. Hazewinkel (Proc. BIBOS IV) eds., S. Riedel, Dordrecht (1986).Google Scholar
  44. 44.
    P. Meakin, Phys. Rev. A34:710 (1986); Phys. Rev. Axx:xxxx (1987).Google Scholar
  45. 45.
    B. B. Mandelbrot, J. Fluid Mech. 62: 331 (1974).ADSMATHCrossRefGoogle Scholar
  46. 46.
    H. G. E. Hentschel and I. Procaccia, Physica 8D: 440 (1983).MathSciNetGoogle Scholar
  47. 47.
    T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia and B. Shraiman, Phys. Rev. A33: 1141 (1986).MathSciNetADSMATHCrossRefGoogle Scholar
  48. 48.
    P. Meakin and L. M. Sander, Phys. Rev. Lett. 54: 2053 (1985).ADSCrossRefGoogle Scholar
  49. 49.
    M. Plischke and Z. Racz.Google Scholar
  50. 50.
    P. Meakin and T. A. Witten, Phys. Rev. A28: 2985 (1983).ADSCrossRefGoogle Scholar
  51. 51.
    P. Meakin and L. M. Sander, unpublished; P. Meakin, unpublished.Google Scholar
  52. 52.
    J. M. Deutch and P. Meakin, J. Chem. Phys. 78: 2093 (1983).ADSCrossRefGoogle Scholar
  53. 53.
    P. Meakin, Phys. Rev. A27: 604 (1983).MathSciNetADSCrossRefGoogle Scholar
  54. 54.
    T. A. Witten, Proceedings “MACRO ’82”, p. 88, IUPAC, Amherst, Mass., June (1982).Google Scholar
  55. 55.
    R. F. Voss, Phys. Rev. B30:334 (1984); J. Stat. Phys. 36: 861 (1984).ADSCrossRefGoogle Scholar
  56. 56.
    R. F. Voss and M. Tomkiewicz, J. Electrochem. Soc. 132: 371 (1985).CrossRefGoogle Scholar
  57. 57.
    P. Meakin and J. M. Deutch, J. Chem. Phys. 80: 2115 (1984).ADSCrossRefGoogle Scholar
  58. 58.
    T. A. Witten and L. M. Sander, Phys. Rev. B27: 5686 (1983).MathSciNetADSGoogle Scholar
  59. 59.
    L. Niemeyer, L. Pietronero and A. J. Wiesmann, Phys. Rev. Lett. 52: 1033 (1984).MathSciNetADSCrossRefGoogle Scholar
  60. 60.
    M. Matsushita, K. Honda, H. Toyoki, Y. Hayakawa and H. Kondo, J. Phys. Soc. Japan 55: 2618 (1986).ADSCrossRefGoogle Scholar
  61. 61.
    L. Pietronero, private communication (1984).Google Scholar
  62. 62.
    P. Meakin, Phys. Rev. A33: 3371 (1986).MathSciNetADSCrossRefGoogle Scholar
  63. 63.
    K. J. Maloy, F. Boger, J. Feder, T. Jossang and P. Meakin, preprint.Google Scholar
  64. 64.
    S. K. Friedlander, Smoke, Dust and Haze, Wiley Interscience, New York (1977).Google Scholar
  65. 65.
    A. A. Lushnikov and V. N. Piskunov, Dolklady Phys. Chem. 231: 1166 (1976).Google Scholar
  66. 66.
    J. Silk and S. D. White, Astrophys. J. 22: L59 (1978).ADSCrossRefGoogle Scholar
  67. 67.
    T. Vicsek and F. Family, Phys. Rev. Lett. 52: 1669 (1984).ADSCrossRefGoogle Scholar
  68. 68.
    M. Kolb, Phys. Rev. Lett. 53: 1653 (1984).CrossRefGoogle Scholar
  69. 69.
    R. Botet and R. Jullien, J. Phys. A17: 2517 (1984).MathSciNetADSMATHGoogle Scholar
  70. 70.
    S. Mijazima, P. Meakin and F. Family, preprint.Google Scholar
  71. 71.
    M. Von Smoluchowski, Z. Phys. 17: 585 (1916).Google Scholar
  72. 72.
    D. Toussaint and F. Wilczek, J. Chem. Phys. 78: 2642 (1983).ADSCrossRefGoogle Scholar
  73. 73.
    P. Meakin and H. E. Stanley, Phys. Rev. A17: L173 (1984).Google Scholar
  74. 74.
    K. Kang and S. Redner, Phys. Rev. Lett. 52: 955 (1984).MathSciNetADSCrossRefGoogle Scholar
  75. 75.
    K. Kang and S. Redner, Phys. Rev. A32: 435 (1985).ADSCrossRefGoogle Scholar
  76. 76.
    P. G. J. Van Dongen and M. H. Ernst, Phys. Rev. Lett. 54: 1396 (1985).ADSCrossRefGoogle Scholar
  77. 77.
    F. Leyvraz “On Growth and Form: Fractal and Nonfractal Patterns in Physics”. H. E. Stanley and N. Ostrowsky, eds., Martinus Nijhoff, Dordrecht (1986), p. 136.Google Scholar
  78. 78.
    P. Meakin, Phys. Rev. B29: 2930 (1984).ADSGoogle Scholar
  79. 79.
    R. Jullien, M. Kolb and R. Botet “Kinetics of Aggregation and Gelation”. F. Family and D. P. Landau, eds., Elsevier, North-Holland Amsterdam (1984).Google Scholar
  80. 80.
    J. G. Kirkwood and J. Riseman, J. Chem. Phys. 16: 565 (1948).ADSCrossRefGoogle Scholar
  81. 81.
    P. Meakin, Z.-Y. Chen and J. M. Deutch, J. Chem. Phys. 82: 3786 (1985).ADSCrossRefGoogle Scholar
  82. 82.
    P. Meakin, T. Vicsek and F. Family, Phys. Rev. A31: 564 (1985).ADSCrossRefGoogle Scholar
  83. 83.
    R. M. Ziff, E. D. McGrady and P. Meakin, J. Chem. Phys. 82: 5269 (1985).ADSCrossRefGoogle Scholar
  84. 84.
    J. L. Spouge, J. Colloid Interface Sci. 107: 38 (1985).CrossRefGoogle Scholar
  85. 85.
    J. H. Jeans, “The Dynamic Theory of Gases”, Dover, New York (1954).Google Scholar
  86. 86.
    B. Donn, private communication.Google Scholar
  87. 87.
    P. Meakin and B. Donn, unpublished.Google Scholar
  88. 88.
    P. Meakin, J. Chem. Phys. 81: 4637 (1984).ADSCrossRefGoogle Scholar
  89. 89.
    R. C. Ball, D. A. Weitz, T. A. Witten and F. Leyvraz, Phys. Rev. Lett. 58: 274 (1987).ADSCrossRefGoogle Scholar
  90. 90.
    R. M. Ziff, J. Stat. Phys. 23: 241 (1980).MathSciNetADSCrossRefGoogle Scholar
  91. 91.
    H. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, J. Chem. Phys. 21: 1089 (1953).ADSCrossRefGoogle Scholar
  92. 92.
    T. Vicsek, P. Meakin and F. Family, Phys. Rev. A32: 1122 (1985).ADSCrossRefGoogle Scholar
  93. 93.
    P. Meakin, J. Chem. Phys. 83:3645 (1985); P. Meakin, unpublished.Google Scholar
  94. 94.
    K. Kang, S. Redner, P. Meakin and F. Leyvraz, Phys. Rev. A33: 1171 (1986).ADSCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1987

Authors and Affiliations

  • Paul Meakin
    • 1
  1. 1.Central Research and Development DepartmentE. I. du Pont de Nemours & CompanyWilmingtonUSA

Personalised recommendations