Advertisement

Journal of Statistical Physics

, Volume 63, Issue 1–2, pp 167–202 | Cite as

Random sequential addition: A distribution function approach

  • Gilles Tarjus
  • Pierre Schaaf
  • Julian Talbot
Articles

Abstract

Random sequential addition (RSA) of hard objects is an irreversible process defined by three rules: objects are introduced on a surface (or ad-dimensional volume) randomly and sequentially, two objects cannot overlap, and, once inserted, an object is clamped in its position. The configurations generated by an RSA can be characterized, in the macroscopic limit, by a unique set of distribution functions and a density. We show that these “nonequilibrium” RSA configurations can be described in a manner which, in many respects, parallels the usual statistical mechanical treatment of equilibrium configurations: Kirkwood-Salsburg-like hierarchies for the distribution functions, zero-separation theorems, diagrammatic expansions, and approximate equations for the pair distribution function. Approximate descriptions valid for low to intermediate densities can be combined with exact results already derived for higher densities close to the jamming limit of the process. Similarities and differences between the equilibrium and the RSAconfigurations are emphasized. Finally, the potential application of RSA processes to the study of glassy phases is discussed.

Key words

Random sequential addition hard-core particles distribution functions nonequilibrium configurations Kirkwood-Salsburg-like equation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    P. J. Flory,J. Am. Chem. Soc. 61:1518 (1939).Google Scholar
  2. 2.
    J. K. Mackenzie,J. Chem. Phys. 37:723 (1962).Google Scholar
  3. 3.
    J. J. Gonzalez, P. C. Hemmer, and J. C. Høye,Chem. Phys. 3:228 (1974).Google Scholar
  4. 4.
    L. Finegold and J. T. Donnell,Nature 278:443 (1979).Google Scholar
  5. 5.
    M. Hasegawa and M. Tanemura, inRecent Developments in Statistical Interference and Data Analysis, K. Matusita, ed. (North-Holland, Amsterdam, 1980).Google Scholar
  6. 6.
    J. Feder and I. Giaver,J. Colloid Interface Sci. 78:144 (1980).Google Scholar
  7. 7.
    J. Feder,J. Theor. Biol. 87:237 (1980).Google Scholar
  8. 8.
    E. M. Tory and W. S. Jodrey, inAdvances in the Mechanics and the Flow of Granular Materials, M. Shahinpoor, ed. (Trans. Tech., Clausthat-Zellerfeld, 1983), Vol.I, and references therein.Google Scholar
  9. 9.
    E. Roman and N. Majlis,Solid State Commun. 47:259 (1983).Google Scholar
  10. 10.
    K. Gotoh, M. Nakagawa, and M. Matsuoka,Paniculate Sci. Technol. 3:27 (1985).Google Scholar
  11. 11.
    R. S. Nord and J. W. Evans,J. Chem. Phys. 82:2785 (1985), and references therein.Google Scholar
  12. 12.
    G. Y. Onoda and E. G. Liniger,Phys. Rev. A 33:715 (1986).Google Scholar
  13. 13.
    B. Widom,J. Chem. Phys. 44:3888 (1966).Google Scholar
  14. 14.
    P. Schaaf and J. Talbot,J. Chem. Phys. 91:4401 (1989).Google Scholar
  15. 15.
    D. K. Hoffman,J. Chem. Phys. 65:95 (1976).Google Scholar
  16. 16.
    J. W. Evans,Physica 123A:297 (1984).Google Scholar
  17. 17.
    J. W. Evans,J. Chem. Phys. 87:3038 (1987).Google Scholar
  18. 18.
    G. C. Barker and M. J. Grimson,Mol. Phys. 63:145 (1988).Google Scholar
  19. 19.
    P. Schaaf, J. Talbot, H. M. Rabeony, and H. Reiss,J. Phys. Chem. 92:4826 (1988).Google Scholar
  20. 20.
    A. Baram and D. Kutasov,J. Phys. A: Math. Gen. 22:L251 (1989).Google Scholar
  21. 21.
    A. Rényi,Publ. Math. Inst. Hung. Acad. Sci. 3:109 (1958) [Translated inSelected Transl. Math. Stat. Prob. 4:203 (1963)].Google Scholar
  22. 22.
    E. L. Hinrichsen, J. Feder, and T. Jøssang,J. Stat. Phys. 44:793 (1986).Google Scholar
  23. 23.
    D. W. Cooper,J. Colloid Interface Sci. 119:442 (1986).Google Scholar
  24. 24.
    D. W. Cooper,Phys. Rev. A 38:522 (1988).Google Scholar
  25. 25.
    R. D. Vigil and R. M. Ziff,J. Chem. Phys. 91:2599 (1989).Google Scholar
  26. 26.
    J. Talbot, G. Tarjus, and P. Schaaf,Phys. Rev. A 40:4808 (1989).Google Scholar
  27. 27.
    Y. Pomeau,J. Phys. A: Math. Gen. 13:L193 (1980).Google Scholar
  28. 28.
    R. H. Swendsen,Phys. Rev. A 24:504 (1981).Google Scholar
  29. 29.
    P. Schaaf and J. Talbot,Phys. Rev. Lett. 62:175 (1989).Google Scholar
  30. 30.
    G. Stell, inThe Wonderful World of Stochastics, M. F. Schlesinger and G. H. Weiss, eds. (Elsevier, Amsterdam, 1985).Google Scholar
  31. 31.
    H. Reiss and P. Schaaf,J. Chem. Phys. 91:2514 (1989).Google Scholar
  32. 32.
    J. Kirkwood and Z. Salsburg,Disc. Faraday Soc. 15:28 (1953).Google Scholar
  33. 33.
    L. D. Landau and E. M. Lifshitz,Statistical Physics, 2nd ed. (Addison-Wesley, Reading, Massachusetts, 1969).Google Scholar
  34. 34.
    R. Brout,Phys. Rev. 115:824 (1959).Google Scholar
  35. 35.
    H. Reiss, H. L. Frisch, and J. L. Lebowitz,J. Chem. Phys. 31:369 (1959).Google Scholar
  36. 36.
    S. Torquato and G. Stell,J. Chem. Phys. 78:3262 (1983).Google Scholar
  37. 37.
    J. W. Evans,Phys. Rev. Lett. 62:2642 (1989).Google Scholar
  38. 38.
    B. Barboy and W. M. Gelbart,J. Stat. Phys. 22:685 (1980).Google Scholar
  39. 39.
    G. Stell, inThe Equilibrium Theory of Classical Fluids, H. L. Frisch and J. L. Lebowitz, eds. (Benjamin, New York, 1964).Google Scholar
  40. 40.
    E. Meeron and A. J. F. Siegert,J. Chem. Phys. 48:3139 (1968).Google Scholar
  41. 41.
    E. W. Grundke and D. Henderson,Mol. Phys. 24:269 (1972).Google Scholar
  42. 42.
    J. P. Hansen and I. R. McDonald,Theory of Simple Liquids, 2nd ed. (Academic Press, London, 1986).Google Scholar
  43. 43.
    J. E. Mayer and E. Montroll,J. Chem. Phys. 9:2 (1941).Google Scholar
  44. 44.
    E. E. Salpeter,Ann. Phys. 5:183 (1958).Google Scholar
  45. 45.
    S. A. Rice and P. Gray, Supplement to I. Z. Fisher,Statistical Theory of Liquids (University of Chicago Press, Chicago, Illinois, 1964).Google Scholar
  46. 46.
    A. Baram and D. Kutasov,J. Phys. A: Math. Gen. 22:L855 (1989).Google Scholar
  47. 47.
    G. Tarjus, J. Talbot, and P. Schaaf,J. Phys. A: Math. Gen. 23:837 (1990).Google Scholar
  48. 48.
    G. S. Cargill,J. Appl. Phys. 41:2248 (1970); J. L. Finney,Nature 266:309 (1977); P. H. Gaskell, inGlassy Metals, H. Beck and H.-J. Giintherodt, eds. (Springer, Berlin, 1983), Vol. II.Google Scholar
  49. 49.
    L. V. Woodcock,J. Chem. Soc. Faraday II 72:1667 (1976);74:11 (1978); L. V. Woodcock and C. A. Angell,Phys. Rev. Lett. 47:1129 (1981).Google Scholar
  50. 50.
    E. L. Hinrichsen, J. Feder, and T. Jøssang,Phys. Rev. A 41:4199 (1990); B. D. Lubachevsky and F. H. Stillinger,J. Stat. Phys. 60:561 (1990).Google Scholar

Copyright information

© Plenum Publishing Corporation 1991

Authors and Affiliations

  • Gilles Tarjus
    • 1
  • Pierre Schaaf
    • 2
  • Julian Talbot
    • 3
  1. 1.Laboratoire de Physique Théorique des LiquidesUniversité Pierre et Marie CurieParis Cédex 05France
  2. 2.Institut Charles Sadron (CRM-EAHP)CNRS-ULPStrasbourg CédexFrance
  3. 3.School of Chemical EngineeringPurdué UniversityWest Lafayette

Personalised recommendations