Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Stationary states of crystal growth in three dimensions

  • 47 Accesses

  • 13 Citations

Abstract

A new Markov process describing crystal growth in three dimensions is introduced. States of the process are configurations of the crystal surface, which has a terrace-edge-kink structure. The states are continuous along edges but discrete across edges, in accordance with the very different rates for the two types of captures of particles. Stationary distributions, describing steady crystal growth, are found in general. To our knowledge, these are the first examples of stationary distributions for layered crystal growth in three dimensions. The steady growth rate and other quantities are obtained explicitly for two interacting edges. For many interacting edges, growth behavior is determined (a) in various asymptotic regimes including thermodynamic limits, (b) via simulations, and (c) using series (cluster) expansions in the slope of the surface, the first three coefficients being computed. The theoretical growth rates show a marked dependence on surface dimensions. This may contribute to the size dependence and dispersion in the observed growth rate of small crystals.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    C. H. Bennett, M. Bütticker, R. Landauer, and H. ThomasJ. Stat. Phys. 24:419 (1981).

  2. 2.

    H. van Beijeren and I. Nolden, InStructure and Dynamics of Surfaces II, W. Schommers and P. von Blanckenhagen, eds. (Springer, Berlin, 1987), Chapter 7, p. 259.

  3. 3.

    E. T. Copson,An Introduction to the Theory of Functions of a Complex Variable (Clarendon, Oxford, 1962).

  4. 4.

    W. Feller,An Introduction to Probability Theory and its Applications, Vol. 1 (Wiley, New York, 1958).

  5. 5.

    W. Feller,An Introduction to Probability Theory and its Applications, Vol. 2 (Wiley, New York, 1958).

  6. 6.

    F. C. Frank,Disc. Faraday Soc. 5:48 (1949).

  7. 7.

    F. C. Frank,J. Crystal Growth 22:233 (1974).

  8. 8.

    C. Garrod,J. Stat. Phys. 63:987 (1991).

  9. 9.

    J. Garside, InCrystal Growth and Materials, E. Kaldis and H. J. Scheel, eds. (North-Holland, Amsterdam, 1977), p. 484.

  10. 10.

    D. J. Gates,J. Stat. Phys. 52:245 (1988).

  11. 11.

    D. J. Gates and O. Penrose,Commun. Math. Phys. 16:231 (1970).

  12. 12.

    D. J. Gates and M. Westcott,Proc. R. Soc. London. A 416:443 (1988).

  13. 13.

    D. J. Gates and M. Westcott,Proc. R. Soc. Lond. A 416:463 (1988).

  14. 14.

    D. J. Gates and M. Westcott,J. Stat. Phys. 59:73 (1990).

  15. 15.

    D. J. Gates and M. Westcott, Markov models of steady crystal growth,Ann. Appl. Prob. 3:339 (1993).

  16. 16.

    D. J. Gates and M. Westcott,J. Stat. Phys. 77:199 (1994).

  17. 17.

    P. G. de Gennes,J. Chem. Phys. 48:2257 (1968).

  18. 18.

    G. H. Gilmer,J. Crystal Growth 49:465–474 (1980).

  19. 19.

    G. H. Gilmer and K. A. Jackson,Crystal Growth and Materials, E. Kaldis and H. J. Scheel, eds. (North-Holland, Amsterdam, 1976), p. 80.

  20. 20.

    N. Goldenfeld,J. Phys. A 17:2807 (1984).

  21. 21.

    J. D. Hoffman, G. T. Davis, and J. I. Lauritzen, Jr., InTreatise on Solid-State Chemistry, Vol. 3.Crystalline and Non-crystalline Solids, N. B. Hanay ed. (Plenum Press, New York, 1976), p. 335.

  22. 22.

    J. D. Hoffman,Polymer 24:3 (1983).

  23. 23.

    B. Joos, T. L. Einstein, and N. C. Bartelt,Phys. Rev. B 43(10):8153 (1991).

  24. 24.

    S. Karlin and J. McGregor,Pac. J. Math. 19:1141 (1959).

  25. 25.

    F. Khoury and E. Passaglia, Ref. 21, p. 497.

  26. 26.

    M. Kotrla and M. Levi,J. Stat. Phys.,64:579 (1991).

  27. 27.

    J. I. Lauritzen, Jr.,J. Appl. Phys. 44:4353 (1973).

  28. 28.

    A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev,Integrals and Series, Vol. 1:Elementary Functions (Gordon and Breach, New York, 1986).

  29. 29.

    D. Ruelle,Statistical Mechanics (Benjamin New York, 1969).

  30. 30.

    D. M. Sadler,Polymer 28:1440 (1987).

  31. 31.

    J. Villain and P. Bak,J. Phys. (Paris)42:657 (1981).

  32. 32.

    J. D. Weeks and G. H. Gilmer,Adv. Chem. Phys. 40:157 (1979).

  33. 33.

    P. Whittle, Reversibility in Markov processes, Unpublished manuscript (1955).

  34. 34.

    P. Whittle, Systems in Stochastic Equilibrium (Wiley, Chichester, 1986).

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Gates, D.J., Westcott, M. Stationary states of crystal growth in three dimensions. J Stat Phys 81, 681–715 (1995). https://doi.org/10.1007/BF02179253

Download citation

Key Words

  • Crystal growth
  • Markov process
  • stationary distribution
  • growth rate
  • thermodynamic limit
  • cluster expansion
  • coincidence probability
  • simulation
  • solid-on-solid