Advertisement

Statistical Physics Theory of Ostwald Ripening

  • Michio Tokuyama

Abstract

We review recent theoretical developments in our understanding of the late-stage processes of phase separation in binary alloys. When a system is quenched into a metastable state, phase separation occurs by nucleation and growth1,2 in the late stage, known as Ostwald ripening3, the minority phase takes the form of spherical droplets whose growth and dissolution proceeds by an evaporation-condensation mechanism. There are two theoretical aspects in understanding of the dynamics of such a phase separation, depending on what processes we are interested in. The first is to study the causal motion which is described by a single droplet size distribution function f(R,t) with radius R, and which is experimentally observable by an electron microscope. This was first done in the monumental works by Lifshitz and Slyozov4 and independently by Wagner5. They found the celebrated scaling law f(R,t)= [n(t)/R(t)]po(R/R(t)), where the average droplet radius grows as R(t)~t1/3 and the number density of droplets decays as n(t)~t~l. The relative droplet size distribution function po(p) is a time-independent function of p. Although their works were the origin or later theoretical studies of Ostwald ripening, their results were valid only in the limit of zero volume fraction of the minority phase and did not agree with experimental observations where the volume fraction is not infinitely zero. After their works, many attempts6–10 to extend their theory to the case of finite volume fraction have been proposed.

Keywords

Structure Function Spherical Droplet Minority Phase Recent Theoretical Development Hierarchy Equation 
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.
    J.D. Gunton, M, San Miguel and P.S. Shani, in: “Phase Transition and Critical Phenomena,” vol.8, C. Domb and J.L. Lebowitz, eds. Academic press, London,(1983).Google Scholar
  2. 2.
    K. Binder and D. Stauffer, Phys. Rev. Lett. 33:1006 (1974).CrossRefGoogle Scholar
  3. 3.
    W. Ostwald, Z. Phys. Chem. 37:385 (1901).Google Scholar
  4. 4.
    I.M. Lifshitz and V.V. Slyozov, J. Phys. Chem. Solids 19:35 (1961)CrossRefGoogle Scholar
  5. 5.
    C. Wagner, Z. Electrochem. 65:581 (1961).Google Scholar
  6. 6.
    A.J. Ardell, Acta. Met. 20:61 (1972).CrossRefGoogle Scholar
  7. 7.
    J.J. Wein and J.W. Cahn, Materals Research 6:151 (1973).Google Scholar
  8. 8.
    A.D. Brailsford and P. Wynblatt, Acta. Met. 27:489 (1979).CrossRefGoogle Scholar
  9. 9.
    P.W. Voorhees and M.E. Glicksman. Acta. Met. 32:2001,2013 (1984).Google Scholar
  10. 10.
    J.A. Marqusee and J. Ross, J. Chem. Phys. 80:536 (1984).CrossRefGoogle Scholar
  11. 11.
    J.L. Lebowitz, J. Marro and M.H. Kalos, Acta. Met. 30:297 (1982).CrossRefGoogle Scholar
  12. 12.
    M. Hennion, D. Ronzaud and P. Guyot, Acta. Met. 30:599 (1982).CrossRefGoogle Scholar
  13. 13.
    P. Fratzl, J.L. Lebowitz, J. Marro and M.H. Kalos, Acta. Met. 31:1849 (1983).CrossRefGoogle Scholar
  14. 14.
    S. Katano and M. Iizumi, Phys. Rev. Lett. 52:835 (1984).CrossRefGoogle Scholar
  15. 15.
    S. Komura, K. Osamura, H. Fujii and T. Takeda, Phys. Rev. B31:1278 (1985).CrossRefGoogle Scholar
  16. 16.
    K. Osamura, H. Okuda and S. Ochiai, Preprint.Google Scholar
  17. 17.
    H. Furukawa, Prog. Theor. Phys. 59:1072 (1978).CrossRefGoogle Scholar
  18. 18.
    P.A. Rikvold and J.D. Gunton, Phys. Rev. Lett. 49:286 (1982).CrossRefGoogle Scholar
  19. 19.
    T. Ohta, Ann. of Phys. 158:31 (1984).CrossRefGoogle Scholar
  20. 20.
    H. Tomita, Prog. Theor. Phys. 71:1405 (1984).CrossRefGoogle Scholar
  21. 21.
    M. Tokuyama and K. Kawasaki, Physica 123A:386 (1984).CrossRefGoogle Scholar
  22. 22.
    M. Tokuyama and R.I. Cukier, J. Chem. Phys. 76:6202 (1982).CrossRefGoogle Scholar
  23. 23.
    K. Kawasaki and T. Ohta, Physica, 118A:175 (1983).CrossRefGoogle Scholar
  24. 24.
    C.W.J. Beenakker, Preprint.Google Scholar
  25. 25.
    Y. Enomoto, K. Kawasaki and M. Tokuyama, Acta. Met. 35:907 (1987).CrossRefGoogle Scholar
  26. 26.
    M. Tokuyama, K. Kawasaki and Y. Enomoto, Physica, 134A:323(1986).CrossRefGoogle Scholar
  27. 27.
    Y. Enomoto, K. Kawasaki and M. Tokuyama, Acta. Met. 35:915 (1987).CrossRefGoogle Scholar
  28. 28.
    Y. Enomoto, M. Tokuyama and K. Kawasaki, Acta. Met. 34:2119(1986).CrossRefGoogle Scholar
  29. 29.
    M. Tokuyama, Y. Enomoto and K. Kawasaki, Physica 143A:183 (1987).CrossRefGoogle Scholar
  30. 30.
    P.K. Rastogi and A.J. Ardell, Acta. Met. 19:321 (1971).CrossRefGoogle Scholar
  31. 31.
    T. Hirata and D.H. Kirkwood, Acta. Met. 25:1425 (1977).CrossRefGoogle Scholar
  32. 32.
    T. Eguchi, Y. Tomokiyo and S. Matsumura, Preprint.Google Scholar
  33. 33.
    K. Osamura, these proceedingsGoogle Scholar

Copyright information

© Plenum Press, New York 1988

Authors and Affiliations

  • Michio Tokuyama
    • 1
  1. 1.General EducationTohwa UniversityFukuoka 815Japan

Personalised recommendations