Skip to main content
SpringerLink
Log in

The Equations of Ostwald Ripening for Dilute Systems

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We consider a dilute mixture in 3D of a finite number of particles initially close to spherical, but of varying sizes, and representing one of the phases evolving according to the quasistatic dynamics. Under the scaling hypotheses that (1) typical size/typical distance and (2) deviation from sphericity/typical size are small, we associate centers and radii to each particle for the whole evolution and derive rigorously a set of ODEs fo the radii which we relate to the Lifschitz–Slyosov–Wagner theory of coarsening.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. W. Ostwald, Z. Phys. Chem. 37:385 (1901).

    Google Scholar 

  2. J. Rubinstein and P. Sternberg, IMA J. Appl. Math. 48:249–264 (1992).

    Google Scholar 

  3. I. M. Lifschitz and V. V. Slyosov, J. Phys. Chem. Solids 19:35–50 (1961).

    Google Scholar 

  4. C. Wagner, Z. Electrochem. 65:581–594 (1961).

    Google Scholar 

  5. N. D. Alikakos and G. Fusco, preprint (1998).

  6. B. Niethammer, University of Bonn Technical Report 453 (1996).

  7. J. W. Cahn, Acta Metall. 9:795–801 (1961).

    Google Scholar 

  8. J. W. Cahn and J. E. Billiard, J. Phys. Chem. 28:258–267 (1958).

    Google Scholar 

  9. R. L. Pego, Proc. Roy. Soc. London, Ser. A 422:261–278 (1989).

    Google Scholar 

  10. N. D. Alikakos, P. Bates, and X. Chen, Arch. Rat. Mech. Anal. 128:165–205 (1994).

    Google Scholar 

  11. W. W. Mullins and R. F. Sekerka, J. Appl. Phys. 34:323–329 (1963).

    Google Scholar 

  12. V. Alexiades, and A. D. Solomon, Mathematical Modeling of Melting and Freezing Processes, hemisphere Publishing Corporation (1993).

  13. X. Chen, X. Hong, and F. Yi, preprint (1997).

  14. X. Chen, Arch. Rational Mech. Anal. 123:117–151.

  15. P. Constantine and M. Pugh, Nonlinearity 6:393–415 (1993).

    Google Scholar 

  16. J. Escher and G. Simonett, J.D.E. 143:267–292 (1998).

    Google Scholar 

  17. X. Chen, Journal of Differential Geometry (to appear).

  18. J. Escher and G. Simonett, Advances in Differential Equations 2:619–642 (1997).

    Google Scholar 

  19. N. D. Alikakos, J. Escher, G. Fusco, and G. Simonett (in progress).

  20. P. Fife, Dynamical Aspects of the Cahn–Hilliard Equations (Barrett lectures, University of Tennessee, Spring 1991).

  21. G. Bellettini and G. Fusco, preprint (1997).

  22. N. D. Alikakos, P. W. Bates, X. Chen, and G. Fusco, preprint (1997).

  23. G. Huisken and S. T. Yau, Inventiones Mathematicae 281–311 (1996).

  24. R. Ye, Pacific Journal of Mathematics 147:381–396 (1991).

    Google Scholar 

  25. G. Huisken, J. Reine Angew. Math. 382:35–48 (1987).

    Google Scholar 

  26. C. Miranda, Lincei-Memorie Sc. Fisiche, esc.––S.VIII VII, Sez. 1, 9 303–336 (1965).

    Google Scholar 

  27. S. Agmon, A. Douglis, and L. Nirenberg, Comm. Pure Appl. Math. 12:623–727 (1959).

    Google Scholar 

  28. G. Da Prato and P. Grisvard, Ann. Mat. Pura Appl. (4) 120:329–396 (1979).

    Google Scholar 

  29. A. Lunardi, Analytic Semigroups and Optimal Regularity for Parabolic Problems (Birkhauser, Basel, 1995).

    Google Scholar 

  30. H. Amann, Linear and Quasilinear Parabolic Problems, Vol. I (Birkhauser, Basel, 1995).

    Google Scholar 

  31. S. B. Angenent, Proc. Roy. Soc. Edinburgh A 115:91–107 (1990).

    Google Scholar 

  32. J. Zhu, X. Chen and T. X. Hou, J. Comput. Phys. 126 (1996).

  33. O. Penrose, J. L. Lebowitz, J. Murro, M. H. Kallos, and A. Sur, J. Stat. Phys. 19:243–267 (1978).

    Google Scholar 

  34. M. Kalos, J. L. Lebowitz, O. Penrose, and A. Sur, J. Stat. Phys. 18:39–51 (1978).

    Google Scholar 

  35. M. Soner.

  36. P. W. Voorhees, Journal of Statistical Physics 38:231–252 (1985); Ann. Rev. Mater. Sc. 22:197–215 (1992).

    Google Scholar 

  37. P. W. Voorhees and M. E. Glicksman, Theory, Acta Metall. 32:2001–2011 (1984).

    Google Scholar 

  38. B. Niethammer and R. Pego, Non-Self-Similar Behavior in the LSW theory of Ostwald Ripening, preprint, 1998.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Tennessee, Knoxville, Tennessee, 37996-1300, and

    Nicholas D. Alikakos

  2. University of Athens, Panepistimiopolis, GR-15784, Athens, Greece

    Nicholas D. Alikakos

  3. Dipartimento di Matematica, Universita Di l', Aquila, Italy

    Giorgio Fusco

Authors
  1. Nicholas D. Alikakos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giorgio Fusco
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Alikakos, N.D., Fusco, G. The Equations of Ostwald Ripening for Dilute Systems. Journal of Statistical Physics 95, 851–866 (1999). https://doi.org/10.1023/A:1004594131850

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1004594131850

Search

Navigation