Abstract
The classical Lifshitz–Slyozov–Wagner theory of domain coarsening predicts asymptotically self-similar behavior for the size distribution of a dilute system of particles that evolve by diffusional mass transfer with a common mean field. Here we consider the long-time behavior of measure-valued solutions for systems in which particle size is uniformly bounded, i.e., for initial measures of compact support. We prove that the long-time behavior of the size distribution depends sensitively on the initial distribution of the largest particles in the system. Convergence to the classically predicted smooth similarity solution is impossible if the initial distribution function is comparable to any finite power of distance to the end of the support. We give a necessary criterion for convergence to other self-similar solutions, and conditional stability theorems for some such solutions. For a dense set of initial data, convergence to any self-similar solution is impossible.
Similar content being viewed by others
REFERENCES
C. Berg and G. Forst, Potential Theory on Locally Compact Abelian Groups (Springer-Verlag, New York, 1975).
N. H. Bingham, C.M. Goldie, and J. L. Teugels, Regular Variation, Encycl. Math. Appl., Vol. 27 (Cambridge University Press, Cambridge, 1987).
L. C. Brown, A new examination of classical coarsening theory, Acta Metall. 37: 71–77 (1989).
L. C. Brown, Reply to further comments by Hillert, Hunderi and Ryum on “A new examination of classical coarsening theory, ” Scripta Metall. Mater. 26:1939–1942 (1992).
L. C. Brown, A new examination of volume fraction effects during particle coarsening, Acta Metall. Mater. 40:1293–1303 (1992).
J. Carr and O. Penrose, Asymptotic behavior in a simplified Lifshitz–Slyozov equation, Physica D 124:166–176 (1998).
J.-F. Collet and T. Goudon, On solutions of the Lifshitz–Slyozov model, preprint.
B. Giron, B. Meerson, and P. V. Sasorov, Weak selection and stability of localized distributions in Ostwald ripening, Phys. Rev. E 58:4213–6 (1998).
I. M. Lifshitz and V. V. Slyozov, The kinetics of precipitation from supersaturated solid solutions, J. Phys. Chem. Solids 19:35–50 (1961).
B. Meerson and P. V. Sasorov, Domain stability, competition, growth and selection in globally constrained bistable systems, Phys. Rev. E 53:3491–4 (1996).
B. Niethammer, Derivation of the LSW theory for Ostwald ripening by homogenization methods, to appear in Arch Rat. Mech. Anal.
B. Niethammer and R. L. Pego, On the initial value problem in the Lifshitz–Slyozov–Wagner theory of Ostwald ripening, to appear in SIAM J. Math. Anal.
O. Penrose, The Becker–Do–ring equations at large times and their connection with the LSW theory of coarsening, J. Stat. Phys. 89:305–320 (1997).
E. Seneta, Regularly Varying Functions, Lec. Notes in Math., Vol. 508 (Springer-Verlag, New York, 1976).
P. W. Voorhees, The theory of Ostwald ripening, J. Stat. Phys. 38:231–252 (1985).
P. W. Voorhees, Ostwald ripening of two-phase mixtures, Ann. Rev. Mater. Sci. 22:197–215 (1992).
C. Wagner, Theorie der Alterung von Niederschlägen durch Umlösen, Z. Elektrochem. 65:581–594 (1961).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Niethammer, B., Pego, R.L. Non-Self-Similar Behavior in the LSW Theory of Ostwald Ripening. Journal of Statistical Physics 95, 867–902 (1999). https://doi.org/10.1023/A:1004546215920
Issue Date:
DOI: https://doi.org/10.1023/A:1004546215920