Scaling functions, self-similarity, and the morphology of phase-separating systems

P. Fratzl, J. L. Lebowitz, O. Penrose, and J. Amar
Phys. Rev. B 44, 4794 – Published 1 September 1991
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Abstract

In the late stages of phase separation in liquids or solids with negligible coherency stresses, the structure function S(k,t) is known to follow a scaling behavior in the form S(k,t)∼km3(t)F(k/km(t)), where km(t) is the value of k that maximizes S at a given t. Previous work has shown that, for many real systems and for three-dimensional computer models, the scaling function F(x) depends only on the volume fraction φ of the minority phase but not on the temperature T for a given φ. Results from a Monte Carlo simulation of the two-dimensional Ising model, and also from a recently published numerical solution of the two-dimensional Cahn-Hilliard equation, are shown here to give a scaling function that can be fitted, as in the three-dimensional case, by an analytical expression containing just one adjustable parameter γ̃, independent of T but dependent on φ. We analyze and interpret some universal features of these scaling functions, including their behavior at small x and at large x, and their dependence on φ. Our discussion is based on a two-phase model, i.e., a mixture of two types of domains separated by thin interfaces, with kinetics based on the Cahn-Hilliard equation. We introduce an assumption of self-similar evolution (in the sense of self-similar probability ensembles) and show that it leads to the well-known t1/3 growth rate for the average domain size and to the above-mentioned universal properties of the scaling function. Simple geometric considerations also allow the calculation of the parameter γ̃, so that the scaling function may be obtained without any adjustment of parameters. The influence of droplet-size distributions on the scaling function, the limit of very dilute alloys, and the temperature dependence of the coarsening rate are also considered.

  • Received 29 March 1991

DOI:https://doi.org/10.1103/PhysRevB.44.4794

©1991 American Physical Society

Authors & Affiliations

P. Fratzl

  • Departments of Mathematics and Physics, Rutgers University, Busch Campus, New Brunswick, New Jersey 08903
  • Institut für Festkörperphysik der Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria

J. L. Lebowitz

  • Departments of Mathematics and Physics, Rutgers University, Busch Campus, New Brunswick, New Jersey 08903

O. Penrose

  • Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4A5, United Kingdom

J. Amar

  • Department of Physics, Emory University, Atlanta, Georgia 30322

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Vol. 44, Iss. 10 — 1 September 1991

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