Coarsening rates in off-critical mixtures

We study coarsening of a binary mixture within the Mullins--Sekerka evolution in the regime where one phase has small volume fraction $\phi \ll1$. Heuristic arguments suggest that the energy density, which represents the inverse of a typical length scale, decreases as $\phi t^{-1/3}$ as $t \to \inft...

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Bibliographic Details
Published in:SIAM journal on mathematical analysis 2006-01, Vol.37 (6), p.1732-1741
Main Authors: CONTI, Sergio, NIETHAMMER, Barbara, OTTO, Felix
Format: Article
Language:English
Subjects:
Energy
Exact sciences and technology
Heat treating
Phase transitions: general studies
Physics
Statistical physics, thermodynamics, and nonlinear dynamical systems
Thermodynamics
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Summary:We study coarsening of a binary mixture within the Mullins--Sekerka evolution in the regime where one phase has small volume fraction $\phi \ll1$. Heuristic arguments suggest that the energy density, which represents the inverse of a typical length scale, decreases as $\phi t^{-1/3}$ as $t \to \infty$. We prove rigorously a corresponding weak lower bound. Moreover, we establish a stronger result for the two-dimensional case, where we find a lower bound of the form $ \phi(\ln \phi^{-1})^{1/3}t^{-1/3}$. Our approach follows closely the analysis in [R. V. Kohn and F. Otto, Comm. Math. Phys., 229 (2002), pp. 375-395], which exploits the relation between two suitable length scales. Our main contribution is an isoperimetric inequality in the two-dimensional case.
ISSN:0036-1410
1095-7154
DOI:10.1137/040620059