https://doi.org/10.1140/epjb/e2008-00020-6
Exact results for two-dimensional coarsening
1
Instituto de Física, Universidade Federal do
Rio Grande do Sul, CP 15051, 91501-970 Porto Alegre RS, Brazil
2
School of Physics and Astronomy, The University of Manchester,
Manchester M13 9PL, UK
3
Université Pierre et Marie Curie – Paris VI, LPTHE UMR 7589,
4 Place Jussieu, 75252 Paris Cedex 05, France
4
LPTENS UMR 8549 24, Rue Lhomond, 75321 Paris Cedex 05,
France
Corresponding author: a alan.bray@manchester.ac.uk
Received:
17
September
2007
Published online:
18
January
2008
We consider the statistics of the areas enclosed by domain
boundaries (`hulls') during the curvature-driven coarsening dynamics
of a two-dimensional nonconserved scalar field from a disordered
initial state. We show that the number of hulls per unit area,
, with enclosed area in the range (A, A+dA), is
described, for large time t, by the scaling form
, demonstrating the validity of dynamical
scaling in this system. Here
is a universal
constant associated with the enclosed area distribution of percolation
hulls at the percolation threshold, and
is a material
parameter. The distribution of domain areas,
, is apparently
very similar to that of hull areas up to very large values of
. Identical forms are obtained for coarsening from a
critical initial state, but with ch replaced by
. The
similarity of the two distributions (of areas enclosed by hulls, and
of domain areas) is accounted for by the smallness of ch. By
applying a `mean-field' type of approximation we obtain the form
, where t0 is a microscopic timescale and
, for a disordered initial state, and a
similar result for a critical initial state but with
and
. We also find that
and
. These
predictions are checked by extensive numerical simulations and found
to be in good agreement with the data.
PACS: 64.60.Cn – Order-disorder transformations; statistical mechanics of model systems / 64.60.Ht – Dynamic critical phenomena
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2008