1,440 research outputs found
Coarsening Dynamics of a One-Dimensional Driven Cahn-Hilliard System
We study the one-dimensional Cahn-Hilliard equation with an additional
driving term representing, say, the effect of gravity. We find that the driving
field has an asymmetric effect on the solution for a single stationary
domain wall (or `kink'), the direction of the field determining whether the
analytic solutions found by Leung [J.Stat.Phys.{\bf 61}, 345 (1990)] are
unique. The dynamics of a kink-antikink pair (`bubble') is then studied. The
behaviour of a bubble is dependent on the relative sizes of a characteristic
length scale , where is the driving field, and the separation, ,
of the interfaces. For the velocities of the interfaces are
negligible, while in the opposite limit a travelling-wave solution is found
with a velocity . For this latter case () a set of
reduced equations, describing the evolution of the domain lengths, is obtained
for a system with a large number of interfaces, and implies a characteristic
length scale growing as . Numerical results for the domain-size
distribution and structure factor confirm this behavior, and show that the
system exhibits dynamical scaling from very early times.Comment: 20 pages, revtex, 10 figures, submitted to Phys. Rev.
Coarsening versus pattern formation
It is known that similar physical systems can reveal two quite different ways
of behavior, either coarsening, which creates a uniform state or a large-scale
structure, or formation of ordered or disordered patterns, which are never
homogenized. We present a description of coarsening using simple basic models,
the Allen-Cahn equation and the Cahn-Hilliard equation, and discuss the factors
that may slow down and arrest the process of coarsening. Among them are pinning
of domain walls on inhomogeneities, oscillatory tails of domain walls, nonlocal
interactions, and others. Coarsening of pattern domains is also discussed.Comment: 14 pages. To appear in a Comptes Rendus Physique special issue on
"Coarsening Dynamics", see
https://sites.google.com/site/ppoliti/crp-special-issu
Convective nonlocal Cahn-Hilliard equations with reaction terms
We introduce and analyze the nonlocal variants of two Cahn-Hilliard type
equations with reaction terms. The first one is the so-called
Cahn-Hilliard-Oono equation which models, for instance, pattern formation in
diblock-copolymers as well as in binary alloys with induced reaction and type-I
superconductors. The second one is the Cahn-Hilliard type equation introduced
by Bertozzi et al. to describe image inpainting. Here we take a free energy
functional which accounts for nonlocal interactions. Our choice is motivated by
the work of Giacomin and Lebowitz who showed that the rigorous physical
derivation of the Cahn-Hilliard equation leads to consider nonlocal
functionals. The equations also have a transport term with a given velocity
field and are subject to a homogenous Neumann boundary condition for the
chemical potential, i.e., the first variation of the free energy functional. We
first establish the well-posedness of the corresponding initial and boundary
value problems in a weak setting. Then we consider such problems as dynamical
systems and we show that they have bounded absorbing sets and global
attractors
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