110 research outputs found
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
Doubly Degenerate Diffuse Interface Models of Anisotropic Surface Diffusion
We extend the doubly degenerate Cahn-Hilliard (DDCH) models for isotropic
surface diffusion, which yield more accurate approximations than classical
degenerate Cahn-Hilliard (DCH) models, to the anisotropic case. We consider
both weak and strong anisotropies and demonstrate the capabilities of the
approach for these cases numerically. The proposed model provides a variational
and energy dissipative approach for anisotropic surface diffusion, enabling
large scale simulations with material-specific parameters.Comment: 15 pages; 6 figure
Nonlinear dynamics of phase separation in thin films
We present a long-wavelength approximation to the Navier-Stokes Cahn-Hilliard
equations to describe phase separation in thin films. The equations we derive
underscore the coupled behaviour of free-surface variations and phase
separation. We introduce a repulsive substrate-film interaction potential and
analyse the resulting fourth-order equations by constructing a Lyapunov
functional, which, combined with the regularizing repulsive potential, gives
rise to a positive lower bound for the free-surface height. The value of this
lower bound depends on the parameters of the problem, a result which we compare
with numerical simulations. While the theoretical lower bound is an obstacle to
the rupture of a film that initially is everywhere of finite height, it is not
sufficiently sharp to represent accurately the parametric dependence of the
observed dips or `valleys' in free-surface height. We observe these valleys
across zones where the concentration of the binary mixture changes sharply,
indicating the formation of bubbles. Finally, we carry out numerical
simulations without the repulsive interaction, and find that the film ruptures
in finite time, while the gradient of the Cahn--Hilliard concentration develops
a singularity.Comment: 26 pages, 20 figures, PDFLaTeX with RevTeX4 macros. A thorough
analysis of the equations is presented in arXiv:0805.103
A multiphase Cahn-Hilliard system with mobilities and the numerical simulation of dewetting
We propose in this paper a new multiphase Cahn-Hilliard model with doubly
degenerate mobilities. We prove by a formal asymptotic analysis that it
approximates with second order accuracy the multiphase surface diffusion flow
with mobility coefficients and surface tensions. To illustrate that it lends
itself well to numerical approximation, we propose a simple and effective
numerical scheme together with a very compact Matlab implementation. We provide
the results of various numerical experiments to show the influence of mobility
and surface tension coefficients. Thanks to its second order accuracy and its
good suitability for numerical implementation, our model is very handy for
tackling notably difficult surface diffusion problems. In particular, we show
that it can be used very effectively to simulate numerically the dewetting of
thin liquid tubes on arbitrary solid supports without requiring nonlinear
boundary conditions.Comment: 35 page
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