Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T11:03:51.514Z Has data issue: false hasContentIssue false

On the van der Waals–Cahn–Hilliard phase-field model and its equilibria conditions in the sharp interface limit

Published online by Cambridge University Press:  08 December 2010

Wolfgang Dreyer
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany (dreyer@wias-berlin.de; kraus@wias-berlin.de)
Christiane Kraus
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany (dreyer@wias-berlin.de; kraus@wias-berlin.de)

Abstract

We study the thermodynamic consistency of phase-field models, which include gradient terms of the density ρ in the free-energy functional such as the van der Waals–Cahn–Hilliard model. It is well known that the entropy inequality admits gradient and higher-order gradient terms of ρ in the free energy only if either the energy flux or the entropy flux is represented by a non-classical form. We identify a non-classical entropy flux, which is not restricted to isothermal processes, so that gradient contributions are possible.

We then investigate equilibrium conditions for the van der Waals–Cahn–Hilliard phase-field model in the sharp interface limit. For a single substance thermodynamics provides two jump conditions at the sharp interface, namely the continuity of the Gibbs free energies of the adjacent phases and the discontinuity of the corresponding pressures, which is balanced by the mean curvature. We show that these conditions can be also extracted from the van der Waals–Cahn–Hilliard phase-field model in the sharp interface limit. To this end we prove an asymptotic expansion of the density up to the first order. The results are based on local energy estimates and uniform convergence results for the density.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)