WIAS Preprint No. 1103, (2006)

The sharp interface limit of the van der Waals--Cahn--Hilliard phase model for fixed and time dependent domains



Authors

  • Dreyer, Wolfgang
  • Kraus, Christiane

2010 Mathematics Subject Classification

  • 82B26 49Q05 35R35

Keywords

  • Van der Waals-Cahn-Hilliard theory of phase transitions, two-phase fluid, asymptotic expansion of the density, local energy estimates, mechanical equilibrium condition, phase equilibrium condition, Gibbs-Thompson relation, surface tension, curvature, perimeter, minimal area, entropy, thermodynamic consistency

Abstract

We study the equilibria of liquid--vapor phase transitions of a single substance at constant temperature and relate the sharp interface model of classical thermodynamics to a phase field model that determines the equilibria by the stationary van der Waals--Cahn--Hilliard theory.
For two reasons we reconsider this old problem. 1. Equilibria in a two phase system can be established either under fixed total volume of the system or under fixed external pressure. The latter case implies that the domain of the two--phase system varies. However, in the mathematical literature rigorous sharp interface limits of phase transitions are usually considered under fixed volume. This brings the necessity to extend the existing tools for rigorous sharp interface limits to changing domains since in nature most processes involving phase transitions run at constant pressure. 2. Thermodynamics provides for a single substance two jump conditions at the sharp interface, viz. the continuity of the specific Gibbs free energies of the adjacent phases and the discontinuity of the corresponding pressures, which is balanced by the mean curvature. The existing estimates for rigorous sharp interface limits show only the first condition. We identify the cause of this phenomenon and develop a strategy that yields both conditions up to the first order.
The necessary information on the equilibrium conditions are achieved by an asymptotic expansion of the density which is valid for an arbitrary double well potential. We establish this expansion by means of local energy estimates, uniform convergence results of the density and estimates on the Laplacian of the density.

Appeared in

  • Proc. Roy. Soc. Edinburgh Sect. A, 140 A (2010) pp. 1161--1186.

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