Higher-order phase transitions with line-tension effect

Department of Mathematics and Statistics, McMaster University, Hamilton ON L8S 4K1, Canada. beni@mcmaster.ca

Received: 13 April 2009
Revised: 13 July 2009
Revised: 8 November 2009

Abstract

The behavior of energy minimizers at the boundary of the domain is of great importance in the Van de Waals-Cahn-Hilliard theory for fluid-fluid phase transitions, since it describes the effect of the container walls on the configuration of the liquid. This problem, also known as the liquid-drop problem, was studied by Modica in [Ann. Inst. Henri Poincaré, Anal. non linéaire 4 (1987) 487–512], and in a different form by Alberti et al. in [Arch. Rational Mech. Anal. 144 (1998) 1–46] for a first-order perturbation model. This work shows that using a second-order perturbation Cahn-Hilliard-type model, the boundary layer is intrinsically connected with the transition layer in the interior of the domain. Precisely, considering the energies

where u is a scalar density function and W and V are double-well potentials, the exact scaling law is identified in the critical regime, when .