Higher-order phase transitions with line-tension effect

TL;DR

This paper investigates higher-order phase transitions influenced by line-tension effects at boundaries, revealing a connection between boundary layers and interior transition layers using a second-order Cahn-Hilliard model.

Contribution

It establishes the exact scaling law for boundary effects in a second-order perturbation model, linking boundary and interior transition layers in phase transition analysis.

Findings

Identifies the critical scaling regime where boundary and interior layers interact.

Derives the energy scaling law in the critical regime.

Connects boundary layer behavior with interior transition layers.

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 [21], and in a different form by Alberti, Bouchitte, and Seppecher in [2] 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 $$ \mathcal{F}_{\varepsilon}(u) := \varepsilon^{3} \int_{\Omega} |D^{2}u|^{2} + \frac{1}{\varepsilon} \int_{\Omega} W (u) + \lambda_{\varepsilon} \int_{\partial \Omega} V(Tu), $$ 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 $\varepsilon \lambda_{\varepsilon}^{{2/3}} \sim 1$.