We make the connection between the geometric model for capillarity with line tension and the Cahn-Hilliard model of two-phase fluids. To this aim we consider the energies $$ F_\epsilon(u) := \epsilon \int_\Omega |Du|^2 + {1\over\epsilon} \int_\Omega W(u) + \lambda \int_{\partial\Omega} V(u) $$ where $u$ is a scalar density function on a $3$-dimensional domain $\Omega$ and $W$ and $V$ are double-well potentials. We show that the behavior of $F_\epsilon$ in the limit $\epsilon\to 0$ and $\lambda\to\infty$ depends on the limit of $\epsilon \log\lambda$. If this limit is finite and strictly positive, then the singular limit of the energies $F_\epsilon$ lead to a coupled problem of bulk and surface phase transitions, and under certain assumptions agrees with the relaxation of the capillary energy with line tension.
Phase transition with line-tension effect
ALBERTI, GIOVANNI;
1998-01-01
Abstract
We make the connection between the geometric model for capillarity with line tension and the Cahn-Hilliard model of two-phase fluids. To this aim we consider the energies $$ F_\epsilon(u) := \epsilon \int_\Omega |Du|^2 + {1\over\epsilon} \int_\Omega W(u) + \lambda \int_{\partial\Omega} V(u) $$ where $u$ is a scalar density function on a $3$-dimensional domain $\Omega$ and $W$ and $V$ are double-well potentials. We show that the behavior of $F_\epsilon$ in the limit $\epsilon\to 0$ and $\lambda\to\infty$ depends on the limit of $\epsilon \log\lambda$. If this limit is finite and strictly positive, then the singular limit of the energies $F_\epsilon$ lead to a coupled problem of bulk and surface phase transitions, and under certain assumptions agrees with the relaxation of the capillary energy with line tension.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
