Nonlinear dynamics of phase separation in thin films

TL;DR

This paper develops a mathematical model for phase separation in thin films, analyzing the coupled dynamics of free-surface variations and concentration changes, and investigates conditions preventing film rupture.

Contribution

It introduces a long-wavelength approximation of the Navier-Stokes Cahn-Hilliard equations with a substrate interaction potential, providing new insights into film stability and rupture mechanisms.

Findings

Theoretical lower bounds prevent rupture in certain conditions.

Numerical simulations show rupture occurs without repulsive interactions.

Formation of concentration-driven valleys indicates bubble formation.

Abstract

We present a long-wavelength approximation to the Navier-Stokes Cahn-Hilliard equations to describe phase separation in thin films. The equations we derive underscore the coupled behaviour of free-surface variations and phase separation. We introduce a repulsive substrate-film interaction potential and analyse the resulting fourth-order equations by constructing a Lyapunov functional, which, combined with the regularizing repulsive potential, gives rise to a positive lower bound for the free-surface height. The value of this lower bound depends on the parameters of the problem, a result which we compare with numerical simulations. While the theoretical lower bound is an obstacle to the rupture of a film that initially is everywhere of finite height, it is not sufficiently sharp to represent accurately the parametric dependence of the observed dips or `valleys' in free-surface height. We observe these valleys across zones where the concentration of the binary mixture changes sharply, indicating the formation of bubbles. Finally, we carry out numerical simulations without the repulsive interaction, and find that the film ruptures in finite time, while the gradient of the Cahn--Hilliard concentration develops a singularity.