Relaxation and Intermediate Asymptotics of a Rectangular Trench in a Viscous Film

TL;DR

This paper investigates the capillary-driven leveling dynamics of rectangular trenches in viscous nanofilms, revealing how boundary conditions influence the evolution, with theoretical and experimental agreement on self-similar profiles and energy dissipation crossover.

Contribution

It provides a comprehensive analysis of how boundary conditions affect trench leveling in viscous films, including a crossover in dissipation behavior and universal self-similar attractors.

Findings

Theoretical and experimental agreement on height profiles and flow dynamics.

Identification of a crossover in energy dissipation from noninteracting to interacting steps.

Profiles converge to a universal self-similar attractor.

Abstract

The surface of a thin liquid film with nonconstant curvature flattens as a result of capillary forces. While this leveling is driven by local curvature gradients, the global boundary conditions greatly influence the dynamics. Here, we study the evolution of rectangular trenches in a polystyrene nanofilm. Initially, when the two sides of a trench are well separated, the asymmetric boundary condition given by the step height controls the dynamics. In this case, the evolution results from the leveling of two noninteracting steps. As the steps broaden further and start to interact, the global symmetric boundary condition alters the leveling dynamics. We report on full agreement between theory and experiments for: the capillary-driven flow and resulting time dependent height profiles; a crossover in the power-law dependence of the viscous energy dissipation as a function of time as the trench evolution transitions from two noninteracting to interacting steps; and the convergence of the profiles to a universal self-similar attractor that is given by the Green's function of the linear operator describing the dimensionless linearized thin film equation.