Dynamics of fixed-volume pinned film -- dealing with a non-self-adjoint thin film problem

TL;DR

This paper presents a theoretical and experimental analysis of the unique dynamics of finite, pinned thin liquid films, highlighting the importance of boundary conditions and non-self-adjoint problems in accurately modeling their behavior.

Contribution

It provides the first time-dependent analytical solution for the linearized non-self-adjoint thin film problem with pinning boundary conditions and validates it through experiments.

Findings

Boundary conditions significantly influence film response time and deformation.

The analytical solution matches experimental measurements of confined film dynamics.

Non-self-adjoint modeling is essential for accurate description of finite film behavior.

Abstract

The use of thin liquid films has expanded beyond lubrication and coatings, and into applications in actuators and adaptive optical elements. In contrast to their predecessors, whose dynamics can be typically captured by modelling infinite or periodic films, these applications are characterized by a finite amount of liquid in an impermeable domain. The global mass conservation constraint, together with common boundary conditions (e.g., pinning) create quantitatively and qualitatively different dynamics than those of infinite films. Mathematically, this manifests itself as a non-self-adjoint problem. This work presents a combined theoretical and experimental study for this problem. We provide a time-dependent closed-form analytical solution for the linearized non-self-adjoint system that arises from these boundary conditions. We highlight that, in contrast to self-adjoint problems, here special care should be given to deriving the adjoint problem to reconstruct the solution based on the eigenfunctions properly. We compare these solutions with those obtained for permeable and periodic boundary conditions, representing common models for self-adjoint thin-film problems. We show that while the initial dynamics are nearly identical, the boundary conditions eventually affect the film deformation as well as its response time. To experimentally illustrate the dynamics and to validate the theoretical model, we fabricated an experimental setup that subjects a thin liquid film to a prescribed normal force distribution through dielectrophoresis, and used high-frame-rate digital holography to measure the film deformation in real-time. The experiments agree well with the model and confirm that confined films exhibit different behaviour which could not be predicted by existing models.