A general model for spin coating on a non-axisymmetric curved substrate

TL;DR

This paper develops a comprehensive asymptotic model for thin film flow on complex curved surfaces during spin coating, incorporating effects like Coriolis force and disjoining pressure, and analyzes their influence on film spreading and fingering instability.

Contribution

It introduces a generalized model for spin coating on non-axisymmetric curved substrates, including effects previously neglected such as Coriolis force and disjoining pressure.

Findings

Coriolis force has minimal impact at low angular velocities on flat substrates.

At high angular velocities, fingers deflect against the rotation direction.

On curved substrates, increased angular velocity induces fingering dominated by centrifugal force.

Abstract

We derive a generalised asymptotic model for the flow of a thin fluid film over an arbitrarily-parameterised non-axisymmetric curved substrate surface based on the lubrication approximation. In addition to surface tension, gravity, and centrifugal force, our model incorporates the effects of the Coriolis force and disjoining pressure, together with a non-uniform initial condition, which have not been widely considered in existing literature. We use this model to investigate the impact of the Coriolis force and fingering instability on the spreading of a non-axisymmetric spin-coated film at a range of substrate angular velocities, first on a flat substrate, and then on parabolic cylinder- and saddle-shaped curved substrates. We show that, on flat substrates, the Coriolis force has a negligible impact at low angular velocities, and at high angular velocities results in a small deflection of fingers formed at the contact line against the direction of substrate rotation. On curved substrates, we demonstrate that as the angular velocity is increased, spin coated films transition from being dominated by gravitational drainage with no fingering to spreading and fingering in the direction with the greatest component of centrifugal force tangent to the substrate surface. For both curved substrates and all angular velocities considered, we show that the film thickness and total wetted substrate area remain similar over time to those on a flat substrate, with the key difference being the shape of the spreading droplet.