# Effects of geometric confinement on a droplet between two parallel planes

## TL;DR

This paper investigates how geometric confinement between two parallel planes influences droplet morphology and wetting behavior, providing analytical and asymptotic solutions that enhance understanding of confined droplet physics.

## Contribution

It develops explicit asymptotic expressions for droplet shape and physical parameters under various confinement regimes, advancing analytical understanding of wetting in confined geometries.

## Key findings

- Analytical expressions for droplet shape valid for arbitrary contact angles.
- Asymptotic formulas for Laplace pressure, volume, and surface energy in different regimes.
- Identification of logarithmic behaviors in liquid Hertzian contact.

## Abstract

When a droplet (which size is characterized by R) is confined between two parallel planes, its morphology will change accordingly to either varying the volume of the droplet or the separation (characterized by h) between the planes. We are aiming at investigating how such a geometric confinement affects the wetting behaviours of a droplet. Our focus lies on two distinguished regimes: (1) a pancake shape in a Hele-Shaw cell when the droplet is highly compressed (i.e. h/R << 1), in which particular attention is paid on nonwetting and wetting cases, respectively; and (2) a liquid Hertzian contact rendered by a slight confinement (i.e. h/(2R) --> 1) in a nonwetting case. To realize this aim, we first develop strict analytical expressions of the shape of the droplet which are available for arbitrary contact angles between the liquid and the solid planes, but in which the elliptic integrals indicate that these solutions are implicit. By employing asymptotic methods, we are able to give expressions of relevant geometrical and physical parameters (the Laplace pressure, droplet volume, surface energy and external force) in terms of sole functions of R and h in an explicit manner. Comparisons suggest that over a large range of h/R, our asymptotic results quantitatively agree well with the numerical solutions of the analytical expressions. This systematic study of the parameter space allows a comprehensive understanding of the geometric confinement on wetting, especially a wide existence of logarithmic behaviours in a liquid Hertzian contact, to be identified.