New exact solutions for the evaporation flux density of a small droplet on a flat horizontal substrate with a contact angle in the range of 135-180 degrees

TL;DR

This paper derives new exact solutions for the evaporation flux density of small droplets with contact angles between 135 and 180 degrees, simplifying modeling for specific hydrophobic cases.

Contribution

It provides explicit exact formulas for evaporation flux density at contact angles 135 and 150 degrees, extending previous complex integral-based models.

Findings

Explicit formulas for 135 and 150-degree contact angles

Simplified expressions suitable for modeling

Applicable as approximate solutions near specified angles

Abstract

Previously [arXiv:2103.15582v3], an expression was proposed for the evaporation flux density of a small liquid droplet having the shape of an axisymmetric spherical segment deposited on a horizontal substrate. The dependence of the flux density on the polar angle was established for arbitrary contact angles. This formula has the form of an integral and is rather complicated for use in modeling algorithms. An approximate expression was obtained for the evaporation flux density at small contact angles. However, the question of which simplified formulas should be appropriate to apply in other ranges of contact angles, for example, in the case of obtuse angles remains open. In this paper, we propose new exact solutions for the set of discrete "hydrophobic" contact angles. As an example, very simple exact expressions are obtained explicitly for the evaporation flux density for droplets with contact angles 135 and 150 degrees that do not contain integral dependencies. They can also be used as approximate solutions for a narrow range of contact angles around the specified values.