A generalized Young's equation to bridge a gap between the experimentally measured and the theoretically calculated line tensions

TL;DR

This paper derives a generalized Young's equation incorporating curvature and contact angle corrections to better reconcile experimental and theoretical line tension measurements, highlighting the importance of these factors in nanoscale and larger droplets.

Contribution

It introduces a comprehensive correction to the classical Young's equation, accounting for curvature and contact angle effects, to improve the accuracy of line tension comparisons.

Findings

Corrections can be as small as nN, affecting apparent line tension sign.

Gravitational line tension is negligible for small droplets but significant for millimeter-scale droplets.

Naive comparisons between experimental and theoretical line tensions can be misleading without these corrections.

Abstract

A generalized Young's equation, which takes into account two corrections to the line tension by the curvature dependence of the liquid-vapor surface tension and by the contact angle dependence of the intrinsic line tension, is derived from the thermodynamic free-energy minimization. The correction from the curvature dependence can be qualitatively estimated using Tolman's formula. The correction from the contact angle dependence can be estimated for nanometer-scale droplets for which the analytical formula for the intrinsic line tension determined from the van der Waals interaction is available. The two corrections to the apparent line tension of this van der Waals nano-droplets are as small as nN, and lead to either a positive or a negative apparent line tension. The gravitational line tension for millimeter-scale droplets by the gravitational acceleration is also considered. The gravitational line tension is of the order of $\mu$N so that the correction from the curvature dependence can be neglected. Yet, the contact angle dependence is so large that the apparent line tension becomes always negative though the intrinsic line tension without the correction is always positive. These two examples demonstrate clear distinction between the theoretical calculated intrinsic line tension and the experimentally determined apparent line tension which includes these two corrections. Naive comparison of the experimentally determined and the theoretically calculated line tension is not always possible.