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

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 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 theoretically 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.

Keywords:

• Line tension
• contact angle
• Young’s formula
• Tolman’s formula

Acknowledgements

A part of this work was conducted as a visiting scientist at the Department of Physics, Tokyo Metropolitan University. The author is grateful to Prof. Hiroyuki Mori and Prof. Yutaka Okabe (Tokyo Metropolitan Univ.) for continuous support and encouragement. The author is also grateful to Dr. Lothar Schimmele and Prof. Siegfried Dietrich (Max-Planck Institute for Intelligent Systems, Stuttgart) for their useful comments on the initial version of the manuscript. He is also grateful to Prof. Dmitry V. Tatyanenko and Prof. Alexander K. Shchekin (St. Petersburg State Univ.) for useful discussion about their new result in reference [48]. He also thanks to the two reviewers for their useful suggestions.

Notes

No potential conflict of interest was reported by the author.