Generalized Young equation for a spherical droplet inside a smooth and homogeneous cone involved by quadratic parabola

Long Zhou, Guang-Hua Sun, Guo-Qiang Chen, Ai-Jun Hu, Xiao-Song Wang


We thermodynamically investigate the wetting characteristics of a spherical droplet in a smooth and homogeneous cone rotated by the quadratic parabola through the mechanisms of both Gibbs’s dividing surfaces and Rusanov’s dividing line. For the triple phase system including the solid, liquid and vapor phases, the derivation of a generalized Young equation containing the influences of the line tension is successfully carried out. Additionally, we as well analyze various approximate forms for this generalized Young equation by using the corresponding assumptions.


Liquid droplet; generalized Young equation; line tension; quadratic parabola; contact angle; cone

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T. Young, “An essay on the cohesion of fluids”, Philos. Trans. Roy. Soc. London, 95, 65-87 (1805)

J. S. Rowlinson and B. Widom, Molecular theory of capillarity (Clarendon Press, Oxford, 1982)

J. W. Gibbs, On the Equilibrium of Heterogeneous Substances (Yale University Press, New Haven, 1961) p326-327

J. Drelich, J. L. Wilbur, J. D. Miller, and G. M. Whitesides, “Contact angles for liquid drops at a model heterogeneous surface consisting of alternating and parallel hydrophobic/hydrophilic strips”, Langmuir, 12, 1913-1922 (1996)

U. Öpik, “Contact-angle hysteresis caused by a random distribution of weak heterogeneities on a solid surface”, Journal of Colloid and Interface Science, 223, 143-166 (2000)

J. Long, M. N. Hyder, R. Y. M. Huang, and P. Chen, “Thermodynamic modeling of contact angles on rough, heterogeneous surfaces”, Advances in Colloid and Interface Science, 118, 173-190 (2005)

S. A. Kulinich and M. Farzaneh, “Effect of contact angle hysteresis on water droplet evaporation from super-hydrophobic surfaces”, Applied Surface Science, 255, 4056-4060 (2009).

D. Klarman, D. Andelman, and M. Urbakh, “A model of electrowetting, reversed electrowetting, and contact angle saturation”, Langmuir, 27, 6031-6041 (2011)

J. W. Jung and J. M. Wan, “Supercritical CO2 and ionic strength effects on wettability of silica surfaces: equilibrium contact angle measurements”, Energy fuels, 26, 6053-6059 (2012)

A. O. Maksimov, A. M. Kaverin, and V. G. Baidakov, “Heterogeneous vapor bubble nucleation on a rough surface”, Langmuir, 29, 3924-3934 (2013)

X. S. Wang, S. W. Cui, L. Zhou, S. H. Xu, Z. W. Sun, and R. Z. Zhu, “A generalized Young’s equation for contact angles of droplets on homogeneous and rough substrates”, Journal of Adhesion Science and Technology, 28, 161-170 (2014)

W. Tillmann, J. Pfeiffer, and L. Wojarski, “The apparent contact angle and wetted area of active alloys on silicon carbide as a function of the temperature and the surface roughness: a multivariate approach”, Metallurgical and Materials Transactions A, 46, 3592-3600 ( 2015)

B. A. Pethica, “The contact angle equilibrium”, Journal of Colloid and Interface Science, 62, 567-569 (1977)

A. I. Rusanov, A. K. Shchekin, and D. V. Tatyanenko, “The line tension and the generalized Young equation: the choice of dividing surface”, Colloids and Surfaces A, 250, 263-268 (2004)

R. Tadmor, “Line energy, line tension and drop size”, Surface Science, 602, 108-111 (2008)

H. C. Kang and A. M. Jacobi, “Equilibrium contact angles of liquid droplets on ideal rough solids”, Langmuir, 27, 14910-14918 (2011)

H. Peng, G. R. Birkett, and A. V. Nguyen, “The impact of line tension on the contact angle of nanodroplets”, Molecular Simulation, 40, 934-941 (2014)

M. Iwamatsu, “Line-tension effects on heterogeneous nucleation on a spherical substrate and in a spherical cavity”, Langmuir, 31, 3861-3868 (2015)

S. Vafaei and M. Z. Podowski, “Theoretical analysis on the effect of liquid droplet geometry on contact angle”, Nuclear Engineering and Design, 235, 1293-1301 (2005)

E. Bormashenko, Y. Bormashenko, G. Whyman, R. Pogreb, A. Musin, R. Jager, and Z. Barkay, “Contact angle hysteresis on polymer substrates established with various experimental techniques, its interpretation, and quantitative characterization”, Langmuir, 24, 4020-4025 (2008)

S. Srinivasan, G. H. McKinley, and R. E. Cohen, “Assessing the accuracy of contact angle measurements for sessile drops on liquid-repellent surfaces”, Langmuir, 27, 13582-13589 (2011)

C. W. Extrand and S. I. Moon, “Indirect methods to measure wetting and contact angles on spherical convex and concave surfaces”, Langmuir, 28, 7775-7779 (2012)

G. Viswanadam and G. G. Chase, “Contact angles of drops on curved superhydrophobic surfaces”, Journal of Colloid and Interface Science, 367, 472-477 (2012)

A. Gauthier, M. Rivetti, J. Teisseire, and E. Barthel, “Finite size effects on textured surfaces: recovering contact angles from vagarious drop edges”, Langmuir, 30, 1544-1549 (2014)

C. Tan, P. Cai, L. Xu, N. Yang, Z. X. Xi, and Q. Li, “Fabrication of superhydrophobic surface with controlled adhesion by designing heterogeneous chemical composition”, Applied Surface Science, 349, 516-523 (2015)


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