Thermodynamic derivations of the mechanical equilibrium conditions for fluid surfaces: Young’s and Laplace’s equations

The fundamental laws governing the mechanical equilibrium of solid-fluid systems were formulated in 1805 and 1806. They are Laplace’s law, which describes the pressure drop across an interface, and Young’s equation for the contact angle. At that time, these laws were justified on purely mechanical grounds. In 1880 Gibbs used thermodynamics to show that these laws were necessary conditions for the equilibrium of heterogeneous systems. We revisit Gibbs’ derivation and simplify it for possible use at the undergraduate level. In addition, we present derivations of Young’s and Laplace’s equations, which involve energy balance on a volume element located at the surface. In particular, it is shown that the derivations are simpler, allow the analysis of nonequilibrium situations, and give a natural identification of the surface energy with the surface tension of the liquid-vapor interface.

[1]  J. Drelich.  The significance and magnitude of the line tension in three-phase (solid-liquid-fluid) systems , 1996 .

[2]  H. Fujii,et al.  Effect of gravity on contact angle , 1995 .

[3]  B. Widom,et al.  Young's law with gravity , 1995 .

[4]  Rulon E. Johnson.  Conflicts between Gibbsian Thermodynamics and Recent Treatments of Interfacial Energies in Solid-Liquid-Vapor , 1959 .

[5]  P. Roura,et al.  Local thermodynamic derivation of Young's equation. , 2004, Journal of colloid and interface science.

[6]  P. Gennes.  Wetting: statics and dynamics , 1985 .

[7]  A. W. Neumann,et al.  Generalization of the classical theory of capillarity , 1977 .

[8]  The physical description of elementary surface phenomena: Thermodynamics versus mechanics , 1995 .

[9]  Ta-Tsien Li.  Hydrostatics in Various Gravitational Fields , 1962 .

[10]  The profile of a dew drop , 1996 .

[11]  C. M. Holmes,et al.  Kinetics of wetting of surfaces by polymers , 1969 .

[12]  S. Herminghaus,et al.  Three-phase contact line energetics from nanoscale liquid surface topographies , 2000, Physical review letters.

[13]  R. Tolman.  Consideration of the Gibbs Theory of Surface Tension , 1948 .

[14]  Thomas Young,et al.  An Essay on the Cohesion of Fluids , 1800 .

[15]  T. Young.  III. An essay on the cohesion of fluids , 1805, Philosophical Transactions of the Royal Society of London.

[16]  Frederick M. Fowkes,et al.  ADDITIVITY OF INTERMOLECULAR FORCES AT INTERFACES. I. DETERMINATION OF THE CONTRIBUTION TO SURFACE AND INTERFACIAL TENSIONS OF DISPERSION FORCES IN VARIOUS LIQUIDS1 , 1963 .

[17]  J. E. McNutt,et al.  Relationship of the Contact Angle to Interfacial Energies , 1959 .

[18]  R. E. Collins,et al.  Fundamental basis for the contact angle and capillary pressure , 1959 .