On the Thermodynamics of Thin Films. The Frumkin–Derjaguin Equation

Abstract

The Frumkin–Derjaguin equation, which is a fundamental relationship in the theory of thin films, relates all surface tensions associated with a wetting film to its disjoining pressure and contact angle. The published ways of proving this relationship have been reviewed and their thermodynamic drawbacks have been analyzed in this work. The relationship has been rigorously derived on the basis of two approaches. The first entails the use of a new thermodynamic potential (J potential), which is defined for fluid systems as the grand thermodynamic potential combined with the product of the system volume and some pressure $p{\kern 1pt} '$ (here, the pressure in the mother phase of the thin film is used as $p{\kern 1pt} '$). The second approach is based on the Gibbs adsorption equation and entails the use of two dividing surfaces. These approaches yield identical results and suggest that not only temperature, but also all chemical potentials of the film must be fixed when calculating the work of film thinning. The dependence of the contact angle on the disjoining pressure has been considered on the basis of the Young equation. It has been shown that, as the disjoining pressure rises, the thermodynamic surface tension of the film-containing interface increases, while the contact angle decreases. The problems encountered when calculating the contact angles via the disjoining pressure isotherms have been noted.