Thermodynamics of interfaces extended to nanoscales by introducing integral and differential surface tensions.

As a system shrinks down in size, more and more molecules are found in its surface region, so surface contribution becomes a large or even a dominant part of its thermodynamic potentials. Surface tension is a venerable scientific concept; Gibbs defined it as the excess of grand potential of an inhomogeneous system with respect to its bulk value per interface area [J. W. Gibbs, "The Collected Works" in (1928), Vol. 1]. The mechanical definition expresses it in terms of pressure tensor. So far, it has been believed the two definitions always give the same result. We show that the equivalence can break down for fluids confined in narrow pores. New concepts of integral and differential surface tensions, along with integral and differential adsorptions, need to be introduced for extending Gibbs thermodynamics of interfaces. We derived two generalized Gibbs adsorption equations. These concepts are indispensable for an adequate description of nanoscale systems. We also find a relation between integral surface tension and Derjaguin's disjoining pressure. This lays down the basis for measuring integral and differential surface tensions from disjoining pressure by using an atomic force microscope.