A Nanothermodynamic Approach to the Shuttleworth and Lippman Equations

The Shuttleworth and Lippman equations are well-known equations used to link surface tension and stress (Shuttleworth) and surface tension and electric surface potentials (Lippmann). We show that the Shuttleworth and Lippman equations have a common thermodynamic basis, common to systems that possess a relatively large interfacial energy. This is relevant for problems of droplet stability, colloidal suspensions, electrode surfaces and more. Both equations are derived for systems that are not Euler homogeneous in the manner classical systems are. Hill's thermodynamics for small systems is used to address this problem. Small in this context refers to systems with interfacial energies that are size- or shape-dependent. The resulting Hill-Gibbs-Duhem equation, an extension of the classical Gibbs-Duhem's equation, gives the common basis for the Shuttleworth and Lippman equations. Hill's thermodynamics enables us to rigorously define two types of surface tension, the differential surface tension and the integral surface tension. These surface tensions are linked by the system's subdivision potential. From Helfrich's equation we obtain a scaling law for the subdivision potential as function of the interfacial curvature. The dependence of the resulting subdivision potential on the system curvature is predicted. A critical analysis of the literature about the Shuttleworth and Lippman equations is given.