Hill's nano-thermodynamics is equivalent with Gibbs' thermodynamics for curved surfaces

We review first how properties of curved surfaces can be studied using Hill's thermodynamics, also called nano-thermodynamics. We proceed to show for the first time that Hill's analysis is equivalent to Gibbs for curved surfaces. This simplifies the study of surfaces that are curved on the nano-scale, and opens up a possibility to study non-equilibrium systems in a systematic manner.