Incompatibility of the Shuttleworth equation with Hermann's mathematical structure of thermodynamics

We show that three independent derivations of the Shuttleworth equation, arguably the second most important equation in surface physics, are inconsistent with Hermann's mathematical structure of thermodynamics [R. Hermann, Geometry, Physics and Systems, Marcel Dekker Inc., New York, USA, 1973 (Chapter 6)]. The possible implications of this result are discussed.