Equilibrium and adhesion of Nb/sapphire: the effect of oxygen partial   pressure
  We derive a formula, useful for first-principles calculations, which relates the free energy of an oxide/metal interface to the free energies of surfaces and the work of separation of the interface. We distinguish the latter {\it mechanical} quantity from the {\it thermodynamic} work of adhesion, and we describe explicitly how both may be calculated. Our formulae for interfacial and surface energies are cast in terms of quantities which can be calculated or looked up in tables, and include as additional parameters the ambient temperature and partial pressure of oxygen $P_{O_2}$. From total energy calculations for the Nb(111)/alpha-alumina(0001) interface, free Nb and alumina surfaces, we obtain firstly numerical estimates of the works of separation, which are independent of $P_{O_2}$. We then obtain surface energies, interfacial energies and the equilibrium work of adhesion as a function of $P_{O_2}$. 
Iskander G. Batirev, Ali Alavi and Michael W. Finnis