Derivation of the generalized Young-Laplace equation of curved interfaces in nanoscaled solids

In nanoscaled solids, the mathematical behavior of a curved interface between two different phases with interface stress effects can be described by the generalized Young-Laplace equations [T. Young, Philos. Trans. R. Soc. London 95, 65 (1805); P. S. Laplace, Traite de Mechanique Celeste (Gauthier-Villars, Paris, 1805), Vol. 4, Supplements au Livre X]. Here we present a geometric illustration to prove the equations. By considering a small element of the curved thin interface, we model the interface stresses as in-plane stresses acting along its edges, while on the top and bottom faces of the interface the tractions are contributed from its three-dimensional bulk neighborhood. With this schematic illustration, simple force balance considerations will give the Young-Laplace equations across the interface. Similar procedures can be applied to conduction phenomena. This will allow us to reconstruct one type of imperfect interfaces, referred to as highly conducting interfaces.