The role of Minkowski functionals in the thermodynamics of two-phase systems

Within this work quite old concepts from integral geometry are applied to classical equilibrium thermodynamics of two-phase systems. In addition to the area as basic interfacial quantity the full geometric characterization of the interface is used, which includes the two remaining Minkowski functionals, the mean curvature integral and the Euler Poincaré characteristic. The basic energetic characteristic of the interface (i.e. the interfacial tension) is extended by two additional properties: edge force as (up to a factor 4/π) the work necessary to form a right-angled edge of unit length, and item energy as the work to form an additional item in the phase morphology. Both quantities are of increasing importance, when going to micro- and nano-scales. They are subsequently used for interfaces of arbitrary shape to derive a relationship extending the classical Young-Laplace equation. The supplementary contribution is proportional to the Gaussian curvature, with the edge force as proportionality constant. Furthermore, both edge force and item energy are shown to be applicable to the description of crystal nucleation in liquids (extending the classical Becker Döring theory). It turns out, that even above the thermodynamic melting temperature stable nuclei can be present in the liquid phase. They immediately are able to grow when quenched to a temperature below a characteristic temperature. This temperature of spontaneous homogeneous nucleation is simply connected to the edge force, whereas the number of stable clusters per unit volume is dominated by the item energy. Finally, the additional energetic interfacial properties are used in a similar way to characterize the stability of emulsions.