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- ItemThe equilibria of vapour-liquid systems revisited(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Dreyer, Wolfgang; Kraus, ChristianeWe study equilibrium conditions of liquid-vapour phase transitions for a single substance at constant temperature. The phase transitions are modelled by a classical sharp interface model with boundary contact energy. We revisit this old problem mainly for the following reasons. Equilibria in a two-phase system can be established either under fixed external pressure or under fixed total volume. These two different settings lead to distinct equilibria, a fact that is usually ignored in the literature. In nature and in most technical processes, the approach of a two-phase system to equilibrium runs at constant pressure, whereas mathematicians prefer to study processes in constant domains, i.e. at constant volume. Furthermore, in the literature the sharp interface of the liquid and the vapour phase is usually described by a surface with high symmetry like a plane interface or a radially symmetric interface which has the shape of the boundary of a ball. In this paper we establish equilibrium conditions for pressure control as well as for volume control with arbitrary shapes of the interface. The results are derived by methods of differential geometry. Further, the common features and differences of pressure and volume control are worked out for some simple cases.
- ItemAsymptotic expansions of the contact angle in nonlocal capillarity problems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Dipierro, Serena; Maggi, Francesco; Valdinoci, EnricoWe consider a family of nonlocal capillarity models, where surface tension is modeled by exploiting a family of fractional interaction kernels The fractional Young's law (contact angle condition) predicted by these models coincides, in the limit, with the classical Young's law determined by the Gauss free energy. Here we refine this asymptotics by showing that, for s close to 1, the fractional contact angle is always smaller than its classical counterpart when the relative adhesion coefficient is negative, and larger if it is positive. In addition, we address the asymptotics of the fractional Young's law in the limit case s close to 0 of interaction kernels with heavy tails. Interestingly, forsmall s, the dependence of the contact angle from the relative adhesion coefficient becomes linear.