Giuliana Fusco

We continue the variational study of the discrete-to-continuum evolution of lattice systems of Blume-Emery-Griffith type which model two immiscible phases in the presence of a surfactant. In our previous work, we analyzed the case of a completely wetted crystal and described how the interplay between surfactant evaporation and mass conservation leads to a transition between crystalline mean curvature flow and pinned evolutions. In the present paper, we extend the analysis to the regime of partial wetting, where the surfactant occupies only a portion of the interface. Within the minimizing-movements scheme, we rigorously derive the continuum evolution and show how partial wetting introduces a complex coupling between interfacial motion and redistribution of surfactant. The resulting evolution exhibits new features absent in the fully wetted case, including the coexistence of moving and pinned facets or the emergence and long-lived metastable states. This provides, to our knowledge, the first discrete-to-continuum variational description of partially wetted crystalline interfaces, bridging the gap between microscopic lattice models and experimentally observed surfactant-induced pinning phenomena in immiscible systems.

We study the discrete-to-continuum evolution of a lattice system consisting of two immiscible phases labelled by -1 and +1 in presence of a surfactant phase labelled by 0. The system's energy is described by the classical Blume-Emery-Griffith model on the lattice epsilon Z^2, and its continuum evolution is obtained as epsilon tends to zero through a minimizing-movements scheme with a time step proportional to epsilon. The dissipation functional we choose contains two contributions: a standard Almgren-Taylor-Wang type term penalizing the distance between successive configurations of the +1 phase, and a term penalizing the variation of the surfactant mass and modeling surfactant evaporation. The latter term depends on a scaling parameter gamma > 0, which determines whether the surfactant mass is conserved at each time step. We focus on the case in which the initial configuration consists of a single crystal of phase 1 completely wetted by the surfactant. For gamma > 2 the surfactant can lose mass and the evolution reduces to the crystalline mean curvature flow of an Ising-type model, while for gamma < 2 the conservation of the surfactant mass leads to a more complex evolution characterized by stronger non-uniqueness and partial pinning.