Phase transitions in presence of surfactants: From discrete to continuum

Abstract

We study by-convergence the discrete-to-continuum limit of the Blume-Emery-Griffiths model describing the phase transition of a binary mixture in presence of a third surfactant phase. In the case of low surfactant concentration we study the dependence of the surface tension on the density of the surfactant and we describe the microstructure of the ground states. We then consider more general (n-dimensional) energies modeling phase transitions in presence of different species of surfactants and, in the spirit of homogenization theory, we provide an integral representation result for theirlimit. As an application we study the ground states of these systems for prescribed volume fractions of the phases.

$Q$-Tensor Continuum Energies as Limits of Head-to-Tail Symmetric Spin Systems

“…Following this approach it has been possible, e.g., to prove compactness and integral representation theorems for volume and surface integrals [1,3,5] that follow the localization arguments of classical results for continuum energies [14,20]. On the other hand the constraints given by the discrete nature of the parameters, often entail interesting features of the limit energies (for example, optimality properties for discrete linear elastic composites [16], multi-phase limits for next-to-nearest neighbor scalar spin systems [2], 'surfactant'-type theories [7], etc. ).…”

On the Variational Limits of Lattice Energies on Prestrained Elastic Bodies

“…In this paper, we pose the following question: is it possible to derive an equivalent continuum mechanics model starting from an appropriate discrete description, by means of a homogenization procedure when the mesh size goes to 0? Discrete-tocontinuum limits of this type have been investigated by means of Γ-convergence in a number of areas of application, including nonlinear elasticity [1,28,21,2,33,34] and others (see for example [3,30,14,32]). Discrete lattices may model both the atomic structures and mechanical trusses.…”

Differential Geometry and Continuum Mechanics

“…In this chapter, we pose the following question: is it possible to derive an equivalent continuum mechanics model starting from an appropriate discrete description, by means of a homogenization procedure when the mesh size goes to 0? Discrete-to-continuum limits of this type have been investigated by means of -convergence in a number of areas of application, including nonlinear elasticity [AC04, MPR12, DR13, ACG11, Sch08, Sch09] and others (see for example [ACS12,Ort12,EKCO13,SS09]). Discrete lattices may model both the atomic structures and mechanical trusses.…”