The Discrete Legendre-Fenchel Transform and its application to phase separation in electrolytes

Lorenzo Contento (1)

1 DIMI - Dipartimento di Matematica e Informatica - Universita Udine

The study of equilibrium electrolytes in a porous and electrically charged medium is of interest in the design of semiconductor devices or the study of clay rocks at the nanometre scale. The bulk free-energy f of these systems, which is a function of the ionic concentrations, contains a non-convex contribution due to the electrostatic interactions between the ions; it has been shown that for small ion diameters this contribution is strong enough to make the function f non-convex. In such cases the region T of the state space where f differs from its convex hull is thermodynamically unstable and T divides the state space in different phases. For a one-dimensional state space (only one ionic species), the convex hull of the bulk free-energy f can be determined using Maxwell equal area rule. For more than one ionic species, we need to determine the convex hull of a multi-dimensional function f. In the thesis we study the Legendre-Fenchel transform and we propose an algorithm to approximate the convex hull, which is based on its discrete version. This approach is known in the literature, but here we develop and test an improved version of it. We apply this algorithm to the bulk free energy in the symmetric and asymmetric case, finding the shape of the region T. Finally we solve the model equations by a finite elements discretization, obtaining solutions in which phase separation clearly arises.

Domaines: Analyse numérique [math.NA]

Lorenzo Contento. The Discrete Legendre-Fenchel Transform and its application to phase separation in electrolytes. 2012. ⟨hal-00806085v2⟩