Crystalline motion of discrete interfaces in the Blume-Emery-Griffiths model

TL;DR

This paper investigates the discrete-to-continuum evolution of a lattice system with two phases and surfactant, revealing how different surfactant mass conservation regimes affect the resulting crystalline motion and flow behavior.

Contribution

It introduces a novel continuum evolution scheme for the Blume-Emery-Griffiths model incorporating surfactant dynamics and analyzes the effects of surfactant mass conservation on interface evolution.

Findings

For gamma > 2, surfactant mass loss leads to crystalline mean curvature flow.

For gamma < 2, surfactant conservation results in complex, non-unique evolution.

The model captures the impact of surfactant evaporation on interface dynamics.

Abstract

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.