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

Marco Cicalese, Giuliana Fusco, Giovanni Savaré

TL;DR

The work rigorously derives the discrete-to-continuum limit for the Blume–Emery–Griffiths surfactant model in a two-dimensional lattice under a minimizing-movements scheme with dissipation split into interface-change and surfactant-mass-variation parts. The limiting dynamics depend critically on the surfactant evaporation parameter $\\gamma$: for $\\gamma>2$ the system undergoes a crystalline mean curvature-type flow where an initial octagonal crystal shrinks while diagonals remain pinned; for $\\gamma<2$ mass conservation induces pinning and a two-stage evolution, first with pinned diagonals and then with a transition to an $\\varepsilon$-quasi-rectangle whose continuum limit is described by a coupled octagon-rectangle geometry with explicit velocity rules. The analysis hinges on preserving discrete geometric structures (octagons, staircase sets) through the minimization steps and establishing connectedness and containment properties, ultimately yielding precise continuum descriptions of the evolving interfaces. This provides a rigorous bridge between a three-phase lattice model with surfactant and its geometric continuum counterpart, clarifying how evaporation vs conservation of surfactant mass shapes interfacial motion.

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.

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

TL;DR

The work rigorously derives the discrete-to-continuum limit for the Blume–Emery–Griffiths surfactant model in a two-dimensional lattice under a minimizing-movements scheme with dissipation split into interface-change and surfactant-mass-variation parts. The limiting dynamics depend critically on the surfactant evaporation parameter : for the system undergoes a crystalline mean curvature-type flow where an initial octagonal crystal shrinks while diagonals remain pinned; for mass conservation induces pinning and a two-stage evolution, first with pinned diagonals and then with a transition to an -quasi-rectangle whose continuum limit is described by a coupled octagon-rectangle geometry with explicit velocity rules. The analysis hinges on preserving discrete geometric structures (octagons, staircase sets) through the minimization steps and establishing connectedness and containment properties, ultimately yielding precise continuum descriptions of the evolving interfaces. This provides a rigorous bridge between a three-phase lattice model with surfactant and its geometric continuum counterpart, clarifying how evaporation vs conservation of surfactant mass shapes interfacial motion.

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.
Paper Structure (12 sections, 18 theorems, 194 equations, 24 figures)

This paper contains 12 sections, 18 theorems, 194 equations, 24 figures.

Table of Contents

  1. Introduction
  2. Formulation of the problem
  3. Notation
  4. Setting of the problem
  5. Discrete sets
  6. Connection and inclusion of the minimizer
  7. Preliminaries
  8. Connectedness
  9. The case gamma>2
  10. The case gamma<2
  11. First stage of the minimizing movements: pinning of the diagonals.
  12. Second stage of the minimizing movements: the minimizer is an $\varepsilon$-quasi-rectangle.

Key Result

Lemma 3.4

[Shape optimality] Let $u_0,\,u_1\in{\mathcal{A}}_\varepsilon$ be such that $u_1$ is a minimizer of $\mathcal{F}^{\tau, \gamma}_\varepsilon(\cdot, u_0)$. Then, for every $\varepsilon>0$ the following properties hold true. Now let $I$ be a strongly connected component of $I_1$, suppose that $I_1\subset I_0$, that $I$ is a staircase set, and let $H^{i} = \llbracket p^{i}, q^{i} \rrbracket,\,i=1, \

Figures (24)

  • Figure 1: The case $\gamma >2$: Discrete flow of a Wulff-type set (in red $u=1$ and in blue $u=0$).
  • Figure 2: The case $\gamma<2$. The first stage: pinning of the diagonals (in red $\{u=1\}$ and in blue $\{u=0\}$).
  • Figure 3: The case $\gamma<2$: one diagonal disappears (in light gray, we represent the surfactant on the short diagonal at step $j$; in dark gray, we represent such surfactant that build up along the upper diagonal at the step $j+1$).
  • Figure 4: The case $\gamma<2$: all the sloped sides are short and $Z^\varepsilon_{j+1}\cup I^\varepsilon_{j+1}\subset R_{I^\varepsilon_{j+1}\cup \partial^+I^\varepsilon_{j+1}}$ (in light gray, we represent the surfactant at step $j$ that is not adjacent to the sides parallel to the coordinate axes at step $j+1$. A possible rearrangement of such surfactant particles at step $j+1$ is represented in dark gray).
  • Figure 5: The case $\gamma<2$: all the sloped sides are short and $Z^\varepsilon_{j+1}\cup I^\varepsilon_{j+1}\subsetneq R_{I^\varepsilon_{j+1}\cup \partial^+I^\varepsilon_{j+1}}$. To make it easier for the reader, $Z^\varepsilon_{j}\cup I^\varepsilon_{j}$ and $Z^\varepsilon_{j+1}\cup I^\varepsilon_{j+1}$ are shown non-overlapping, differently from the other figures (in light gray, we represent the surfactant at step $j$ that is rearranged at step $j+1$ as the dark gray particles on the right hand side).
  • ...and 19 more figures

Theorems & Definitions (57)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Lemma 3.4
  • ...and 47 more