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Crystalline Motion of discrete interfaces in the Blume-Emery-Griffiths Model: partial wetting

Marco Cicalese, Giuliana Fusco, Giovanni Savaré

TL;DR

This work develops a rigorous discrete-to-continuum description for the Blume-Emery-Griffiths surfactant model of two immiscible phases, focusing on partial wetting and non-evaporating surfactant under a minimizing-movements scheme. For $oldsymbol{\gamma}<2$, the evolving crystal interface remains, up to defects, a quasi-octagon and undergoes three distinct stages governed by surfactant availability and geometry, including a nonlocal velocity regime that captures redistributed surfactant effects. The analysis introduces novel discrete barrier arguments and exact nonlocal motion laws for crystalline interfaces, linking lattice energetics to anisotropic mean-curvature-type flows with pinning phenomena and metastable states. In the critical case $oldsymbol{\gamma}=2$, the dynamics depend on sharp parameter thresholds, with transitions to previously analyzed regimes or to mixed behaviors, thereby providing a comprehensive framework bridging microscopic BEG lattice models and experimentally observed surfactant-driven pinning in immiscible systems.

Abstract

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 \cite{CFS}, 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.

Crystalline Motion of discrete interfaces in the Blume-Emery-Griffiths Model: partial wetting

TL;DR

This work develops a rigorous discrete-to-continuum description for the Blume-Emery-Griffiths surfactant model of two immiscible phases, focusing on partial wetting and non-evaporating surfactant under a minimizing-movements scheme. For , the evolving crystal interface remains, up to defects, a quasi-octagon and undergoes three distinct stages governed by surfactant availability and geometry, including a nonlocal velocity regime that captures redistributed surfactant effects. The analysis introduces novel discrete barrier arguments and exact nonlocal motion laws for crystalline interfaces, linking lattice energetics to anisotropic mean-curvature-type flows with pinning phenomena and metastable states. In the critical case , the dynamics depend on sharp parameter thresholds, with transitions to previously analyzed regimes or to mixed behaviors, thereby providing a comprehensive framework bridging microscopic BEG lattice models and experimentally observed surfactant-driven pinning in immiscible systems.

Abstract

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 \cite{CFS}, 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.
Paper Structure (13 sections, 17 theorems, 152 equations, 20 figures)

This paper contains 13 sections, 17 theorems, 152 equations, 20 figures.

Table of Contents

  1. Introduction
  2. Formulation of the problem and preliminary results
  3. Notation
  4. Setting of the problem
  5. Preliminary results
  6. Discrete sets
  7. Connection and inclusion of the minimizer
  8. The partial wetting in the case gamma<2
  9. Stage one: "pinning of diagonals".
  10. Stage two: "surfactant dependent side velocities".
  11. Stage three: "nonlocal averaging of velocities by surfactant redistribution".
  12. Partial wetting with negligible surfactant
  13. The case gamma=2

Key Result

Proposition 2.11

Let $\gamma>0$ and let $\mu\in(0, 1/4)$. Let $u_0,u_1\in{\mathcal{A}}_\varepsilon$ be such that $u_0$ verifies H and $u_1$ is a minimizer of $\mathcal{F}^{\tau, \gamma}_\varepsilon(\cdot, u_0)$. Then there exists $\bar{\varepsilon}$ such that the set $I_1$ is a connected staircase set contained in $

Figures (20)

  • Figure 1: The octagon $A \subset \mathbb{R}^2$. The lengths of the sides parallel to the coordinate axes are denoted by $P_i$, while the lengths of the diagonal sides are denoted by $D_i$, with indices ordered clockwise, for $i=1,\dots, 4$.
  • Figure 2: Stage one of the evolution at $\varepsilon$ scale in the case $\gamma<2$ in the non complete wetting regime. We represent with a thicker black line the minimizer at time step $j+1$.
  • Figure 3: Stage twp of the evolution at $\varepsilon$ scale in the case $\gamma<2$ in the non complete wetting regime. We represent with a thicker black line the minimizer at time step $j+1$.
  • Figure 4: Example \ref{['ex:degenerate_octagon_evolution']}: stage three of the evolution at $\varepsilon$ scale in the case $\gamma<2$ in the non complete wetting regime when $I_j$ is a degenerate octagon. On the left we illustrate the displacement of a subset of the sides of $I_j$; the dashed line denotes the corresponding subset of the sides of $I_{j+1}$. On the right side we represent $I_{j+1}$ and the particles that are on its boundary.
  • Figure 5: Example \ref{['ex:surfactant_on_diagonals']}: stage three of the evolution at $\varepsilon$ scale in the case $\gamma<2$ in the non complete wetting regime when $I_j$ is a (non degenerate) octagon with diagonals completely wetted by surfactant particles.
  • ...and 15 more figures

Theorems & Definitions (59)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 49 more