Cross-diffusion systems coupled via a moving interface

TL;DR

The paper develops a one-dimensional moving-interface cross-diffusion model coupling solid-phase size-exclusion diffusion with a gaseous Stefan–Maxwell phase via Butler–Volmer type interface laws. It establishes thermodynamic consistency through an entropy structure, characterizes stationary states, and analyzes stability in a simplified setting. A structure-preserving finite-volume moving-mesh scheme is introduced, and its discrete dissipation, mass conservation, and positivity are proven, with numerical experiments confirming energy decay and convergence to equilibria. The work provides a rigorous mathematical and numerical framework for moving-boundary diffusion systems with coupled phases, relevant to vapor deposition and related materials processes.

Abstract

We propose and study a one-dimensional model which consists of two cross-diffusion systems coupled via a moving interface. The motivation stems from the modelling of complex diffusion processes in the context of the vapor deposition of thin films. In our model, cross-diffusion of the various chemical species can be respectively modelled by a size-exclusion system for the solid phase and the Stefan-Maxwell system for the gaseous phase. The coupling between the two phases is modelled by linear phase transition laws of Butler-Volmer type, resulting in an interface evolution. The continuous properties of the model are investigated, in particular its entropy variational structure and stationary states. We introduce a two-point flux approximation finite volume scheme. The moving interface is addressed with a moving-mesh approach, where the mesh is locally deformed around the interface. The resulting discrete nonlinear system is shown to admit a solution that preserves the main properties of the continuous system, namely: mass conservation, nonnegativity, volume-filling constraints, decay of the free energy and asymptotics. In particular, the moving-mesh approach is compatible with the entropy structure of the continuous model. Numerical results illustrate these properties and the dynamics of the model.

Figures (7)

• Figure 1: A virtual mesh displacement between $t^{p-1} = (p-1) \Delta t$ and $t^p = p \Delta t$, where $K := K^{p-1}$.
• Figure 2: Trivial case: no interface movement
• Figure 3: Equilibrium case with monotone interface
• Figure 4: Equilibrium case with non-monotone interface
• Figure 5: Non-equilibrium case

Theorems & Definitions (17)

• Remark 1: Physical variables
• Remark 2: Extension of the model
• Definition 1
• Proposition 1: Stationary states
• proof
• Proposition 2
• proof
• Lemma 1
• proof
• Lemma 2