Two-Phase Diffusion Model

• The two-phase diffusion model is defined as a framework for describing transport in systems with distinct phases and a moving interfacial boundary.
• It applies cross-diffusion, mass conservation, and thermodynamic consistency to simulate phenomena such as vapor deposition, thin film formation, and multiphase flows.
• The model leverages entropy variational principles and structure-preserving finite-volume schemes to ensure stability, convergence, and accurate mass conservation in numerical simulations.

A two-phase diffusion model describes transport phenomena in systems consisting of two contiguous domains (phases) with distinct physical properties, where the interface between the phases evolves dynamically and plays an essential role in coupling the underlying processes. Such models arise across physics, engineering, and applied mathematics, with key applications in vapor deposition, thin film formation, solidification, multiphase flows, and multicomponent transport in heterogeneous media. The mathematical structure typically involves systems of partial differential equations (PDEs) for conserved quantities (e.g., chemical species) in each phase, together with cross-diffusive couplings, moving interfaces governed by phase transition kinetics, and rigorous thermodynamic underpinnings.

1. Model Structure and Formulation

A canonical two-phase diffusion model, as introduced by Cancès, Cauvin-Vila, Chainais-Hillairet, and Ehrlacher, considers a one-dimensional domain partitioned by a moving interface at position $X(t)\in[0,1]$, with distinct cross-diffusion systems acting in each subdomain $(0,X(t))$ (solid phase) and $(X(t),1)$ (gas phase). The state consists of molar concentrations $c=(c_1,...,c_n)$, subject to a local volume-filling constraint: $\sum_{i=1}^n c_i(t,x) = 1 \qquad x\in(0,1), ~ t>0$ In each phase, mass conservation and cross-diffusion are governed by: $\partial_t c_i + \partial_x J_i = 0$ with cross-diffusive fluxes given by: $$J = \begin{cases} -A_s(c) \partial_x c & \text{in } (0,X(t)) \ -A_g(c) \partial_x c & \text{in } (X(t),1) \end{cases}$$ where $A_s(c)$ is a size-exclusion diffusivity matrix (solid phase), and $A_g(c)$ is the Stefan–Maxwell mobility matrix (gaseous phase). Both matrices satisfy normalization and coercivity properties required for well-posedness and thermodynamic consistency: $$A_\alpha(c)\mathbf{1}=0, ~ M_\alpha(c):=A_\alpha(c)\,\mathrm{diag}(c)^{-1}\succeq C_\alpha$$ for $\alpha\in\{s,g\}$.

Zero-flux boundary conditions are imposed at $x=0,1$. The interface at $x = X(t)$ is coupled via a conservation-plus-reaction law: $$-J^s_i + X'\,c^s_i = -J^g_i + X'\,c^g_i = F_i, \qquad F_i = \sqrt{\beta_i^*}\,c^g_i - \frac{1}{\sqrt{\beta_i^*}\,c^s_i}$$ with phase transition parameters $\beta_i^* = \exp(\mu_i^{*,g} - \mu_i^{*,s})$. The collective interface velocity is prescribed by: $X'(t) = \sum_{i=1}^n F_i(t)$ The initial state $(c_i(0,x), X(0))$ is selected from the (n-1)-dimensional simplex consistent with the volume-filling constraint.

2. Entropy Variational Structure and Thermodynamics

The model is endowed with a variational structure based on a global free-energy (entropy) functional: $$\mathcal{E}[c,X] = \int_0^{X} h_s(c)\,dx + \int_X^{1} h_g(c)\,dx$$ where each phase specifies its local free-energy density: $$h_\alpha(c) = \sum_{i=1}^n c_i(\log c_i + \mu_i^{*,\alpha}) - c_i + 1$$ and chemical potentials $\mu_i^\alpha(c) = \log c_i + \mu_i^{*,\alpha}$.

A key feature is the dissipation law, establishing $\mathcal{E}$ as a Lyapunov functional: $$\frac{d}{dt}\mathcal{E}[c,X] + \int_0^X \partial_x \mu_s^T M_s(c) \partial_x \mu_s\,dx + \int_X^1 \partial_x \mu_g^T M_g(c) \partial_x \mu_g\,dx + F^T (\mu^g - \mu^s) = 0$$ Due to the Butler–Volmer kinetics at the interface ($F_i\,\Delta\mu_i\ge 0$), one always has monotonic energy decay: $\frac{d}{dt}\mathcal{E}[c,X] \le 0$ ensuring long-time stability and thermodynamic consistency.

Stationary (equilibrium) states are characterized by piecewise-constant concentration profiles, mass conservation, and vanishing interfacial fluxes: $X\,c^s + (1-X)\,c^g = m^0,\qquad F_i=0,~\forall i$ Nontrivial (two-phase) equilibria exist and are unique if and only if: $$\min\left\{\sum m_i^0 \beta_i^*, ~ \sum m_i^0 \beta_i^{*-1}\right\} > 1$$ outside of which pure-phase equilibria dominate.

3. Numerical Discretization: Finite-Volume Moving-Mesh Scheme

Discretization employs a two-point flux approximation (TPFA) on a finite-volume mesh partitioned to accurately track the moving interface. The discrete state consists of cell-averaged concentrations and the current interface location. The update in each phase is: $$\frac{|C_K^{p,\star}|\,c_{i,K}^{p,\star} - |C_K^{p-1}|\,c_{i,K}^{p-1}}{\Delta t} + J^p_{i,K+1/2} - J^p_{i,K-1/2} = 0$$ for each cell $K$ and species $i$, with consistent discretization of the cross-diffusion matrices adapted to the phase (solid or gas). The interface is located in a "cut-cell," requiring careful local mesh deformation and averaging to preserve mass conservation and nonnegativity.

Interface fluxes adopt the discrete Butler–Volmer form: $$J^p_{i,K+1/2} = -F_i^p,~~F_i^p = \sqrt{\beta_i^*}\,c_{i,K^{p-1}+1}^{p,\star} - \frac{1}{\sqrt{\beta_i^*}\,c_{i,K^{p-1}}^{p,\star}}$$ The interface motion is time-stepped semi-implicitly: $X^p = X^{p-1} + \Delta t \sum_i F_i^p$ After each time-step, a mass-conserving projection reshapes the local mesh.

The discrete scheme rigorously preserves:

• Nonnegativity and volume-filling: $c^p_{i,K} \geq 0,\quad \sum_i c^p_{i,K}=1$
• Mass conservation
• Decay of the discrete free energy: $$\mathcal{E}^p(c^p, X^p) + \Delta t \mathscr{D}^p \leq \mathcal{E}^{p-1}$$ for a discrete dissipation $\mathscr{D}^p$.

Under a Courant–Friedrichs–Lewy (CFL) time-step restriction, Theorem 4.7 ensures existence of solutions preserving all these properties at the fully discrete level.

4. Analytical Results and Long-Time Behavior

The analytical framework guarantees global-in-time existence and well-posedness for the fully discrete scheme, with the key invariants reflected at the discrete level. Notably:

• The discrete free energy serves as a Lyapunov function, guaranteeing asymptotic stability and control of solution norms.
• Stationary states are exponentially attractive if the existence criterion is met.
• In parameter regimes violating the two-phase existence criterion, the interface migrates to the boundary in finite time, resulting in a pure-phase solution.

Mesh-refinement studies confirm first-order spatial convergence in both cell concentrations and interface position.

5. Numerical Experiments and Applications

Numerical results demonstrate:

• Homogenization and energy relaxation for the trivial (no interface motion) case.
• Nontrivial interface dynamics and convergence to unique two-phase steady states for well-chosen parameters.
• Finite-time interface disappearance in unfavored regimes.
• Robustness and conservation under mesh refinement, with $\ell^1$ error convergence consistent with theory.

The model provides a detailed and predictive description of complex vapor deposition phenomena, multicomponent surface growth, and cross-diffusive transport in multi-phase systems, establishing a computationally effective and mathematically sound approach to dynamically coupled two-phase diffusion processes (Cancès et al., 2024).

6. Relation to Generalized and Higher-Dimensional Two-Phase Diffusion Models

The model exemplifies broader classes of two-phase and multi-component diffusion systems:

• Sharp-interface cut-cell methods for stationary and moving interfaces (Libat et al., 22 Dec 2025).
• Variational and thermodynamically consistent phase-field models for two-phase flows, including cross-diffusion, capillarity, and viscoelastic effects (Abels et al., 8 May 2025, Cheng et al., 29 Sep 2025).
• Bulk-interface coupling conditions, typically enforcing normal-flux balance and (weighted) value jumps, often with interface motion governed by Onsager-type kinetics.
• Application domains extending from materials science (alloy solidification, grain boundary evolution) to bioengineering and porous medium hydrodynamics.

The structure-preserving discretization strategy—combining phase-specific cross-diffusion, dynamic interface laws, and entropy stability—represents the state-of-the-art for high-fidelity simulation of two-phase microstructure evolution and cross-coupled reactive transport.

7. Significance and Research Directions

The rigorous derivation, variational framework, and structure-preserving numerics developed for two-phase cross-diffusion systems underpin a class of models applicable to a diverse array of coupled multi-phase transport phenomena. The approach in (Cancès et al., 2024):

• Guarantees global existence and stability for initial-boundary value problems with moving interfaces,
• Provides concrete conditions for emergence and uniqueness of two-phase equilibria,
• Enables systematic construction of entropy-stable, mass-conserving numerical schemes,
• Informs further research into multi-dimensional extensions, complex interface geometries, and kinetic regimes beyond linear Butler–Volmer coupling,
• Forms a mathematical basis for physically consistent upscaling of reactive transport in heterogeneous and evolving media.

The framework sets a benchmark for future analytical and computational studies into multiphase cross-diffusive systems with dynamic interface evolution.