Diffuse-Interface Model Fundamentals

• Diffuse-interface models are mathematical frameworks that represent interfaces as finite-thickness regions with continuous phase fields, capturing complex multiphase phenomena.
• They couple energetic formulations with systems like the Cahn–Hilliard and Navier–Stokes equations to accurately simulate phase separation, capillarity, and interface motion.
• These models have broad applications in multiphase flows, solidification, and reactive systems, validated by adaptive numerical methods and asymptotic analyses.

A diffuse-interface model is a mathematical and computational framework in which interfaces between distinct phases (fluids, solids, elastic media, etc.) are described not as sharply defined hypersurfaces but as spatial regions of finite thickness wherein an order parameter, phase field, or set of concentration variables varies smoothly and continuously. This approach allows for thermodynamically consistent modeling of multiphase phenomena—including phase separation, coarsening, capillarity, and interface motion—in a manner amenable to rigorous analysis and efficient numerical solution. Diffuse-interface models are grounded in variational principles, typically via gradient-flow structures or energetic formulations, and generalize the classical Cahn–Hilliard, Ginzburg–Landau, and van der Waals theories to complex fluids, solids, and their interactions.

1. Energetic Foundations and Free-Energy Functionals

The cornerstone of a diffuse-interface model is a total free-energy functional that penalizes both deviations from the pure phases and gradients of the order parameter. For binary fluid mixtures, a prototypical example is the nonlocal Cahn–Hilliard energy (Frigeri et al., 2013):

$$E[\varphi] = \int_\Omega \left( \frac{1}{2} a(x)\,\varphi(x)^2 - \frac{1}{2} \int_\Omega J(x-y)\,\varphi(x)\,\varphi(y)\,dy + F(\varphi(x)) \right) dx,$$

where $\varphi(x) \in [-1, 1]$ is the phase field, $J(\cdot)$ is an even, integrable interaction kernel, $a(x) = \int_\Omega J(x-y) dy$, and $F(\varphi)$ is a double-well bulk potential. The energy may equivalently be rewritten in a quadratic “square-difference” form.

In Ginzburg–Landau or local Cahn–Hilliard models, the gradient term is explicit:

$$E[\varphi] = \int_\Omega \left( \frac{\varepsilon}{2} |\nabla \varphi|^2 + \frac{1}{\varepsilon} W(\varphi) \right) dx,$$

with $W(\varphi)$ a double-well potential and $\varepsilon$ the interface thickness parameter (Demont et al., 2022).

Extensions to multicomponent or multiphase mixtures introduce multiple order parameters, Onsager-structured dissipation, and in some cases, higher-rank Korteweg matrices to capture cross-gradient couplings and interfacial energies between arbitrary pairs of phases (Brannick et al., 2014, Benilov, 2022). Ternary systems and gas-liquid-solid models incorporate additional penalty and wetting terms to enforce correct near-wall behavior and static contact angles (Zhan et al., 2023).

2. Governing Equations and Coupling Mechanisms

Diffuse-interface models are typically formulated as coupled systems of (i) convective Cahn–Hilliard or Allen–Cahn equations for the evolution of phase fields, and (ii) Navier–Stokes–type equations for velocity and pressure, with capillary or Korteweg stress contributions. For incompressible isothermal two-phase mixtures (matched densities), the governing equations are ((Frigeri et al., 2013), Model H class):

• Navier–Stokes:

$$\partial_t u - \nu \Delta u + (u \cdot \nabla) u + \nabla\pi = \mu \nabla \varphi + h,\quad \nabla \cdot u = 0$$

• Convective nonlocal Cahn–Hilliard:

$$\partial_t \varphi + \nabla \cdot (u\varphi) = \nabla \cdot (m(\varphi)\nabla \mu),\quad \mu = a(x)\varphi - J * \varphi + F'(\varphi)$$

where $m(\varphi)\geq 0$ is the (possibly degenerate) mobility, $\mu$ is the chemical potential, and $h$ is an external force.

For variable-density models, such as Abels–Garcke–Grün, the momentum equation is further coupled to interpolation laws for density and viscosity in the phase field, and the momentum equation contains a capillary body force $-\mu \nabla \varphi$ (Demont et al., 2022). Magnetohydrodynamic and electrohydrodynamic extensions incorporate Lorentz forces and electrowetting terms (Zhang, 2021, Nochetto et al., 2011).

In reactive or elastoplastic systems, the conserved variable set is extended to include reaction progress, plastic strain, and multiple fractions, with unifying finite-volume evolution on regular grids regardless of material distinctions (Wallis et al., 2020).

3. Analytical Properties: Existence, Weak Solutions, and Attractors

Well-posedness theory for diffuse-interface PDEs typically involves weak solution frameworks in function spaces corresponding to incompressible hydrodynamics ($H_\sigma, V_\sigma$) and phase fields ($H=L^2, V=H^1$) (Frigeri et al., 2013). The abstract Cauchy problem is addressed via Faedo–Galerkin schemes, a priori energy estimates leveraging the dissipativity (Lyapunov property) of the total energy, and compactness techniques (Aubin–Lions, Banach–Alaoglu).

For sufficiently regular or singular (logarithmic) potentials $F$, and for non-degenerate or admissibly degenerate mobility functions $m$, the existence of global weak solutions is rigorously established. In two-dimensional settings, strict Lyapunov inequalities and asymptotic compactness yield the existence of compact global attractors in the appropriate phase spaces, often via Ball’s theory of generalized semiflows. For certain 3D convective CH systems, connected global attractors also exist under weak uniqueness (Frigeri et al., 2013).

4. Sharp-Interface Limits and Asymptotic Analysis

A central property of well-constructed diffuse-interface models is their convergence, in the limit of vanishing interface thickness ($\varepsilon \to 0$), to sharp-interface (free-boundary) models. Using matched asymptotic expansions, it is shown that, depending on the scaling of the mobility $m(\varepsilon)$, the limiting dynamics can reproduce classical two-phase Navier–Stokes with surface tension (Young–Laplace law), convective–Mullins–Sekerka dynamics, or surface-diffusion-controlled motion (Abels et al., 2012, Abels et al., 2010, Zhang, 2021).

In full generality, for $m(\varepsilon)\sim\varepsilon$ or larger, varifold (geometric measure-theoretic) convergence of weak solutions can be established, guaranteeing Laplace pressure jumps across interfaces and reproduction of the correct interfacial velocity law. However, if $m(\varepsilon)$ decays faster than $\varepsilon^3$, spurious behavior emerges: the diffuse-interface solution no longer enforces the correct pressure jump (Young–Laplace law fails), yielding physically incorrect dynamics (Abels et al., 2012).

Similar matched-asymptotic analyses quantify the relation between model parameters and experimentally measurable interfacial quantities (e.g., surface tension, contact angle), both in binary (Abels et al., 2010) and multicomponent (Benilov, 2022) scenarios.

5. Numerical Methods and Adaptive Strategies

Numerical realization of diffuse-interface models leverages both the regularity and the conservation properties of the field equations. Adaptive frameworks exploit hierarchical or local mesh refinement near the thin interfacial region, guided by a-posteriori error estimates to ensure resolution of the interface at all times while maintaining computational efficiency (Demont et al., 2022).

Key techniques include:

• $\varepsilon$-continuation: Inflating the diffuse interface thickness $\varepsilon$ and scaling mobility appropriately on coarse meshes, then refining $\varepsilon$ as adaptivity proceeds for robust Newton convergence and avoidance of severe time-step restrictions.
• Partitioned Newton–GMRES solvers: Decomposition of the Jacobian of the NSCH system into sub-blocks associated with Navier–Stokes and Cahn–Hilliard subproblems, using block-triangular preconditioners for enhanced robustness of iterative solvers.
• Time-stepping: Convex–concave splitting of the double-well potential for unconditional stability in Backward Euler steps during mesh coarsening, switching to Crank–Nicolson schemes for temporal accuracy at fine levels.
• Interface-sharpening techniques: High-order algebraic THINC-BVD reconstruction to reduce numerical interface thickness while preserving strict conservation and stability (Wallis et al., 2020).

Validation studies show error decay with interface width, and best-practice guidelines specify interface thicknesses and blending kernels for optimal accuracy in complex or irregular geometries (Treeratanaphitak et al., 2021).

6. Thermodynamic Consistency and Extensions

Diffuse-interface models are constructed to enforce both local and global dissipation (second law of thermodynamics), ensuring that the total (kinetic plus interfacial) free energy is non-increasing along solution trajectories. The energy dissipation identities are crucial both for mathematical analysis (existence, stability) and for numerical robustness (energy-stable schemes).

Inclusion of additional physics—such as electrostatics, magnetohydrodynamics, evaporation or condensation, elasticity, plasticity, reaction kinetics, and solid–fluid interaction—follows by extension of the free energy and corresponding variational or Onsager-structured dissipative functionals (Brannick et al., 2014, Zhan et al., 2023, Schreter-Fleischhacker et al., 2024, Nochetto et al., 2011). Practical models for rapid evaporation, for example, require careful density interpolation (reciprocal mixtures), corrected constitutive relations near interfaces to avoid unphysical pressure artifacts, and mass-conserving interface transport (Schreter-Fleischhacker et al., 2024).

7. Applications and Generalizations

Diffuse-interface models are used in a remarkably wide array of scientific and engineering applications, including:

• Multiphase flows with complex topology (coalescence, breakup, moving contact lines)
• Phase separation and coarsening (spinodal decomposition of binary/alloy mixtures)
• Solidification and dendritic growth with melt convection (Subhedar et al., 2019)
• Capillary-driven flows and wetting/dewetting phenomena (dynamic and equilibrium contact angles) (Brannick et al., 2014, Benilov, 2022)
• Electrowetting and electrically controlled drop manipulation (Nochetto et al., 2011)
• Tumor growth in biological tissue under reaction–diffusion/Ginzburg–Landau frameworks (Dai et al., 2015)
• Electro- and magnetohydrodynamic multiphase flows (Zhang, 2021)
• Multiphase flows in porous media and complex solid–liquid geometries (Zhan et al., 2023, Treeratanaphitak et al., 2021)
• Large-deformation elastoplastic impact, detonation, and fluid–structure interaction (Wallis et al., 2020)
• Extended to non-isothermal flow, concentration-dependent properties, and even multicomponent/multiphase flows including vapor, liquid, and solid phases (Benilov, 2022, Zhan et al., 2023).

Wherever direct front-tracking is impractical due to topological changes or unresolved geometric complexity, diffuse-interface models provide a mathematically rigorous, computationally robust, and thermodynamically sound alternative.

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References:

• Frigeri, Grasselli, Rocca, "A diffuse interface model for two-phase incompressible flows with nonlocal interactions and nonconstant mobility" (Frigeri et al., 2013).
• "A Robust and Accurate Adaptive Approximation Method for a Diffuse-Interface Model of Binary-Fluid Flows" (Demont et al., 2022).
• "Diffuse Interface Methods for Multiple Phase Materials: An Energetic Variational Approach" (Brannick et al., 2014).
• "A consistent diffuse-interface model for two-phase flow problems with rapid evaporation" (Schreter-Fleischhacker et al., 2024).
• "On Sharp Interface Limits for Diffuse Interface Models for Two-Phase Flows" (Abels et al., 2012).
• "Thermodynamically Consistent Diffuse Interface Models for Incompressible Two-Phase Flows with Different Densities" (Abels et al., 2010).
• "A thermodynamically consistent and conservative diffuse-interface model for gas-liquid-solid multiphase flows" (Zhan et al., 2023).
• "The multicomponent diffuse-interface model and its application to water/air interfaces" (Benilov, 2022).
• "A diffuse interface model for smoothed particle hydrodynamics" (Xu et al., 2018).
• "Analysis of a diffuse interface model of multispecies tumor growth" (Dai et al., 2015).