Three-Phase Cahn–Hilliard/Allen–Cahn Model

• The topic introduces a unified variational framework coupling conserved composition fields with non-conserved structural order parameters to simulate complex phase transitions.
• Advanced numerical discretizations like spectral and finite element methods enable high-fidelity simulation of interface dynamics and triple junction evolution.
• Physics-informed neural operators complement traditional solvers by significantly reducing PDE residual loss, enhancing scalability in multiphase material simulations.

A three-phase Cahn–Hilliard/Allen–Cahn (CH/AC) system describes the mesoscale evolution of a material with three distinct phases, coupling one or more conserved order parameters (composition fields) and two or more non-conserved structural order parameters under a unified variational framework. Such systems enable the modeling of complex phase transitions, interfacial dynamics, and microstructural evolution—including elastic effects and multiple order parameters—within a rigorous thermodynamic and mathematical structure. Analytical formulations and numerical discretizations for the three-phase CH/AC system, as well as recent physics-informed machine learning approaches for rapid resolution of these dynamics, are established in foundational literature (Gangmei et al., 24 Jul 2025, Wu et al., 2016).

1. Variational Formulation and Free-Energy Functionals

The underlying physics of three-phase CH/AC systems is constructed through a free-energy functional that incorporates chemical, gradient, and, in some applications, elastic energy terms. Two canonical forms arise:

• For composition-order-parameter systems (as in "Learning coupled Allen–Cahn and Cahn–Hilliard phase-field equations using Physics-informed neural operator (PINO)" (Gangmei et al., 24 Jul 2025)):

$$\mathcal{F}[c,\,\eta_{1},\,\eta_{2}] = \int_\Omega \bigg[ f(c,\,\eta_{1},\,\eta_{2}) + \kappa_c |\nabla c|^2 + \sum_{i=1}^2 \kappa_{\eta_i} |\nabla\eta_i|^2 + F_{el}(c,\,\eta_{1},\,\eta_{2}) \bigg]\,dV$$

where $f$ is a bulk chemical free-energy density with terms that couple $c$ and $\eta_i$, $\kappa_c$ and $\kappa_{\eta_i}$ are gradient coefficients, and $F_{el}$ captures elastic interactions.

• For multiphase concentration fields (as in (Wu et al., 2016)):

$$E[\phi] = \int_\Omega \left[ \frac{1}{2}\eta\, (\nabla\phi)^T \Lambda (\nabla\phi) + \frac{1}{\eta}F(c) \right]\;dx$$

with $c = (c_1, c_2, c_3)^T$, $\sum_i c_i=1$, capillarity matrix $\Lambda$, and bulk potential $F(c)$. Phase variables $\phi$ facilitate definition of chemical potentials and the implementation of the constraint.

Both frameworks encode the interaction between phases, gradient-penalized interfaces, and can accommodate elastic or coupling energy contributions.

2. Governing Coupled PDE System

The prototype three-phase CH/AC system comprises (using (Gangmei et al., 24 Jul 2025) notation):

• Cahn–Hilliard equation for conserved field $c(x,y,t)$:

$$\frac{\partial c}{\partial t} = M \nabla^2 \left(\frac{\partial f}{\partial c} - 2\kappa_c \nabla^2 c\right)$$

or, in expanded form,

$$\frac{\partial c}{\partial t} = M \left( \partial_x^2(\partial f/\partial c) + \partial_y^2(\partial f/\partial c) \right) - 2\kappa_c M (\partial_x^4 c + \partial_y^4 c)$$

• Allen–Cahn equations for non-conserved order parameters $\eta_i(x,y,t)$, $i=1,2$:

$$\frac{\partial\eta_i}{\partial t} = -L\left[ \frac{\partial f}{\partial \eta_i} - 2\kappa_{\eta_i}\nabla^2 \eta_i + \frac{\delta F_{el}}{\delta \eta_i} \right]$$

• Elastic coupling, present in $F_{el}(c, \eta_1, \eta_2)$:

$$\frac{\delta F_{el}}{\delta \eta_i} = 2 \eta_i(r) \left( \sum_{p=1}^2 B_{pi}(n)\, \widetilde{\theta}_p(k) \right)_r$$

where $B_{pi}(n)$ incorporates elastic tensors and $\widetilde{\theta}_p(k)$ encodes quadratic order parameter terms in Fourier space.

The coupling terms in the free energy and their variational derivatives produce interaction and feedback between the conserved and non-conserved fields, as well as elastically driven effects.

An alternate classical multiphase formulation for three concentration fields (with $\sum_i c_i = 1$) specifies Allen–Cahn or Cahn–Hilliard–type dynamics for each $c_i$ on the constraint hyperplane, using projection operations to ensure mass conservation and positivity, with all chemical potentials derived variationally as in (Wu et al., 2016).

3. Discretization and Numerical Solution Strategies

• Spatial discretization: Finite difference, finite element, or, for periodic domains, Fourier pseudo-spectral approaches. For periodic boundaries, spectral methods compute derivatives using FFT, yielding high accuracy especially for higher-order derivatives ($\nabla^4 c$) (Gangmei et al., 24 Jul 2025). For non-periodic domains, finite elements or finite differences are employed (Wu et al., 2016).
• Time integration: Schemes rely on convex splitting or implicit/semi-implicit techniques to ensure unconditional or conditional energy stability. Modified Crank–Nicolson schemes have been established to be unconditionally energy-stable for the Allen–Cahn system (Wu et al., 2016).
• Mesh and constraints: Multiphase models (with $N=3$) require that the concentrations $c_i$ satisfy $\sum_i c_i=1$ at all times and appropriate boundary conditions (Neumann or periodic).

The choice of numerical approach is dictated by physics, regularity of the domain, and the presence of high-order derivatives.

4. Coupling, Surface Tension, and Elasticity

In three-phase CH/AC systems, interfacial phenomena and elasticity are introduced through specific terms in the free energy and coupling structure:

• Surface tension and capillarity: The capillarity matrix $\tilde\Lambda$ encodes pairwise surface tensions $\sigma_{ij}$ and determines the SPD property required for well-posedness. The triangle inequalities among $\sigma_{ij}$ ensure existence of a physical solution (Wu et al., 2016).
• Elastic effects: Elastic energy contributions $F_{el}$ are often evaluated in Fourier space for systems with periodic boundary, leading to efficient computation of convolutional elastic couplings, as implemented in (Gangmei et al., 24 Jul 2025).
• Coupling via bulk energy: Terms such as $A_2(1-c)(\eta_1^2 + \eta_2^2)$ generate coupling between composition and order parameters, with variational derivatives encoding feedback in both CH and AC equations.

These elements support simulation of phenomena such as triple junction evolution, shape-driven precipitation growth, and interfacial energy minimization.

5. Machine Learning–Based Operator Approaches

Physics-Informed Neural Operator (PINO) provides an alternative to classical solvers for three-phase CH/AC systems, with network architectures designed to encode both data- and physics-based supervision (Gangmei et al., 24 Jul 2025):

• Network structure: The initial $(c, \eta_1, \eta_2)$ state is lifted into a high-dimensional space, passed through $N=4$ Fourier layers, and decoded to yield the spatiotemporal evolution on the discrete grid. Fourier convolution layers enforce periodic boundary conditions exactly.
• Loss function: Combines mean squared error with physics-informed PDE residuals for both AC and CH equations; spectral differentiation is used to compute all spatial derivatives.
• Advantages: PINO learns the solution operator for coupled second- and fourth-order PDEs, generalizes to new initializations and compositions, and allows cost-free computation of high-order derivatives via FFT-based multiplications.
• Performance: Spectral methods reduce PDE-residual loss for CH by twelve orders of magnitude over finite differences. On $128\times128\times100$ grids, inference time is similar to that of state-of-the-art spectral solvers; gains are expected for higher-dimensional systems.

A summary of PINO and baseline solver performance is given below (as in (Gangmei et al., 24 Jul 2025)):

Method	CH Residual Loss	Inference Time (s)
FDM (central)	$8\times10^{10}$	–
Fourier spectral	$1\times10^{-1}$	0.332 (PINO)
Fourier extension	$3\times10^{-1}$	0.313 (PINO)
Classical solver	–	0.254

This suggests that spectral PINO achieves much improved physical consistency in the loss landscape compared to standard finite difference approaches, particularly as the order of the PDE increases.

6. Energy Stability and Theoretical Guarantees

For the finite element discretization, the Allen–Cahn and Cahn–Hilliard schemes are supplemented by:

• Energy law: Both the Allen–Cahn and Cahn–Hilliard schemes obey a discrete energy-dissipation property, enforced unconditionally for certain time integration schemes (e.g., Crank–Nicolson for AC). Energy decreases with each time step, providing theoretical guarantees of stability (Wu et al., 2016).
• Coefficient matrix properties: For well-posedness, the capillarity coefficient matrix must be SPD on the tangent plane to the concentration simplex, equivalent to triangle inequalities on the pairwise surface tensions (Wu et al., 2016).
• Adaptive time stepping: Near singular topological events (e.g., during rapid interface evolution), time-step sizes must be reduced to maintain numerical stability.

These theoretical results guarantee physical fidelity of the numerical solution with respect to the underlying free energy dissipation.

7. Applications and Quantitative Results

Three-phase CH/AC systems, discretized or surrogate-learned, are applied in mesoscale microstructure evolution, including

• Spinodal decomposition and triple junction evolution: Numerical validation includes initialization of distinct regions, resolution of equilibrium contact angles, and observation of interface dynamics under varying surface tensions. For example, equilibrium angles of $120^\circ$ are observed when $\sigma_{ij}=1$; with altered surface tensions, contact angles satisfy Young's law, and total wetting is observed for degenerate cases (Wu et al., 2016).
• Precipitate growth in alloys: PINO-based models predict the spatiotemporal growth of $\theta'$ precipitates in Al-Cu systems, generalizing to new initial compositions and stochastic seeds (Gangmei et al., 24 Jul 2025).
• Generalization capability: PINO demonstration included relative $L_2$ errors of $5.14\times10^{-2}$ for $c$, $1.41\times10^{-2}$ for $\eta_1$, and $1.15\times10^{-2}$ for $\eta_2$ on unseen data at $t=99$.

Practical deployment of such systems provides a foundation for studies on coarsening, microstructure-controlled properties, and the design of multi-component materials, with an ongoing shift towards scalable, operator-learning methodologies.