Modified Cahn-Hilliard Equation Models

• Modified Cahn-Hilliard equations are advanced models of phase separation that use nonclassical free-energy functionals and modified flow laws to capture complex interfacial phenomena.
• Modifications include the use of nonlocal singular kernels, higher-order potentials, variable gradient coefficients, dynamic boundaries, and stochastic effects to represent heterogeneous and anisotropic systems.
• This framework is pivotal for multiscale modeling in materials science, image inpainting, and biofilm simulation, providing robust theoretical and numerical insights.

The modified Cahn-Hilliard equation refers to any evolutionary model of phase separation based on a nonclassical free-energy functional or modified flow law, distinguished from the standard (isothermal, quartic-potential, constant-k, local Laplacian) Cahn-Hilliard equation. Modifications include nonlocal singular kernels, higher-order or non-polynomial bulk potentials, spatially dependent gradient coefficients, convective or viscous terms, and extended boundary or stochastic frameworks. Such generalizations capture a broader range of physical phenomena—heterogeneity, anisotropy, interfacial interactions, and robust numerical regularization—and they have become central in contemporary analysis, simulation, and multiscale modeling of phase-field systems.

1. Nonlocal Singular Free Energy Formulations

A prominent class of modified Cahn-Hilliard systems is based on nonlocal, singular free energies. The functional considered by Abels, Bosia, and Grasselli (Abels et al., 2013) is

$$\mathcal F[c] = \frac12\int_\Omega\!\!\int_\Omega (c(x)-c(y))^2\,k(x,y,x-y)\,dx\,dy + \int_\Omega f(c(x))\,dx,$$

where the interaction kernel $k(x,y,z)$ is singular of order $\alpha\in(1,2)$, and the Helmholtz energy density

$$f(c) = \frac\theta2[(1+c)\ln(1+c)+(1-c)\ln(1-c)] - \frac{\theta_c}{2}c^2,\quad 0<\theta<\theta_c,$$

exhibits logarithmic singularities. The resulting chemical potential is the variational derivative,

$$\mu(x) = Lc(x) + f'(c(x)),\qquad L\,u(x) = \mathrm{p.v.}\int_\Omega (u(x)-u(y))\,k(x,y,x-y)\,dy,$$

where $L$ is the regional fractional Laplacian associated with $k$. The evolution law,

$\partial_t c = \Delta\,\mu,$

with no-flux boundary condition for $\mu$, generates a gradient flow that conserves mass and dissipates the nonlocal energy.

Notable analytic results include global existence and uniqueness of weak solutions, regularity $c(t)\in C^\beta(\overline\Omega)$ for some $\beta>0$, and the existence of a (connected) global attractor in the phase space $$Z_m=\{c\in H^{\alpha/2}(\Omega):\mathcal F[c]<\infty, \text{mean}(c)=m\}$$. The construction relies on maximal-monotone operator theory and fine decomposition of the singular Helmholtz potential (Abels et al., 2013).

2. Higher-Order Potential and Variable Gradient Energy

For systems exhibiting stronger heterogeneity or interface physics, standard quartic potentials and constant gradient coefficients are insufficient. The model introduced in (Mchedlov-Petrosyan et al., 4 Dec 2024) proposes

$$\begin{aligned} (1 - \nu \partial_{xx})\,\partial_t\phi + \partial_x[\phi^2] &= \partial_x^2\mu,\ \mu &= -\partial_x[\kappa(\phi)\,\partial_x\phi] + f'(\phi), \end{aligned}$$

with a sixth-degree bulk potential

$f(\phi)=a_6\phi^6+a_4\phi^4+a_2\phi^2+a_0,$

and a quadratic gradient energy coefficient

$\kappa(\phi)=\kappa_2\,\phi^2+\kappa_0.$

Exact traveling-kink solutions, $\phi(x,t)=\Phi(\xi=x-ct)$, exist provided algebraic matching conditions among the coefficients are satisfied. These solutions display pronounced sensitivity to asymmetry in $f(\phi)$ and strong dependence on the variable stiffness $\kappa(\phi)$. The existence requires specific inequalities, such as $K^2=\frac{A}{3\kappa_2}\,p>0$, and the wave speed $c=-\alpha(\phi_1+\phi_2)$ is determined by these balances.

This framework enables representation of polymer blends, microemulsions, or alloys with local interfacial heterogeneity and can serve as a rigorous testbed for numerical methods in spatially non-uniform media (Mchedlov-Petrosyan et al., 4 Dec 2024).

3. Modified Evolution Laws, Dynamic Boundaries, and Non-Isothermal Coupling

Beyond modifications to the free energy, altered flow laws are often invoked for thermodynamic or kinetic reasons. Examples include:

• Dynamic boundary conditions coupling bulk and surface phase separation:

$$\begin{aligned} \phi_t &= \Delta\mu, \quad \mu = -\Delta\phi + F'(\phi) \qquad \text{(bulk)},\ (\phi|_\Gamma)_t &= \Delta_\Gamma\,\mu_\Gamma, \quad \mu_\Gamma = -k\,\Delta_\Gamma\phi + \partial_n\phi + G'(\phi) \qquad \text{(boundary)}, \end{aligned}$$

with energetic variational derivation and well-posedness results for global weak and strong solutions (Liu et al., 2017).

• Non-isothermal, viscous, and inertial Cahn-Hilliard systems:

$$\epsilon\,\chi_{tt} + \chi_t - \Delta\mu = 0,\qquad \mu = -\Delta\chi + \alpha\,\chi_t + f(\chi) - \theta,$$

together with hyperbolic heat conduction and dynamic boundary conditions, yield additional dissipation and long-time behavior results via Lojasiewicz-Simon inequalities (Cavaterra et al., 2013).

Such extended systems model kinetic and interfacial effects essential for multi-phase flows, solidification, and biofilm growth.

4. Generalized Interface Limits and Nonlocal-to-Local Convergence

In systems where the kinetic parameter $\varepsilon$ is small, sharp interface limits emerge. The Hilbert expansion methodology (Antonopoulou et al., 2013) demonstrates how generalized Cahn-Hilliard equations

$$\partial_t m^\varepsilon = \Delta\mu^\varepsilon + G_1(x;\varepsilon),\qquad \mu^\varepsilon = -\varepsilon\Delta m^\varepsilon + \varepsilon^{-1}f(m^\varepsilon) - G_2(x;\varepsilon),$$

with external forcing, converge (in the sense of level sets and $L^p$ error estimates) to moving-boundary problems (Mullins-Sekerka with nonconservative source terms). Rigorous spectral-gap analysis establishes error bounds and convergence rates.

Subsequent works (Elbar et al., 4 Jul 2024) extend the nonlocal-to-local convergence theory to degenerate mobility and logarithmic Flory-Huggins potentials. Given

$$\partial_t u_\varepsilon = \nabla\!\cdot(m(u_\varepsilon)\nabla\mu_\varepsilon),\qquad \mu_\varepsilon = B_\varepsilon[u_\varepsilon] + F'(u_\varepsilon),$$

where $m(u) = u(1-u)$, and $B_\varepsilon$ is a convolution operator, it is shown that as the interaction radius tends to zero, the solutions converge in strong norms to the classical local degenerate Cahn-Hilliard system. The analysis relies on uniform energy/entropy estimates and nonlocal compactness results.

5. Stochastic and High-Dimensional Modified Cahn-Hilliard Equations

The nonlocal stochastic Cahn-Hilliard equation (Cornalba, 2015) introduces an Itô-driven evolution over a bounded domain: $$d\phi = (-u\cdot\nabla\phi + \Delta\mu)\,dt + dw(t),\qquad \mu=a\phi-J*\phi+F'(\phi),$$ under specified regularity and covariance conditions on the Wiener process and interaction kernel. The framework establishes existence and uniqueness of weak statistical and strong solutions using variational and probabilistic techniques.

For high-dimensional, stiff settings, stabilized or regularized formulations are necessary. Deng and He (Deng et al., 7 Jan 2024) derive a second-order modified Cahn-Hilliard system via linear stabilization, yielding: $$\begin{cases} \phi_t - L_d\,\Delta\mu + f = 0,\ \mu_t + (\gamma^2/\delta)\Delta\phi - S\Delta\mu + (1/\delta)(\phi-\phi^3) + (S/L_d)f = 0, \end{cases}$$ and solve it via forward-backward stochastic neural networks, showing error control and scalability in up to 100-dimensional domains.

6. Applications: Image Inpainting, Biofilm Modeling, and Anisotropic Surface Evolution

Modified Cahn-Hilliard equations have reached beyond physical phase separation to diverse applications:

• Image inpainting utilizes PDEs of the form

$$u_t = -\Delta(\epsilon\,\Delta u-(1/\epsilon)W'(u)) + \lambda(x)(f-u),$$

with double-well potentials and spatially modulated fidelity (Xian, 2019). Stochastic representation (polynomial chaos, perturbation expansion) quantifies sensitivity to noise in initial conditions.

• Morphological shock filter variants employ evolution laws

$$\partial_t u = \Delta(-\nu\,\arctan(\Delta u)|\nabla u|-\mu\Delta u) + \lambda(u_0-u),$$

with nonlocal, edge-preserving dynamics and convexity-splitting solvers (Mitrovic et al., 2023).

• Biofilm flow modeling incorporates degenerate mobility, advection, and network source terms:

$$\frac{\partial \phi_n}{\partial t} + \nabla\cdot(\phi_n\,\mathbf v) = \nabla\cdot(\Lambda\,\phi_n\,\nabla\mu) + g_n,$$

coupled to viscoelastic stresses and nutrient transport (McClanahan et al., 2018).

• Anisotropic faceted crystal and nanostructure formation is modeled by

$$\partial_t\phi = \nabla\cdot(M\,\nabla\mu),\qquad \mu = \gamma(n)\,f(\phi) - \nabla\cdot Q + (1/2)\epsilon\,\delta G/\delta\phi,$$

with strongly nonlinear angular coefficients and simulation via multiple-relaxation-time lattice Boltzmann schemes (Liu et al., 7 Jan 2025).

7. Classification and Impact

Modified Cahn-Hilliard equations form a flexible and mathematically rigorous framework for modeling phase separation in complex fluids, alloys, imaging, and biological systems. Modifications may occur at the level of the free energy (bulk or interfacial terms), the flow law (mobility, convection, viscous or inertial effects), the domain and boundary conditions, the inclusion of nonlocal interactions, stochastic effects, or higher-order regularization. Analytical progress includes sharp interface limits, well-posedness and attractor theory, convergence of nonlocal models, and entropy methods. Numerically, stabilized schemes, energy-dissipative discretizations, neural-network solvers, and highly scalable parallel implementations have expanded applicability to high-dimensional and stiff regimes.

The ongoing refinement of these models, both in terms of mathematical techniques and physical content, continues to drive innovation in multiscale material simulation, image analysis, and interface science.