Phase-Field – Cahn–Hilliard –

Introduction

There exists a modeling approach known as the phase-field method.
It provides an effective way to represent interface motion and morphological changes using continuous fields. The method is widely used in materials science for analyzing microscopic phase transformations, and it has been applied to many other areas as well.

Within the phase-field framework, typical evolution equations include the Cahn–Hilliard equation, a fourth-order partial differential equation, and the Allen–Cahn equation, a second-order reaction–diffusion equation.
The Cahn–Hilliard equation describes a conserved dynamics: phase separation occurs, but the total concentration remains constant over time. In other words, no reaction takes place; only diffusion-driven transport occurs, and the overall amount of the order parameter is preserved.

In contrast, the Allen–Cahn equation represents a non-conserved dynamics, where the total quantity is allowed to change because the order parameter can vary through a local reaction term.

In this article, we summarize the formulation of the Cahn–Hilliard equation and present a simple numerical experiment.

Cahn Hilliard Equation

Free-Energy Functional

Assuming that the material evolves in the direction that reduces its free energy, the Cahn–Hilliard formulation arises naturally.

F[c] =
\int_\Omega
\left(
\frac{1}{4}(c^2 - 1)^2

* \frac{\kappa}{2}|\nabla c|^2
  \right) dx
  

The first term shows the double-well potential. It states c=±1 are stable, while c=0 is unstable — this drives the phase separation.

The second term penalizes large gradients, ensuring that interfaces remain smooth and preventing abrupt, non-physical transitions.

Chemical Potential

The chemical potential μ is the variational derivative of the free energy:

\mu = \frac{\delta F}{\delta c}

The following is led from above.

\mu = c^3 - c - \kappa \Delta c

The meaning is that

• c^3 - ctendency to relax toward the stable states
• \kappa \Delta c smooths the interface by penalizing curvature

The Cahn–Hilliard equation represents a conserved dynamics

This means that the order parameter (e.g., concentration or composition) evolves in such a way that its total amount remains constant over time. The system only redistributes the quantity through diffusion-like fluxes, without creating or destroying it. As a result, the spatial integral of the order parameter is preserved at all times.

In short, the Cahn–Hilliard model allows the field to move around, but never to increase or decrease in total.

Time Evolution

The evolution of the concentration field is governed by a mass-conserving diffusion equation:

\frac{\partial c}{\partial t} = \nabla \cdot ( M \nabla \mu )

Together with the definition of μ, we obtain the full Cahn–Hilliard system:

\begin{cases}
\displaystyle
\frac{\partial c}{\partial t}
= \Delta \mu, \\
\displaystyle
\mu = c^{3} - c - \kappa \Delta c.
\end{cases}

Weak Form

\int_{\Omega} c^{n+1}(x) v(x) dx
+
\Delta t
\int_{\Omega}
\nabla \mu^{n+1}(x) \cdot \nabla v(x) dx
=
\int_{\Omega} c^{n}(x) v(x) dx

\\
\\

Mc^{n+1} + \Delta t K \mu^{n+1}
= M c^{n}

\int_{\Omega}
\mu^{n+1}(x) q(x) dx
-
\kappa
\int_{\Omega}
\nabla c^{n+1}(x) \cdot \nabla q(x) dx
=
\int_{\Omega}
\bigl((c^{n}(x))^{3} - c^{n}(x)\bigr) q(x) dx
\\
  -\kappa K c^{n+1} + M \mu^{n+1} = F(c^n)

Mass / Stiffness \\
  M_{ij} = \int_\Omega \phi_i \phi_j dx\\
  K_{ij} = \int_\Omega \nabla \phi_i \cdot \nabla \phi_j dx

Block Linear System

Using these, the two weak equations can be written as the block linear system

\begin{bmatrix}
M & \Delta tK \\
-\kappa K & M
\end{bmatrix}
\begin{bmatrix}
c^{n+1} \\ mu^{n+1}
\end{bmatrix}
=
\begin{bmatrix}
M c^n \\ F(c^n)
\end{bmatrix}

• The upper-left block updates the concentration
• The upper-right block couples the concentration to the chemical potential
• The lower-left block enforces the definition of μ\muμ via Laplacian
• The lower-right block is another mass matrix
• The right-hand side contains the previous concentration and the nonlinear term

Result