Phase-Field – Cahn–Hilliard –

About

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

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