Cahn-Hilliard Phase Separation

What is modelled

The conserved Cahn-Hilliard equation moves composition down gradients of a chemical potential. A mixture with a double-well free energy is unstable for compositions inside the spinodal, small ripples of the right wavelength grow fastest and then the two phases coarsen to reduce interface area.

Equations

• ∂c/∂t = M ∇²μ, μ = A (c³ − c) − κ ∇²c, periodic grid.
• Linear growth rate λ(k) = −M k² [A (3c&sub0;² − 1) + κ k²], fastest mode k* = √(A (1 − 3c&sub0;²) / 2κ).
• Equilibrium front c = tanh(x/w), w = √(2κ/A).
• Free energy F = ∑ [A/4 (c² − 1)² + κ/2 |∇c|²].

Checks run in the page

• Measured growth of one cosine mode against the dispersion relation.
• Relaxed 1D front width against sqrt(2 kappa / A).
• Conservation of the mean composition.
• Free energy never increases.

Honest simplifications

The model is dimensionless and qualitative: no elasticity, no temperature-dependent mobility and no noise after the start. It is not a calibrated alloy or polymer solver.

• Press Off-critical droplets to see isolated drops instead of a maze.
• Press Shallow quench: the fastest wavelength is long and growth is slow.
• Press Thick interfaces and see the width follow sqrt(2 kappa / A).