Cahn-Hilliard Conserved Spinodal Growth

worked-example

We linearize the conserved Cahn-Hilliard equation \partial_t φ = M ∇²(aφ + φ³ - κ ∇² φ) around a homogeneous mean field bar(φ). Let's calculate the growth of an infinitesimal Fourier perturbation for specific parameters:

1. Assume the parameters: Quadratic coefficient a = -1, Mean field bar(φ) = 0, Gradient penalty κ = 1, Mobility M = 1.
2. First, calculate the effective free-energy curvature: c = a + 3bar(φ)² = -1 + 3(0)² = -1.
3. The Fourier growth rate for wavenumber k is given by σ(k) = -M k²(c + κ k²).
4. For a probe wavenumber k = 0.5, substitute the values: σ(0.5) = -(1)(0.5)²(-1 + (1)(0.5)²).
5. Calculate intermediate terms: k² = 0.25, so σ(0.5) = -0.25(-1 + 0.25) = -0.25(-0.75).
6. The expected growth rate is σ(0.5) = 0.1875.
7. The fastest growing wavenumber is k_max = √(-c / (2κ)) = √(1 / 2) ≈ 0.7071.

experiment

You can interactively explore the stability regions using the Cahn-Hilliard Conserved Spinodal Growth module.

1. Set Quadratic coefficient to -1, Mean field to 0, Gradient penalty to 1, and Mobility to 1.
2. Set the Probe wavenumber to 0.5.
3. Observe that the growth rate is positive (0.1875), indicating the uniform state is unstable to perturbations of this wavelength, leading to phase separation.
4. Change the Mean field to 0.5. Notice how the curvature becomes less negative (c = -0.25) and the maximum growth rate decreases.
5. Set the Probe wavenumber to 0. The growth rate is exactly 0, confirming that the mean field (k=0 mode) cannot amplify due to conservation of mass.

limitations

The presented growth rate is derived from a linear stability analysis, which is only valid for infinitesimal perturbations. As the amplitude of the concentration fluctuations grows, nonlinear terms become significant, leading to coarsening (Ostwald ripening) which this linear model does not capture. Furthermore, k is treated as a continuous parameter for an infinite or periodic domain, without finite-container discrete mode quantization.

references

For an in-depth derivation and analysis of the Cahn-Hilliard equation, see:

• Phase-Field Methods for Multiphase Fluid Flows - Discusses the thermodynamics and gradient flow structure of the Cahn-Hilliard energy.