The reduced Cahn-Hilliard equation is run on a periodic grid of spacing

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Condensed Matter, Materials & Soft Matter · Non-equilibrium statistical mechanics and stochastic thermodynamics

Soft Matter at the Thermal Scale, Ch Spinodal decomposition and coarsening

Problem

The reduced Cahn-Hilliard equation ∂tψ=∇2(ψ3−ψ−∇2ψ)\partial_t\psi = \nabla^2(\psi^3-\psi-\nabla^2\psi) is run on a 2562256^2 periodic grid of spacing 0.50.5, time step 0.010.01, starting from a uniform mean ψ0\psi_0 plus noise of amplitude 0.010.01 (seed 11), for three mean compositions ψ0=0\psi_0 = 0, 0.30.3 and 0.50.5. Linearising about ψ0\psi_0 gives f′′=3ψ02−1f''=3\psi_0^2-1 and the fastest-growing wavevector qm=(1−3ψ02)/2q_m = \sqrt{(1-3\psi_0^2)/2} (in these reduced units, κg=1\kappa_g=1). ((a)) Find qmq_m and the predicted dominant wavelength λm=2π/qm\lambda_m = 2\pi/q_m for each of the three values of ψ0\psi_0. ((b)) Say in words why the wavelength lengthens as ψ0\psi_0 increases towards 1/31/\sqrt3.

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