In the reduced units used for numerical Cahn-Hilliard runs

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

In the reduced units used for numerical Cahn-Hilliard runs, lengths and time are rescaled so that the mobility and gradient coefficient are M=κg=1M=\kappa_g=1; the length unit is this rescaled one, not the spacing of a simulation grid. The composition variable is ψ=2ϕ−1\psi = 2\phi-1, with the symmetric double-well free energy density f(ψ)=−12ψ2+14ψ4f(\psi) = -\tfrac12\psi^2+\tfrac14\psi^4 (also in these reduced units) and square-gradient free energy F[ψ]=∫dV[f(ψ)+12(∇ψ)2]F[\psi]=\int\mathrm{d}V\left[f(\psi)+\tfrac12(\nabla\psi)^2\right], with conserved flux J=−M∇μ\mathbf J = -M\nabla\mu, μ=δF/δψ\mu=\delta F/\delta\psi. Consider a quench that starts from the fully symmetric, uniform state ψˉ=0\bar\psi=0 (the deepest possible instability for this free energy). ((a)) Derive the linearised growth rate R(q)R(q) for a small sinusoidal perturbation about ψˉ=0\bar\psi=0. ((b)) Find the wavenumber qmq_m that grows fastest and the corresponding maximum growth rate Rmax⁡R_{\max}. ((c)) State the fastest-growing wavelength λm=2π/qm\lambda_m=2\pi/q_m in these reduced units.

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