Topic H — Phase Separation and Coarsening in Polymer Blends and Emulsions

Cahn–Hilliard · linear stability · coarsening law · parallel performance · tier: standard

Motivation

Polymer blends, porous membranes made by phase inversion, emulsions and self-assembled soft materials all form by phase separation: an unstable mixture splits into domains that then coarsen. The Cahn–Hilliard equation describes both stages. Its verification is unusually clean (a dispersion relation for the early stage, a power law for the late stage, conservation and energy decay throughout), and the late stage needs large domains and long times — a natural case for parallel computing.

Related fields: emulsions, foams and porous polymers (morphology control); self-assembly of polymers, liquid crystals and colloids; phase transitions in polymeric systems. Course week: 14 (Cahn–Hilliard, matrix-free solvers, parallel scaling).

Model

Concentration c (volume fraction of one component), chemical potential \mu, mixed form (the DOLFINx Cahn–Hilliard demo): \partial_tc=\nabla\cdot(M(c)\nabla\mu),\qquad \mu=f'(c)-\kappa\Delta c, with f(c)=\frac{c\ln c+(1-c)\ln(1-c)}{N}+\chi c(1-c) (Flory–Huggins, polymer blend; N = chain length) or the quartic double well f=Ac^2(1-c)^2 (the demo), periodic or no-flux boundaries, a random perturbation of a uniform mixture c_0, and a time-adaptive implicit scheme (θ-method or convex splitting).

Design outputs: the characteristic length L(t) (from the structure factor or the interface area) against time, the morphology (droplets vs bicontinuous) against c_0, and the effect of \chi N.

Suggested parameters

Quartic potential first (A=100, \kappa=10^{-2} in the demo’s units), then Flory–Huggins with \chi N = 2.5–5 (\chi N>2 for c_0=0.5 to be unstable); c_0 = 0.3–0.5; domain 64–256 interface widths; M constant, then M=c(1-c).

Required tasks

  1. The demo reproduced and extended: energy and mass printed every step.
  2. Early stage: growth rates of Fourier modes against the linear theory.
  3. Late stage: L(t) on two domain sizes; the exponent; morphology for three c_0.
  4. Parallel runs (1, 2 and more ranks on a large domain): time per step, iterations, and the scaling (Week 14).

Verification specific to this topic

  • Dispersion relation: linearising about c_0, a mode e^{i\mathbf k\cdot\mathbf x+\sigma t} grows with \sigma(k)=-Mk^2\big(f''(c_0)+\kappa k^2\big); the fastest mode has k^*=\sqrt{-f''(c_0)/(2\kappa)}. Seed one mode with a small amplitude and compare the measured growth rate.
  • Conservation and energy: \int c constant to round-off; the free energy \int f(c)+\frac\kappa2|\nabla c|^2 non-increasing (a property your time scheme must keep — convex splitting guarantees it, the θ-method only for small steps).
  • Coarsening law: diffusive coarsening L\propto t^{1/3} (Lifshitz–Slyozov) once the domains are much larger than the interface width.
  • MMS for the mixed system (space and time rates).

Stretch goals

  1. Phase separation in a shear flow (advected Cahn–Hilliard, coupled to Stokes: model H) — elongated domains.
  2. Ternary mixture (membrane formation by non-solvent induced phase separation, NIPS).
  3. Surface-directed spinodal decomposition at a wall with a preferential wetting energy.

Milestones

M1 demo + conservation/energy · M2 dispersion relation · M3 coarsening with two domain sizes · M4 Flory–Huggins and morphology · M5 parallel scaling.

Pitfalls

  • c leaves (0,1) with the logarithmic potential: the logarithm is undefined — clip, regularise near the bounds, or use a variational-inequality solver, and report how often it happens.
  • The interface width \sim\sqrt{\kappa/|f''|} must be resolved by several cells; otherwise the dynamics are mesh-dependent (pinning).
  • A random seed per rank makes runs with different -np incomparable: generate the noise from coordinates (as in AMPS I, topic F).

References

  • J. W. Cahn, J. E. Hilliard, “Free energy of a nonuniform system. I. Interfacial free energy”, J. Chem. Phys. 28 (1958) 258–267.
  • A. J. Bray, “Theory of phase-ordering kinetics”, Adv. Phys. 43 (1994) 357–459.
  • DOLFINx demo: Cahn–Hilliard equation (docs.fenicsproject.org, v0.11).