A diblock copolymer chain has two chemically distinct, covalently-joined blocks (A and B) that want to demix like oil and water but cannot separate on a macroscopic scale — the covalent bond caps domain growth. The result is microphase separation into a periodic pattern whose shape depends only on the A-block volume fraction f and the segregation strength χN (χ = Flory–Huggins interaction parameter, N = chain length). This is the Ohta–Kawasaki phase-field model (1986), the standard coarse-grained description used to design directed self-assembly (DSA) patterns for sub-10 nm chip lithography:
∂φ/∂t = ∇²[ −φ + φ³ − κ∇²φ ] − α(φ − f)
φ(r,t) local A-fraction order parameter, φ∈[−1,1]
κ interface-width parameter, κ ∝ 1/χN
α long-range penalty enforcing block connectivity
→ sets the equilibrium domain period L* ∝ α^(−1/2)
The local Cahn–Hilliard term drives A and B apart (spinodal decomposition); the non-local −α(φ−f) term is the Ohta–Kawasaki correction that stops domains from coarsening past L* — it is exactly the connectivity constraint the covalent A–B bond imposes.
- f — near 0.5 the system forms alternating lamellae (sheets); moving toward the ends of the range gives hexagonally-packed cylinders, then isolated spheres of the minority block.
- χN — raises or lowers κ (interface width). Weak segregation gives diffuse, wavy boundaries; strong segregation gives sharp, well-defined domains.
- α — larger α shrinks the equilibrium period L*, packing more, smaller domains into the same box — this is the lever DSA lithography pulls to hit a target pitch.
- The grid evolves the PDE above every frame; Reseed restarts from small random noise around φ=0, the physical initial condition of a freshly quenched melt.