Block Copolymer Nanoscale Self-Assembly

Segregation Strength

χN (A–B repulsion) 14.0

Block fraction f (A share) 0.50

Thermal noise (kT) 0.30

Live State

Order parameter

Sim time (τ)

0.0

Predicted morphology

Lamellae

A block (hydrophobic-like)

B block (hydrophilic-like)

How it works

An A-B diblock copolymer is a single polymer chain made of two chemically distinct segments joined end to end. Because A and B repel each other (Flory-Huggins interaction χ) but are covalently bonded and cannot fully phase-separate at the macroscale, the melt instead microphase-separates into periodic nanoscale domains a few chain-lengths wide — the same mechanism used industrially in directed self-assembly for semiconductor nanolithography.

Free energy (Flory-Huggins / Leibler):
  F/kT ∝ χN · f(1-f)  +  entropy of chain stretching
Order-disorder transition (symmetric, mean field):
  (χN)_ODT ≈ 10.5
Equilibrium morphology (strong segregation, by A fraction f):
  f ≈ 0.5           → lamellae (alternating sheets)
  0.65 < f < 0.8 (or 0.2–0.35) → hexagonal cylinders
  f < 0.2 or f > 0.8    → spheres on a BCC lattice

This simulator integrates Langevin dynamics for ~960 coarse-grained beads arranged into 8-bead chains. Bonded beads feel a harmonic spring; any two nearby beads feel a short-range soft repulsion whose strength between unlike (A-B) beads is set by the χN slider — the larger it is relative to the ≈10.5 order-disorder threshold, the more strongly A and B beads are pushed apart even though their chains keep them tethered together, driving spontaneous domain formation. The f slider sets what fraction of each chain is A-type, which sets the natural curvature of the A-B interface and therefore which morphology the domains organize into. The order parameter tracks how far the local A-B contact statistics have drifted from a fully random (well-mixed) melt: 0% is disordered, values approaching 100% indicate sharply segregated domains.

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