Colloidal Crystal Self-Assembly

Colloidal hard spheres (with short-range attraction) undergo spontaneous crystallization at high packing fractions. The hexatic order parameter ψ₆ detects six-fold bond orientational order. The pair correlation function g(r) shows sharp peaks at crystallographic distances.

System Parameters

Step0
|ψ₆| (hexatic order)—
Kinetic energy—
PhaseFluid
2D Lennard-Jones model:
U(r) = 4ε[(σ/r)¹² - (σ/r)⁶], r < r_c

Hexatic order parameter:
ψ₆ = (1/N)·Σ_i (1/n_i)·Σ_j e^{6iθ_{ij}}
|ψ₆| → 1: perfect hexagonal crystal
|ψ₆| → 0: disordered fluid

KTHNY theory (2D melting):
Two continuous transitions:
Crystal → Hexatic (at T_m, dislocations)
Hexatic → Liquid (at T_i, disclinations)
This contrasts with 3D first-order transition.