Extended defects in hard disk system and melting criteria

The hard sphere model is widely used in description of fluids and solid media as a zero approximation to real systems. Despite the uniqueness of the model, few analytical results are known for it, both for the 2D and 3D cases. In present research we have investigated melting of the hard disk system by considering accumulation of extended defects of a certain type in the crystaline phase, and jamming of the disk packing. It results in formulation of melting criteria with lower and upper bounds on volume ratio at melting transition: $25/21 \le V/V_0 \le 5/4$. It was found that, in full agreement with the Berezinskii-Kosterlitz-Thouless-Halperin-Nelson-Young theory, the 2D crystal melts into anisotropic liquid. The second transition, which is the transition between anisotropic and isotropic liquid has volume ratio $5/4 \le V/V_0 \le 13/9$.

💡 Research Summary

The paper investigates the melting behavior of the two‑dimensional hard‑disk (hard‑sphere) system by focusing on a specific class of extended defects, which the authors refer to as “cracks”. In an ideal triangular lattice of hard disks (disk radius set to ½), a crack is defined as a planar slip that displaces one half‑plane relative to the other by √3/2 parallel to the crack and ½ normal to it. This creates a locally jammed configuration: individual disks cannot move, yet the two half‑planes can slide past each other. The authors ask how densely such cracks can be packed and what the resulting change in the system’s area (or volume in 2D) is.

Two representative defect networks are constructed. The first, called the “19‑core” configuration, contains 21 disks in a hexagonal patch. The crack network occupies an extra area of 2√3, leading to an area ratio V/V₀ = 25/21 ≈ 1.190. In this state 19 interior disks remain in contact while six corner disks sit in triangular cages; the authors argue that weak attractive perturbations would effectively lock these corner disks, rendering the whole patch “nearly locally jammed”. They identify this configuration with the solid side of the first melting transition.

Increasing the crack density yields the “7‑core” configuration, where the same crack area is retained but the disk cores are reduced to seven. Here the disks lose mutual contacts and can reorient, representing a completely unjammed, isotropic liquid state. The corresponding area ratio is V/V₀ = 13/9 ≈ 1.444.

A middle ground, the “14‑core” arrangement, gives V/V₀ = 5/4 = 1.250. This is interpreted as the anisotropic liquid (a 2D analogue of a liquid crystal) that lies between the solid and the isotropic liquid. Consequently the authors propose two melting criteria:

1. Solid → anisotropic liquid: 25/21 ≤ Vₘ/V₀ ≤ 5/4,
2. Anisotropic liquid → isotropic liquid: 5/4 ≤ Vᵢ/V₀ ≤ 13/9.

These bounds are compared with a large body of simulation and colloidal‑experiment data. Reported anomalies in the equation of state (EOS) for hard disks typically appear at V/V₀ ≈ 1.265–1.285, while orientational‑order analyses locate the solid‑to‑hexatic transition near V/V₀ ≈ 1.237–1.259. Both ranges comfortably sit within the proposed intervals, lending credence to the defect‑based picture. Moreover, the authors note that the second transition (hexatic → isotropic liquid) is often first‑order, consistent with the broader interval they obtain.

To connect the defect picture with thermodynamics, the authors write a combinatorial expression for the Gibbs free‑energy change associated with introducing k defect lines of length l into a system of N particles: ΔF = P ΔV − k k_BT ln