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Extended defects in hard disk system and melting criteria

M. V. Kondrin, Y. B. Lebed, V. V. Brazhkin

Abstract

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 $5/4 \le V/V_0 \le 13/9$.

Extended defects in hard disk system and melting criteria

Abstract

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: . 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 .

Paper Structure

This paper contains 5 equations, 4 figures.

Figures (4)

  • Figure 1: Extended defect (crack) separating two half planes in otherwise intact triangular crystal lattice consisting of hard disks. Yellow arrow shows the shift of blue half plane from positions in the ideal crystal.
  • Figure 2: Network of extended defects (cracks) in 19-core (shown in different colors) nearly jammed configuration. Grey balls stand for corner disks. Yellow and cyan lines are construction used for calculating $V/V_0$ ratio (see text).
  • Figure 3: Similar to Fig. \ref{['f:2']} but with unjammed 7-core configuration.
  • Figure 4: Equation of state of hard disk system (solid blue curve) drawn according to virial expansion and compressibility factor according to Eq. \ref{['eq:x']} in selected points (open circles). The symbols are fitted with smooth curve (dashed red) obtained by substitution $l=\alpha/\sqrt{x}$ in Eq. \ref{['eq:x']}.