Liu hard disk equation of state: Difference between revisions

Latest revision as of 19:53, 28 February 2024

The Liu equation of state for hard disks (2-dimensional hard spheres) is given by Eq. 1, 9 and 13 of ^[1].

For the stable fluid:

Z_{v} = \frac{1 + η^{2} / 8 + η^{3} / 18 - 4 η^{4} / 21}{(1 - η)^{2}}

where the packing fraction is given by $η = π ρ σ^{2} / 4$ where $ρ$ is density and $σ$ is the diameter of the disks.

The EoS for the stable fluid, liquid-hexatic transition region and hexatic:

Z_{l h} = Z_{v} + \frac{b_{1} η^{m_{1}} + b_{2} η^{m_{2}}}{(1 - c η)}

The global EoS for all phases:

$Z = Z_{l h}$ , $η < = 0.72$

$Z = Z_{s o l i d}$ , $η > 0.72$

where: $Z_{s o l i d} = \frac{2}{α} + 1.9 + α - 5.2 α^{2} + 114.48 α^{4}$

and $α = \frac{2}{3^{1 / 2} ρ σ^{2}} - 1$

@@ Line 1: / Line 1: @@
 The '''Liu''' [[Equations of state | equation of state]] for [[hard disks]] (2-dimensional [[hard sphere model | hard spheres]]) is given by Eq. 1, 9 and 13 of
-<ref>[https://arxiv.org/abs/2010.10624]</ref>.
+<ref>[https://arxiv.org/abs/2010.10624 Hongqin Liu "Global equation of state and phase transitions of the hard disc systems", arXiv:2010.10624 (2020)]</ref>.
 For the stable fluid:
-:<math>Z_v = \frac{1 + \eta^2/8 + \eta^4/18 - 4 \eta^4/21}{(1-\eta)^2} </math>
+:<math>Z_v = \frac{1 + \eta^2/8 + \eta^3/18 - 4 \eta^4/21}{(1-\eta)^2} </math>
 where the packing fraction is given by <math>\eta = \pi \rho \sigma^2 /4 </math> where  <math>\rho</math> is density and <math>\sigma</math> is the diameter of the disks.
@@ Line 19: / Line 19: @@
 <math>Z_{solid} = \frac{2}{\alpha} + 1.9 + \alpha - 5.2 \alpha^2 + 114.48 \alpha^4</math>
-and <math>\alpha = \frac{2}{3^{1/2} \rho \sigma^2}</math>
+and <math>\alpha = \frac{2}{3^{1/2} \rho \sigma^2} - 1</math>