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+\eta ^{2}/8+\eta ^{3}/18-4\eta ^{4}/21}{(1-\eta )^{2}}}

where the packing fraction is given by $\eta =\pi \rho \sigma ^{2}/4$ where $\rho$ is density and $\sigma$ is the diameter of the disks.

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

Z_{lh}=Z_{v}+{\frac {b_{1}\eta ^{m_{1}}+b_{2}\eta ^{m_{2}}}{(1-c\eta )}}

The global EoS for all phases:

$Z=Z_{lh}$ , $\eta <=0.72$

$Z=Z_{solid}$ , $\eta >0.72$

where: $Z_{solid}={\frac {2}{\alpha }}+1.9+\alpha -5.2\alpha ^{2}+114.48\alpha ^{4}$

and $\alpha ={\frac {2}{3^{1/2}\rho \sigma ^{2}}}-1$

@@ 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>\sigma</math> is the diameter of the disks.
+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.
 The EoS for the stable fluid, liquid-hexatic transition region and hexatic:
@@ 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>
@@ Line 32: / Line 32: @@
 | <math>m_2</math> || 56
 |-
-| <math>c </math> || 0.75
+| <math>\frac{1}{c}</math> || 0.75
 |}