Liu hard disk equation of state: Difference between revisions

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m (The definition of alpha was missing a -1. See right below Equation (13) on pp. 5 of Liu's paper.)
 
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The '''Liu''' [[Equations of state | equation of state]] for [[hard disks]] (2-dimensional [[hard sphere model | hard spheres]]) is given by Eq. 1 of
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>.


:<math>Z = \frac{1 + \eta^2/8 + \eta^4/18- \4eta^4/21}{(1-\eta)^2} </math>
For the stable fluid:
:<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.
 
The EoS for the stable fluid, liquid-hexatic transition region and hexatic:
:<math>Z_{lh} = Z_v + \frac{b_1 \eta^{m_1} + b_2 \eta^{m_2}}{(1-c \eta)} </math>
 
The global EoS for all phases:
 
<math>Z=Z_{lh} </math>, <math>\eta <= 0.72 </math>
 
<math>Z=Z_{solid} </math>, <math>\eta > 0.72 </math>
 
where:
<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} - 1</math>
 
 
{| border="1"
|-
| <math>b_1</math> || <math>- 1.04191 * 10^8</math>
|-
| <math>b_2</math>|| <math>2.66813 * 10^8</math>
|-
| <math>m_1</math> || 53
|-
| <math>m_2</math> || 56
|-
| <math>\frac{1}{c}</math> || 0.75
|}


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.
==References==
==References==
<references/>
<references/>


[[Category: Equations of state]]
[[Category: Equations of state]]

Latest revision as of 18:53, 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:

where the packing fraction is given by where is density and is the diameter of the disks.

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

The global EoS for all phases:

,

,

where:

and


53
56
0.75

References[edit]

  1. ↑ Hongqin Liu "Global equation of state and phase transitions of the hard disc systems", arXiv:2010.10624 (2020)
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