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>.


For the stable fluid:
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:
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>
<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>




{| border="1"
{| border="1"
|-  
|-  
| <math>b_1</math> || <math>- 1.04191 x 10^8</math>
| <math>b_1</math> || <math>- 1.04191 * 10^8</math>
|-  
|-  
| <math>b_2</math>|| <math>2.66813 x 10^8</math>
| <math>b_2</math>|| <math>2.66813 * 10^8</math>
|-  
|-  
| <math>m_1</math> || 53
| <math>m_1</math> || 53
Line 32: Line 32:
| <math>m_2</math> || 56
| <math>m_2</math> || 56
|-  
|-  
| <math>c </math> || 0.75
| <math>\frac{1}{c}</math> || 0.75
|}
|}


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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_v = \frac{1 + \eta^2/8 + \eta^3/18 - 4 \eta^4/21}{(1-\eta)^2} }

where the packing fraction is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta = \pi \rho \sigma^2 /4 } where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho} is density and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} is the diameter of the disks.

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

The global EoS for all phases:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z=Z_{lh} } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta <= 0.72 }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z=Z_{solid} } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta > 0.72 }

where:

and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = \frac{2}{3^{1/2} \rho \sigma^2} - 1}


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle - 1.04191 * 10^8}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_2}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_1} 53
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_2} 56
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{c}} 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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