SklogWiki - User contributions [en]

Liu hard disk equation of state

2020-11-11T01:59:11Z

Hqliu1018:

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 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^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}</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
|}

==References==
<references/>

[[Category: Equations of state]]

Liu hard sphere equation of state

2020-11-08T23:23:17Z

Hqliu1018:

Hongqin Liu proposed a correction to the C-S EOS which improved accuracy by almost two order of magnitude <ref>[https://arxiv.org/abs/2010.14357]</ref>:

: <math>
Z = \frac{ 1 + \eta + \eta^2 - \frac{8}{13}\eta^3 - \eta^4 + \frac{1}{2}\eta^5 }{(1-\eta)^3 }.
</math>

The conjugate virial coefficient correlation is given by:

: <math>
B_n = 0.9423n^2 + 1.28846n - 1.84615, n > 3.
</math>

The excess Helmholtz free energy is given by:

: <math>
A^{ex} = \frac{ A - A^{id}}{Nk_B}= \frac{ 188\eta - 126\eta^2 - 13\eta^4 }{52(1-\eta)^2} - \frac{5}{13} ln(1-\eta).
</math>

The isothermal compressibility is given by:

: <math>
k_T = (\eta\frac{ dZ}{d\eta} + Z)^{-1} \rho^{-1}.
</math>

where

: <math>
\frac{ dZ}{d\eta} = \frac{ 4 + 4\eta - \frac {11}{13} \eta^2 - \frac{52}{13}\eta^3 + \frac {7}{2}\eta^4 - \eta^5 }{(1-\eta)^4 }.
</math>

Liu hard sphere equation of state

2020-11-08T23:22:25Z

Hqliu1018:

Hongqin Liu proposed a correction to the C-S EOS which improved accuracy by almost two order of magnitude <ref>[https://arxiv.org/abs/2010.14357]</ref>:

: <math>
Z = \frac{ 1 + \eta + \eta^2 - \frac{8}{13}\eta^3 - \eta^4 + \frac{1}{2}\eta^5 }{(1-\eta)^3 }.
</math>

The conjugate virial coefficient correlation is given by:

: <math>
B_n = 0.9423n^2 + 1.28846n - 1.84615, n > 3.
</math>

The excess Helmholtz free energy is given by:

: <math>
A^{ex} = \frac{ A - A^{id}}{Nk_b}= \frac{ 188\eta - 126\eta^2 - 13\eta^4 }{52(1-\eta)^2} - \frac{5}{13} ln(1-\eta).
</math>

The isothermal compressibility is given by:

: <math>
k_T = (\eta\frac{ dZ}{d\eta} + Z)^{-1} \rho^{-1}.
</math>

where

: <math>
\frac{ dZ}{d\eta} = \frac{ 4 + 4\eta - \frac {11}{13} \eta^2 - \frac{52}{13}\eta^3 + \frac {7}{2}\eta^4 - \eta^5 }{(1-\eta)^4 }.
</math>

Liu hard sphere equation of state

2020-11-08T23:21:10Z

Hqliu1018:

Hongqin Liu proposed a correction to the C-S EOS which improved accuracy by almost two order of magnitude <ref>[https://arxiv.org/abs/2010.14357]</ref>:

: <math>
Z = \frac{ 1 + \eta + \eta^2 - \frac{8}{13}\eta^3 - \eta^4 + \frac{1}{2}\eta^5 }{(1-\eta)^3 }.
</math>

The conjugate virial coefficient correlation is given by:

: <math>
B_n = 0.9423n^2 + 1.28846n - 1.84615, n > 3.
</math>

The excess Helmholtz free energy is given by:

: <math>
A^{ex} = \frac{ A - A^{id}}{Nk_b}= \frac{ 188\eta - 126\eta^2 - 13\eta^4 }{52(1-\eta)^2} - \frac{5}{13} ln(1-\eta).
</math>

The isothermal compressibility is given by:

: <math>
k_T = (\eta\frac{ dZ}{d\eta})^{-1} \rho^{-1}.
</math>

where

: <math>
\frac{ dZ}{d\eta} = \frac{ 4 + 4\eta - \frac {11}{13} \eta^2 - \frac{52}{13}\eta^3 + \frac {7}{2}\eta^4 - \eta^5 }{(1-\eta)^4 }.
</math>

Liu hard sphere equation of state

2020-11-08T23:12:01Z

Hqliu1018:

Liu hard sphere equation of state

2020-11-08T23:10:42Z

Hqliu1018:

Liu hard sphere equation of state

2020-11-08T23:09:24Z

Hqliu1018:

Liu hard sphere equation of state

2020-10-28T21:09:07Z

Hqliu1018:

Liu hard sphere equation of state

2020-10-28T21:08:28Z

Hqliu1018:

Carnahan-Starling equation of state

2020-10-28T13:46:05Z

Hqliu1018: /* Liu correction */

[[Image:CS_EoS_plot.png|thumb|350px|right]]
The '''Carnahan-Starling equation of state''' is an approximate (but quite good) [[Equations of state |equation of state]] for the fluid phase of the [[hard sphere model]] in three dimensions. It is given by (Ref <ref name="CH"> [http://dx.doi.org/10.1063/1.1672048 N. F. Carnahan and K. E. Starling,"Equation of State for Nonattracting Rigid Spheres" Journal of Chemical Physics '''51''' pp. 635-636 (1969)] </ref> Eqn. 10).

: <math>
Z = \frac{ p V}{N k_B T} = \frac{ 1 + \eta + \eta^2 - \eta^3 }{(1-\eta)^3 }.
</math>

where:
*<math> Z </math> is the [[compressibility factor]]
*<math> p </math> is the [[pressure]]
*<math> V </math> is the volume
*<math> N </math> is the number of particles
*<math> k_B </math> is the [[Boltzmann constant]]
*<math> T </math> is the absolute [[temperature]]
*<math> \eta </math> is the [[packing fraction]]:

:<math> \eta = \frac{ \pi }{6} \frac{ N \sigma^3 }{V} </math>

*<math> \sigma </math> is the [[hard sphere model | hard sphere]] diameter.
The Carnahan-Starling equation of state is not applicable for packing fractions greater than 0.55 <ref>[https://arxiv.org/abs/cond-mat/0605392 Hongqin Liu "A very accurate hard sphere equation of state over the entire stable and metstable region", arXiv:cond-mat/0605392 (2006)]</ref>.
==Virial expansion==
It is interesting to compare the [[Virial equation of state | virial coefficients]] of the Carnahan-Starling equation of state (Eq. 7 of <ref name="CH"></ref>) with the [[Hard sphere: virial coefficients | hard sphere virial coefficients]] in three dimensions (exact up to <math>B_4</math>, and those of Clisby and McCoy <ref> [http://dx.doi.org/10.1007/s10955-005-8080-0 Nathan Clisby and Barry M. McCoy "Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions", Journal of Statistical Physics '''122''' pp. 15-57 (2006)] </ref>):
{| style="width:40%; height:100px" border="1"
|-
| <math>n</math> ||Clisby and McCoy ||<math>B_n=n^2+n-2</math>
|-
| 2 || 4 || 4
|-
| 3 || 10 || 10
|-
| 4 || 18.3647684 || 18
|-
| 5 || 28.224512 || 28
|-
| 6 || 39.8151475 || 40
|-
| 7 || 53.3444198 || 54
|-
| 8 || 68.5375488 || 70
|-
| 9 || 85.8128384 || 88
|-
| 10 || 105.775104 || 108
|}

==Thermodynamic expressions==
From the Carnahan-Starling equation for the fluid phase
the following thermodynamic expressions can be derived
(Ref <ref>[http://dx.doi.org/10.1063/1.469998 Lloyd L. Lee "An accurate integral equation theory for hard spheres: Role of the zero-separation theorems in the closure relation", Journal of Chemical Physics '''103''' pp. 9388-9396 (1995)]</ref> Eqs. 2.6, 2.7 and 2.8)

[[Pressure]] (compressibility):

:<math>\frac{p^{CS}V}{N k_B T } = \frac{1+ \eta + \eta^2 - \eta^3}{(1-\eta)^3}</math>

Configurational [[chemical potential]]:

:<math>\frac{ \overline{\mu }^{CS}}{k_B T} = \frac{8\eta -9 \eta^2 + 3\eta^3}{(1-\eta)^3}</math>

Isothermal [[compressibility]]:

:<math>\chi_T -1 = \frac{1}{k_BT} \left.\frac{\partial P^{CS}}{\partial \rho}\right\vert_{T} -1 = \frac{8\eta -2 \eta^2 }{(1-\eta)^4}</math>

where <math>\eta</math> is the [[packing fraction]].

Configurational [[Helmholtz energy function]]:

:<math> \frac{ A_{ex}^{CS}}{N k_B T} = \frac{4 \eta - 3 \eta^2 }{(1-\eta)^2}</math>

==The 'Percus-Yevick' derivation==
It is interesting to note (Ref <ref> [http://dx.doi.org/10.1063/1.1675048 G. A. Mansoori, N. F. Carnahan, K. E. Starling, and T. W. Leland, Jr. "Equilibrium Thermodynamic Properties of the Mixture of Hard Spheres", Journal of Chemical Physics '''54''' pp. 1523-1525 (1971)] </ref> Eq. 6) that one can arrive at the Carnahan-Starling equation of state by adding two thirds of the [[exact solution of the Percus Yevick integral equation for hard spheres]] via the compressibility route, to one third via the pressure route, i.e.

:<math>Z = \frac{ p V}{N k_B T} = \frac{2}{3} \left[ \frac{(1+\eta+\eta^2)}{(1-\eta)^3} \right] + \frac{1}{3} \left[ \frac{(1+2\eta+3\eta^2)}{(1-\eta)^2} \right] = \frac{ 1 + \eta + \eta^2 - \eta^3 }{(1-\eta)^3 }</math>

The reason for this seems to be a slight mystery (see discussion in Ref. <ref>[http://dx.doi.org/10.1021/j100356a008 Yuhua Song, E. A. Mason, and Richard M. Stratt "Why does the Carnahan-Starling equation work so well?", Journal of Physical Chemistry '''93''' pp. 6916-6919 (1989)]</ref> ).
== Kolafa correction ==
Jiri Kolafa produced a slight correction to the C-S EOS which results in improved accuracy <ref>[http://dx.doi.org/10.1063/1.4870524 Miguel Robles, Mariano López de Haro and Andrés Santos "Note: Equation of state and the freezing point in the hard-sphere model", Journal of Chemical Physics '''140''' 136101 (2014)]</ref>:

: <math>
Z = \frac{ 1 + \eta + \eta^2 - \frac{2}{3}(1+\eta) \eta^3 }{(1-\eta)^3 }.
</math>
== Liu correction ==
Hongqin Liu proposed a correction to the C-S EOS which improved accuracy by almost two order of magnitude <ref>[https://arxiv.org/abs/2010.14357]</ref>:

: <math>
Z = \frac{ 1 + \eta + \eta^2 - \frac{8}{13}\eta^3 - \eta^4 + \frac{1}{2}\eta^5 }{(1-\eta)^3 }.
</math>

== See also ==
*[[Equations of state for hard spheres]]
*[[Kolafa-Labík-Malijevský equation of state]]

== References ==
<references/>
[[Category: Equations of state]]
[[category: hard sphere]]

Carnahan-Starling equation of state

2020-10-28T13:45:43Z

Hqliu1018: /* Liu correction */

[[Image:CS_EoS_plot.png|thumb|350px|right]]
The '''Carnahan-Starling equation of state''' is an approximate (but quite good) [[Equations of state |equation of state]] for the fluid phase of the [[hard sphere model]] in three dimensions. It is given by (Ref <ref name="CH"> [http://dx.doi.org/10.1063/1.1672048 N. F. Carnahan and K. E. Starling,"Equation of State for Nonattracting Rigid Spheres" Journal of Chemical Physics '''51''' pp. 635-636 (1969)] </ref> Eqn. 10).

: <math>
Z = \frac{ p V}{N k_B T} = \frac{ 1 + \eta + \eta^2 - \eta^3 }{(1-\eta)^3 }.
</math>

where:
*<math> Z </math> is the [[compressibility factor]]
*<math> p </math> is the [[pressure]]
*<math> V </math> is the volume
*<math> N </math> is the number of particles
*<math> k_B </math> is the [[Boltzmann constant]]
*<math> T </math> is the absolute [[temperature]]
*<math> \eta </math> is the [[packing fraction]]:

:<math> \eta = \frac{ \pi }{6} \frac{ N \sigma^3 }{V} </math>

*<math> \sigma </math> is the [[hard sphere model | hard sphere]] diameter.
The Carnahan-Starling equation of state is not applicable for packing fractions greater than 0.55 <ref>[https://arxiv.org/abs/cond-mat/0605392 Hongqin Liu "A very accurate hard sphere equation of state over the entire stable and metstable region", arXiv:cond-mat/0605392 (2006)]</ref>.
==Virial expansion==
It is interesting to compare the [[Virial equation of state | virial coefficients]] of the Carnahan-Starling equation of state (Eq. 7 of <ref name="CH"></ref>) with the [[Hard sphere: virial coefficients | hard sphere virial coefficients]] in three dimensions (exact up to <math>B_4</math>, and those of Clisby and McCoy <ref> [http://dx.doi.org/10.1007/s10955-005-8080-0 Nathan Clisby and Barry M. McCoy "Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions", Journal of Statistical Physics '''122''' pp. 15-57 (2006)] </ref>):
{| style="width:40%; height:100px" border="1"
|-
| <math>n</math> ||Clisby and McCoy ||<math>B_n=n^2+n-2</math>
|-
| 2 || 4 || 4
|-
| 3 || 10 || 10
|-
| 4 || 18.3647684 || 18
|-
| 5 || 28.224512 || 28
|-
| 6 || 39.8151475 || 40
|-
| 7 || 53.3444198 || 54
|-
| 8 || 68.5375488 || 70
|-
| 9 || 85.8128384 || 88
|-
| 10 || 105.775104 || 108
|}

==Thermodynamic expressions==
From the Carnahan-Starling equation for the fluid phase
the following thermodynamic expressions can be derived
(Ref <ref>[http://dx.doi.org/10.1063/1.469998 Lloyd L. Lee "An accurate integral equation theory for hard spheres: Role of the zero-separation theorems in the closure relation", Journal of Chemical Physics '''103''' pp. 9388-9396 (1995)]</ref> Eqs. 2.6, 2.7 and 2.8)

[[Pressure]] (compressibility):

:<math>\frac{p^{CS}V}{N k_B T } = \frac{1+ \eta + \eta^2 - \eta^3}{(1-\eta)^3}</math>

Configurational [[chemical potential]]:

:<math>\frac{ \overline{\mu }^{CS}}{k_B T} = \frac{8\eta -9 \eta^2 + 3\eta^3}{(1-\eta)^3}</math>

Isothermal [[compressibility]]:

:<math>\chi_T -1 = \frac{1}{k_BT} \left.\frac{\partial P^{CS}}{\partial \rho}\right\vert_{T} -1 = \frac{8\eta -2 \eta^2 }{(1-\eta)^4}</math>

where <math>\eta</math> is the [[packing fraction]].

Configurational [[Helmholtz energy function]]:

:<math> \frac{ A_{ex}^{CS}}{N k_B T} = \frac{4 \eta - 3 \eta^2 }{(1-\eta)^2}</math>

==The 'Percus-Yevick' derivation==
It is interesting to note (Ref <ref> [http://dx.doi.org/10.1063/1.1675048 G. A. Mansoori, N. F. Carnahan, K. E. Starling, and T. W. Leland, Jr. "Equilibrium Thermodynamic Properties of the Mixture of Hard Spheres", Journal of Chemical Physics '''54''' pp. 1523-1525 (1971)] </ref> Eq. 6) that one can arrive at the Carnahan-Starling equation of state by adding two thirds of the [[exact solution of the Percus Yevick integral equation for hard spheres]] via the compressibility route, to one third via the pressure route, i.e.

:<math>Z = \frac{ p V}{N k_B T} = \frac{2}{3} \left[ \frac{(1+\eta+\eta^2)}{(1-\eta)^3} \right] + \frac{1}{3} \left[ \frac{(1+2\eta+3\eta^2)}{(1-\eta)^2} \right] = \frac{ 1 + \eta + \eta^2 - \eta^3 }{(1-\eta)^3 }</math>

The reason for this seems to be a slight mystery (see discussion in Ref. <ref>[http://dx.doi.org/10.1021/j100356a008 Yuhua Song, E. A. Mason, and Richard M. Stratt "Why does the Carnahan-Starling equation work so well?", Journal of Physical Chemistry '''93''' pp. 6916-6919 (1989)]</ref> ).
== Kolafa correction ==
Jiri Kolafa produced a slight correction to the C-S EOS which results in improved accuracy <ref>[http://dx.doi.org/10.1063/1.4870524 Miguel Robles, Mariano López de Haro and Andrés Santos "Note: Equation of state and the freezing point in the hard-sphere model", Journal of Chemical Physics '''140''' 136101 (2014)]</ref>:

: <math>
Z = \frac{ 1 + \eta + \eta^2 - \frac{2}{3}(1+\eta) \eta^3 }{(1-\eta)^3 }.
</math>
== Liu correction ==
Hongqin Liu proposed a correction to the C-S EOS which improved accuracy by almost two order of magnitude <ref>[https://arxiv.org/abs/2010.14357]</ref>:

: <math>
Z = \frac{ 1 + \eta + \eta^2 - \frac{8}{13}\eta^3 - \eta^4 + frac{1}{2}\eta^5 }{(1-\eta)^3 }.
</math>

== See also ==
*[[Equations of state for hard spheres]]
*[[Kolafa-Labík-Malijevský equation of state]]

== References ==
<references/>
[[Category: Equations of state]]
[[category: hard sphere]]

Carnahan-Starling equation of state

2020-10-28T13:44:54Z

Hqliu1018:

[[Image:CS_EoS_plot.png|thumb|350px|right]]
The '''Carnahan-Starling equation of state''' is an approximate (but quite good) [[Equations of state |equation of state]] for the fluid phase of the [[hard sphere model]] in three dimensions. It is given by (Ref <ref name="CH"> [http://dx.doi.org/10.1063/1.1672048 N. F. Carnahan and K. E. Starling,"Equation of State for Nonattracting Rigid Spheres" Journal of Chemical Physics '''51''' pp. 635-636 (1969)] </ref> Eqn. 10).

: <math>
Z = \frac{ p V}{N k_B T} = \frac{ 1 + \eta + \eta^2 - \eta^3 }{(1-\eta)^3 }.
</math>

where:
*<math> Z </math> is the [[compressibility factor]]
*<math> p </math> is the [[pressure]]
*<math> V </math> is the volume
*<math> N </math> is the number of particles
*<math> k_B </math> is the [[Boltzmann constant]]
*<math> T </math> is the absolute [[temperature]]
*<math> \eta </math> is the [[packing fraction]]:

:<math> \eta = \frac{ \pi }{6} \frac{ N \sigma^3 }{V} </math>

*<math> \sigma </math> is the [[hard sphere model | hard sphere]] diameter.
The Carnahan-Starling equation of state is not applicable for packing fractions greater than 0.55 <ref>[https://arxiv.org/abs/cond-mat/0605392 Hongqin Liu "A very accurate hard sphere equation of state over the entire stable and metstable region", arXiv:cond-mat/0605392 (2006)]</ref>.
==Virial expansion==
It is interesting to compare the [[Virial equation of state | virial coefficients]] of the Carnahan-Starling equation of state (Eq. 7 of <ref name="CH"></ref>) with the [[Hard sphere: virial coefficients | hard sphere virial coefficients]] in three dimensions (exact up to <math>B_4</math>, and those of Clisby and McCoy <ref> [http://dx.doi.org/10.1007/s10955-005-8080-0 Nathan Clisby and Barry M. McCoy "Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions", Journal of Statistical Physics '''122''' pp. 15-57 (2006)] </ref>):
{| style="width:40%; height:100px" border="1"
|-
| <math>n</math> ||Clisby and McCoy ||<math>B_n=n^2+n-2</math>
|-
| 2 || 4 || 4
|-
| 3 || 10 || 10
|-
| 4 || 18.3647684 || 18
|-
| 5 || 28.224512 || 28
|-
| 6 || 39.8151475 || 40
|-
| 7 || 53.3444198 || 54
|-
| 8 || 68.5375488 || 70
|-
| 9 || 85.8128384 || 88
|-
| 10 || 105.775104 || 108
|}

==Thermodynamic expressions==
From the Carnahan-Starling equation for the fluid phase
the following thermodynamic expressions can be derived
(Ref <ref>[http://dx.doi.org/10.1063/1.469998 Lloyd L. Lee "An accurate integral equation theory for hard spheres: Role of the zero-separation theorems in the closure relation", Journal of Chemical Physics '''103''' pp. 9388-9396 (1995)]</ref> Eqs. 2.6, 2.7 and 2.8)

[[Pressure]] (compressibility):

:<math>\frac{p^{CS}V}{N k_B T } = \frac{1+ \eta + \eta^2 - \eta^3}{(1-\eta)^3}</math>

Configurational [[chemical potential]]:

:<math>\frac{ \overline{\mu }^{CS}}{k_B T} = \frac{8\eta -9 \eta^2 + 3\eta^3}{(1-\eta)^3}</math>

Isothermal [[compressibility]]:

:<math>\chi_T -1 = \frac{1}{k_BT} \left.\frac{\partial P^{CS}}{\partial \rho}\right\vert_{T} -1 = \frac{8\eta -2 \eta^2 }{(1-\eta)^4}</math>

where <math>\eta</math> is the [[packing fraction]].

Configurational [[Helmholtz energy function]]:

:<math> \frac{ A_{ex}^{CS}}{N k_B T} = \frac{4 \eta - 3 \eta^2 }{(1-\eta)^2}</math>

==The 'Percus-Yevick' derivation==
It is interesting to note (Ref <ref> [http://dx.doi.org/10.1063/1.1675048 G. A. Mansoori, N. F. Carnahan, K. E. Starling, and T. W. Leland, Jr. "Equilibrium Thermodynamic Properties of the Mixture of Hard Spheres", Journal of Chemical Physics '''54''' pp. 1523-1525 (1971)] </ref> Eq. 6) that one can arrive at the Carnahan-Starling equation of state by adding two thirds of the [[exact solution of the Percus Yevick integral equation for hard spheres]] via the compressibility route, to one third via the pressure route, i.e.

:<math>Z = \frac{ p V}{N k_B T} = \frac{2}{3} \left[ \frac{(1+\eta+\eta^2)}{(1-\eta)^3} \right] + \frac{1}{3} \left[ \frac{(1+2\eta+3\eta^2)}{(1-\eta)^2} \right] = \frac{ 1 + \eta + \eta^2 - \eta^3 }{(1-\eta)^3 }</math>

The reason for this seems to be a slight mystery (see discussion in Ref. <ref>[http://dx.doi.org/10.1021/j100356a008 Yuhua Song, E. A. Mason, and Richard M. Stratt "Why does the Carnahan-Starling equation work so well?", Journal of Physical Chemistry '''93''' pp. 6916-6919 (1989)]</ref> ).
== Kolafa correction ==
Jiri Kolafa produced a slight correction to the C-S EOS which results in improved accuracy <ref>[http://dx.doi.org/10.1063/1.4870524 Miguel Robles, Mariano López de Haro and Andrés Santos "Note: Equation of state and the freezing point in the hard-sphere model", Journal of Chemical Physics '''140''' 136101 (2014)]</ref>:

: <math>
Z = \frac{ 1 + \eta + \eta^2 - \frac{2}{3}(1+\eta) \eta^3 }{(1-\eta)^3 }.
</math>
== Liu correction ==
Hongqin Liu proposed a correction to the C-S EOS which improved accuracy by almost two order of magnitude <ref>[https://arxiv.org/abs/2010.14357]</ref>:

: <math>
Z = \frac{ 1 + \eta + \eta^2 - \frac{8}{13}\eta^3 - \eta^4 + frac{1}{2}\ata^5 }{(1-\eta)^3 }.
</math>

== See also ==
*[[Equations of state for hard spheres]]
*[[Kolafa-Labík-Malijevský equation of state]]

== References ==
<references/>
[[Category: Equations of state]]
[[category: hard sphere]]

Liu hard disk equation of state

2020-10-24T18:59:35Z

Hqliu1018:

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>.

For the stable fluid:
:<math>Z_v = \frac{1 + \eta^2/8 + \eta^4/18 - 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}</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
|}

==References==
<references/>

[[Category: Equations of state]]

Liu hard disk equation of state

2020-10-24T14:41:34Z

Hqliu1018:

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>.

For the stable fluid:
:<math>Z_v = \frac{1 + \eta^2/8 + \eta^4/18 - 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.

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}</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
|}

==References==
<references/>

[[Category: Equations of state]]

Liu hard disk equation of state

2020-10-24T14:40:50Z

Hqliu1018:

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>.

For the stable fluid:
:<math>Z_v = \frac{1 + \eta^2/8 + \eta^4/18 - 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.

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}</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
|}

==References==
<references/>

[[Category: Equations of state]]

Equations of state for hard disks

2020-10-22T19:15:31Z

Hqliu1018:

Equations of state for the [[Hard disks |hard disk model]]:
*[[Alder-Hoover-Young hard disk equation of state |Alder-Hoover-Young]]
*[[Andrews hard disk equation of state | Andrews]]
*[[Baram and Luban hard disk equation of state | Baram and Luban]]
*[[Boublik 2D hard convex body equation of state | Boublik]]
*[[Scaled-particle theory#Equation of state of hard disks | Helfand, Frisch and Lebowitz]]
*[[Henderson hard disk equation of state | Henderson]]
*[[Liu hard disk equation of state | Liu]]
*[[Santos-Lopez de Haro-Yuste hard disk equation of state | Santos-Lopez de Haro-Yuste]]
*[[Solana hard disk equation of state | Solana]]
*[[Tejero and Cuesta hard disk equation of state |Tejero and Cuesta]]
*[[Kolafa and Rottner equation of state | Kolafa and Rottner]]
*[[Woodcock hard disk equation of state | Woodcock]]
==Computer simulation data==
The following data is taken from Table 1 of Ref. <ref>[http://dx.doi.org/10.1080/00268970600967963 J. Kolafa and M. Rottner "Simulation-based equation of state of the hard disk fluid and prediction of higher-order virial coefficients", Molecular Physics '''104''' pp. 3435 - 3441 (2006)]</ref>
:{| border="1"
|-
| <math>\rho</math> || <math>Z</math>
|-
|0.40 || 2.1514393
|-
|0.45 || 2.4276680
|-
|0.50 || 2.7601235
|-
|0.55 || 3.1647878
|-
|0.60 || 3.663691
|-
|0.65 || 4.287926
|-
|0.70 || 5.082362
|-
|0.75 || 6.113391
|-
|0.80 || 7.476491
|-
|0.83 || 8.494891
|-
|0.84 || 8.866011
|-
|0.85 || 9.245785
|-
|0.86 || 9.621609
|-
|0.87 || 9.96782
|}
==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1007/978-3-540-78767-9_3 A. Mulero, C.A. Galán, M.I. Parra and F. Cuadros "Equations of State for Hard Spheres and Hard Disks", Lecture Notes in Physics '''753''' Chapter 3 pp.37-109 (2008)]
[[Category: Equations of state]]
{{numeric}}

Liu hard disk equation of state

2020-10-22T19:13:10Z

Hqliu1018:

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>.

For the stable fluid:
:<math>Z_v = \frac{1 + \eta^2/8 + \eta^4/18 - 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.

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}</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>c </math> || 0.75
|}

==References==
<references/>

[[Category: Equations of state]]

Liu hard disk equation of state

2020-10-22T19:11:18Z

Hqliu1018:

The '''Liu''' [[Equations of state | equation of state]] for [[hard disks]] (2-dimensional [[hard sphere model | hard spheres]]) is given by Eq. 1 of
<ref>[https://arxiv.org/abs/2010.10624]</ref>.

For the stable fluid:
:<math>Z_v = \frac{1 + \eta^2/8 + \eta^4/18 - 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.

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}</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>c </math> || 0.75
|}

==References==
<references/>

[[Category: Equations of state]]

Liu hard disk equation of state

2020-10-22T19:10:47Z

Hqliu1018:

The '''Liu''' [[Equations of state | equation of state]] for [[hard disks]] (2-dimensional [[hard sphere model | hard spheres]]) is given by Eq. 1 of
<ref>[https://arxiv.org/abs/2010.10624]</ref>.

For the stable fluid:
:<math>Z_v = \frac{1 + \eta^2/8 + \eta^4/18 - 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.

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}</math>

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

==References==
<references/>

[[Category: Equations of state]]

Liu hard disk equation of state

2020-10-22T19:09:28Z

Hqliu1018:

The '''Liu''' [[Equations of state | equation of state]] for [[hard disks]] (2-dimensional [[hard sphere model | hard spheres]]) is given by Eq. 1 of
<ref>[https://arxiv.org/abs/2010.10624]</ref>.

For the stable fluid:
:<math>Z_v = \frac{1 + \eta^2/8 + \eta^4/18 - 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.

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}</math>

{| border="1"
|-
| <math>b_1</math> || <math>- 1.04191 X 10^8</math>
|-
| <math>b_2</math>|| <math>2.66813 X 10^8</math>
|-
| <math>m_1</math> || 53
|-
| <math>m_2</math> || 56
|-
| <math>c </math> || 0.75
|}

==References==
<references/>

[[Category: Equations of state]]

Liu hard disk equation of state

2020-10-22T19:03:12Z

Hqliu1018:

The '''Liu''' [[Equations of state | equation of state]] for [[hard disks]] (2-dimensional [[hard sphere model | hard spheres]]) is given by Eq. 1 of
<ref>[https://arxiv.org/abs/2010.10624]</ref>.

For the stable fluid:
:<math>Z_v = \frac{1 + \eta^2/8 + \eta^4/18 - 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.

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>

where <math>\alpha = \frac{2}{3^{1/2} \rho \sigma^2}</math>

==References==
<references/>

[[Category: Equations of state]]

Liu hard disk equation of state

2020-10-22T19:02:10Z

Hqliu1018:

The '''Liu''' [[Equations of state | equation of state]] for [[hard disks]] (2-dimensional [[hard sphere model | hard spheres]]) is given by Eq. 1 of
<ref>[https://arxiv.org/abs/2010.10624]</ref>.

For the stable fluid:
:<math>Z_v = \frac{1 + \eta^2/8 + \eta^4/18 - 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.

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>

where <math>\alpha = \frac{2}{3^{1/2} \rho}</math>

==References==
<references/>

[[Category: Equations of state]]

Liu hard disk equation of state

2020-10-22T19:01:31Z

Hqliu1018:

The '''Liu''' [[Equations of state | equation of state]] for [[hard disks]] (2-dimensional [[hard sphere model | hard spheres]]) is given by Eq. 1 of
<ref>[https://arxiv.org/abs/2010.10624]</ref>.

For the stable fluid:
:<math>Z_v = \frac{1 + \eta^2/8 + \eta^4/18 - 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.

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>

where <math>\alpha = \frac{2}{3^{1/3} \rho}</math>

==References==
<references/>

[[Category: Equations of state]]

Liu hard disk equation of state

2020-10-22T19:00:51Z

Hqliu1018:

The '''Liu''' [[Equations of state | equation of state]] for [[hard disks]] (2-dimensional [[hard sphere model | hard spheres]]) is given by Eq. 1 of
<ref>[https://arxiv.org/abs/2010.10624]</ref>.

For the stable fluid:
:<math>Z_v = \frac{1 + \eta^2/8 + \eta^4/18 - 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.

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>

where <math>\alpha = \frac{2}{3^{1/3 \rho}</math>

==References==
<references/>

[[Category: Equations of state]]

Liu hard disk equation of state

2020-10-22T18:55:40Z

Hqliu1018:

The '''Liu''' [[Equations of state | equation of state]] for [[hard disks]] (2-dimensional [[hard sphere model | hard spheres]]) is given by Eq. 1 of
<ref>[https://arxiv.org/abs/2010.10624]</ref>.

For the stable fluid:
:<math>Z_v = \frac{1 + \eta^2/8 + \eta^4/18 - 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.

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>

==References==
<references/>

[[Category: Equations of state]]

Liu hard disk equation of state

2020-10-22T18:52:21Z

Hqliu1018:

The '''Liu''' [[Equations of state | equation of state]] for [[hard disks]] (2-dimensional [[hard sphere model | hard spheres]]) is given by Eq. 1 of
<ref>[https://arxiv.org/abs/2010.10624]</ref>.

For the stable fluid:
:<math>Z_v = \frac{1 + \eta^2/8 + \eta^4/18 - 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.

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} = </math>

==References==
<references/>

[[Category: Equations of state]]

Liu hard disk equation of state

2020-10-22T18:51:09Z

Hqliu1018:

The '''Liu''' [[Equations of state | equation of state]] for [[hard disks]] (2-dimensional [[hard sphere model | hard spheres]]) is given by Eq. 1 of
<ref>[https://arxiv.org/abs/2010.10624]</ref>.

For the stable fluid:
:<math>Z_v = \frac{1 + \eta^2/8 + \eta^4/18 - 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.

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>\eta <= 0.72 </math>

<math>Z=Z_{lh} </math>

<math>\eta > 0.72 </math>

<math>Z=Z_{solid} </math>

where:
<math>Z_{solid} = </math>

==References==
<references/>

[[Category: Equations of state]]

Liu hard disk equation of state

2020-10-22T18:50:16Z

Hqliu1018:

The '''Liu''' [[Equations of state | equation of state]] for [[hard disks]] (2-dimensional [[hard sphere model | hard spheres]]) is given by Eq. 1 of
<ref>[https://arxiv.org/abs/2010.10624]</ref>.

For the stable fluid:
:<math>Z_v = \frac{1 + \eta^2/8 + \eta^4/18 - 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.

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>\eta <= 0.72 </math>
<math>Z=Z_{lh} </math>
<math>\eta > 0.72 </math>
<math>Z=Z_{solid} </math>

where:
<math>Z_{solid} = </math>

==References==
<references/>

[[Category: Equations of state]]

Liu hard disk equation of state

2020-10-22T18:47:21Z

Hqliu1018:

Liu hard disk equation of state

2020-10-22T18:44:27Z

Hqliu1018:

Liu hard disk equation of state

2020-10-22T18:43:47Z

Hqliu1018:

Liu hard disk equation of state

2020-10-22T18:42:33Z

Hqliu1018:

Liu hard disk equation of state

2020-10-22T18:41:40Z

Hqliu1018:

Liu hard disk equation of state

2020-10-22T18:39:03Z

Hqliu1018:

Liu hard disk equation of state

2020-10-22T18:38:24Z

Hqliu1018:

Liu hard disk equation of state

2020-10-22T18:38:01Z

Hqliu1018:

Liu hard disk equation of state

2020-10-22T18:36:57Z

Hqliu1018:

Liu hard disk equation of state

2020-10-22T18:32:44Z

Hqliu1018:

Liu hard disk equation of state

2020-10-22T18:32:01Z

Hqliu1018:

Liu hard disk equation of state

2020-10-22T18:30:42Z

Hqliu1018:

Liu hard disk equation of state

2020-10-22T18:29:30Z

Hqliu1018:

The '''Liu''' [[Equations of state | equation of state]] for [[hard disks]] (2-dimensional [[hard sphere model | hard spheres]]) is given by Eq. 1 of
<ref>[https://arxiv.org/abs/2010.10624]</ref>
:<math>Z = \frac{1 + \eta^2/8 - \eta^4/10}{(1-\eta)^2} </math>

:<math>Z = \frac{1 + \eta^2/8 + \eta^4/18 - \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.
==References==
<references/>

[[Category: Equations of state]]

Liu hard disk equation of state

2020-10-22T18:28:54Z

Hqliu1018:

The '''Liu''' [[Equations of state | equation of state]] for [[hard disks]] (2-dimensional [[hard sphere model | hard spheres]]) is given by Eq. 1 of
<ref>[https://arxiv.org/abs/2010.10624]</ref>
:<math>Z = \frac{1 + \eta^2/8 - \eta^4/10}{(1-\eta)^2} </math>

:<math>Z = \frac{1 + \eta^2/8 + \eta^4/18 - \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.
==References==
<references/>

[[Category: Equations of state]]

Liu hard disk equation of state

2020-10-22T18:27:34Z

Hqliu1018:

The '''Liu''' [[Equations of state | equation of state]] for [[hard disks]] (2-dimensional [[hard sphere model | hard spheres]]) is given by Eq. 1 of
<ref>[https://arxiv.org/abs/2010.10624]</ref>
:<math>Z = \frac{1 + \eta^2/8 - \eta^4/10}{(1-\eta)^2} </math>

:<math>Z = \frac{1 + \eta^2/8 + \eta^4/18 - \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.
==References==
<references/>

[[Category: Equations of state]]

Liu hard disk equation of state

2020-10-22T18:26:50Z

Hqliu1018:

The '''Liu''' [[Equations of state | equation of state]] for [[hard disks]] (2-dimensional [[hard sphere model | hard spheres]]) is given by Eq. 1 of
<ref>[https://arxiv.org/abs/2010.10624]</ref>
:<math>Z = \frac{1 + \eta^2/8 - \eta^4/10}{(1-\eta)^2} </math>

:<math>Z = \frac{1 + \eta^2/8 + \eta^4/18- \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.
==References==
<references/>

[[Category: Equations of state]]

Liu hard disk equation of state

2020-10-22T18:24:37Z

Hqliu1018:

Liu hard disk equation of state

2020-10-22T18:22:27Z

Hqliu1018:

Liu hard disk equation of state

2020-10-22T18:18:07Z

Hqliu1018: Created page with "The '''Liu''' equation of state for hard disks (2-dimensional hard spheres) is given by Eq. 1 of <ref>[https://arxiv.org/a..."