Liu hard disk equation of state

The Liu equation of state for hard disks (2-dimensional hard spheres) is given by Eq. 1 of .

For the stable fluid:

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

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

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

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_{lh} = Z_v + \frac{b_1 \eta^{m_1} + b_2 \eta^{m_2}}{(1-c \eta)} }

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: 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_{solid} = \frac{2}{\alpha} + 1.9 + \alpha - 5.2 \alpha^2 + 114.48 \alpha^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 \alpha = \frac{2}{3^{1/3 \rho}}