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:

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

The global EoS for all phases:

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

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

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

where Failed to parse (syntax error): {\displaystyle \alpha = \frac{2}{3^{1/3 \rho}}