Vega equation of state for hard ellipsoids

The Vega equation of state for an isotropic fluid of hard (biaxial) ellipsoids is given by (Eq. 20):

$Z=1+B_2^{*}y+B_3^{*}{y}^2+B_4^{*}{y}^3+B_5^{*}{y}^4+\frac{B_2}{4}(\frac{1+y+y^2-y^3}{(1-y{)}^3}-1-4y-10y^2-18.3648y^3-28.2245y^4)$

where $Z$ is the compressibility factor and $y$ is the volume fraction, given by $y=\rho V$ where $\rho$ is the number density. The virial coefficients are given by the fits

$B_3^{*}=10+13.094756{\alpha }^{\prime }-2.073909{\tau }^{\prime }+4.096689\alpha {\text{'}}^2+2.325342\tau {\text{'}}^2-5.791266{\alpha }^{\prime }{\tau }^{\prime },$

$B_4^{*}=18.3648+27.714434{\alpha }^{\prime }-10.2046{\tau }^{\prime }+11.142963\alpha {\text{'}}^2+8.634491\tau {\text{'}}^2-28.279451{\alpha }^{\prime }{\tau }^{\prime }-17.190946\alpha {\text{'}}^2{\tau }^{\prime }+24.188979{\alpha }^{\prime }\tau {\text{'}}^2+0.74674\alpha {\text{'}}^3-9.455150\tau {\text{'}}^3,$

and

$B_5^{*}=28.2245+21.288105{\alpha }^{\prime }+4.525788{\tau }^{\prime }+36.032793\alpha {\text{'}}^2+59.0098\tau {\text{'}}^2-118.407497{\alpha }^{\prime }{\tau }^{\prime }+24.164622\alpha {\text{'}}^2{\tau }^{\prime }+139.766174{\alpha }^{\prime }\tau {\text{'}}^2-50.490244\alpha {\text{'}}^3-120.995139\tau {\text{'}}^3+12.624655\alpha {\text{'}}^3{\tau }^{\prime },$

where

$B_n^{*}=B_n/V^{n-1}$,

${\tau }^{\prime }=\frac{4\pi R^2}{S}-1,$

and

${\alpha }^{\prime }=\frac{RS}{3V}-1.$

where $V$ is the volume, $S$, the surface area, and $R$ the mean radius of curvature. These can be calculated using this Mathematica notebook file for calculating the surface area and mean radius of curvature of an ellipsoid. For $B_2$ see the page "Second virial coefficient".

References

Related reading

• Carl McBride and Enrique Lomba "Hard biaxial ellipsoids revisited: Numerical results", Fluid Phase Equilibria 255 pp. 37-45 (2007)