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):

$Z=1+B_{2}^{*}y+B_{3}^{*}y^{2}+B_{4}^{*}y^{3}+B_{5}^{*}y^{4}+{\frac {B_{2}}{4}}\left({\frac {1+y+y^{2}-y^{3}}{(1-y)^{3}}}-1-4y-10y^{2}-18.3648y^{3}-28.2245y^{4}\right)$

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

$B_{3}^{*}=10+13.094756\alpha '-2.073909\tau '+4.096689\alpha '^{2}+2.325342\tau '^{2}-5.791266\alpha '\tau ',$

$B_{4}^{*}=18.3648+27.714434\alpha '-10.2046\tau '+11.142963\alpha '^{2}+8.634491\tau '^{2}-28.279451\alpha '\tau '-17.190946\alpha '^{2}\tau '+24.188979\alpha '\tau '^{2}+0.74674\alpha '^{3}-9.455150\tau '^{3},$

and

$B_{5}^{*}=28.2245+21.288105\alpha '+4.525788\tau '+36.032793\alpha '^{2}+59.0098\tau '^{2}-118.407497\alpha '\tau '+24.164622\alpha '^{2}\tau '+139.766174\alpha '\tau '^{2}-50.490244\alpha '^{3}-120.995139\tau '^{3}+12.624655\alpha '^{3}\tau ',$

where

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

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

and

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

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)