Hard ellipsoid model

A prolate ellipsoid.

Interaction Potential

The general ellipsoid, also called a triaxial ellipsoid, is a quadratic surface which is given in Cartesian coordinates by

${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1$

where $a$, $b$ and $c$ define the lengths of the axis.

Overlap algorithm

The most widely used overlap algorithm is that of Perram and Wertheim:

• John W. Perram and M. S. Wertheim "Statistical mechanics of hard ellipsoids. I. Overlap algorithm and the contact function", Journal of Computational Physics 58 pp. 409-416 (1985)

Geometric properties

The mean radius of curvature is given by (Refs. 2 and 3)

$R={\frac {a}{2}}\left[{\sqrt {\frac {1+\epsilon _{b}}{1+\epsilon _{c}}}}+{\sqrt {\epsilon }}_{c}\left\{{\frac {1}{\epsilon _{c}}}F(\varphi ,k_{1})+E(\varphi ,k_{1})\right\}\right],$

and the surface area is given by

$S=2\pi a^{2}\left[1+{\sqrt {\epsilon _{c}(1+\epsilon _{b})}}\left\{{\frac {1}{\epsilon _{c}}}F(\varphi ,k_{2})+E(\varphi ,k_{2})\right\}\right],$

where $F(\varphi ,k)$ is an elliptic integral of the first kind and 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 E(\varphi,k)} is an elliptic integral of the second kind, with the amplitude being

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 \varphi = \tan^{-1} (\sqrt \epsilon_c),}

and the moduli

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 k_1= \sqrt{\frac{\epsilon_c-\epsilon_b}{\epsilon_c}},}

and

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 k_2= \sqrt{\frac{\epsilon_b (1+\epsilon_c)}{\epsilon_c(1+\epsilon_b)}},}

where the anisotropy parameters, $\epsilon _{b}$ and 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 \epsilon_c} , are

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 \epsilon_b = \left( \frac{b}{a} \right)^2 -1,}

and

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 \epsilon_c = \left( \frac{c}{a} \right)^2 -1.}

The volume of the ellipsoid is given by the well known

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 V = \frac{4 \pi}{3}abc.}

Mathematica notebook file for calculating the surface area and mean radius of curvature of an ellipsoid

See also

• Hard ellipsoid equation of state

References

1. D. Frenkel and B. M. Mulder "The hard ellipsoid-of-revolution fluid I. Monte Carlo simulations", Molecular Physics 55 pp. 1171-1192 (1985)
2. Michael P. Allen "Computer simulation of a biaxial liquid crystal", Liquid Crystals 8 pp. 499-511 (1990)
3. Philip J. Camp and Michael P. Allen "Phase diagram of the hard biaxial ellipsoid fluid", Journal of Chemical Physics 106 pp. 6681- (1997)
4. Carl McBride and Enrique Lomba "Hard biaxial ellipsoids revisited: Numerical results", Fluid Phase Equilibria 255 pp. 37-45 (2007)
5. G. S. Singh and B. Kumar "Geometry of hard ellipsoidal fluids and their virial coefficients", Journal of Chemical Physics 105 pp. 2429-2435 (1996)
6. G. S. Singh and B. Kumar "Molecular Fluids and Liquid Crystals in Convex-Body Coordinate Systems", Annals of Physics 294 pp. 24-47 (2001)