Hard ellipsoid model

A prolate ellipsoid.

Hard ellipsoids represent a natural choice for an anisotropic model. Ellipsoids can be produced from an affine transformation of the hard sphere model. However, in difference to the hard sphere model, which has fluid and solid phases, the hard ellipsoid model is also able to produce a nematic phase.

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. 5 and 6)

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

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 $E(\varphi ,k)$ is an elliptic integral of the second kind, with the amplitude being

$\varphi =\tan ^{-1}({\sqrt {\epsilon }}_{c}),$

and the moduli

$k_{1}={\sqrt {\frac {\epsilon _{c}-\epsilon _{b}}{\epsilon _{c}}}},$

and

$k_{2}={\sqrt {\frac {\epsilon _{b}(1+\epsilon _{c})}{\epsilon _{c}(1+\epsilon _{b})}}},$

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

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

and

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

The volume of the ellipsoid is given by the well known

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

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

Maximum packing fraction

Using event-driven molecular dynamics, it has been found that the maximally random jammed (MRJ) packing fraction for hard ellipsoids is $\phi =0.7707$ for models whose maximal aspect ratio is greater than ${\sqrt {3}}$.

1. Aleksandar Donev, Ibrahim Cisse, David Sachs, Evan A. Variano, Frank H. Stillinger, Robert Connelly, Salvatore Torquato, and P. M. Chaikin "Improving the Density of Jammed Disordered Packings Using Ellipsoids", Science 303 pp. 990-993 (2004)
2. Aleksandar Donev, Frank H. Stillinger, P. M. Chaikin and Salvatore Torquato "Unusually Dense Crystal Packings of Ellipsoids", Physical Review Letters 92 255506 (2004)

Equation of state

Main article: Hard ellipsoid equation of state

Virial coefficients

Main article: Hard ellipsoids: virial coefficients

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)