# Hard ellipsoid model

**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[edit]

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

where , and define the lengths of the axis.

## Overlap algorithm[edit]

The most widely used overlap algorithm is that of Perram and Wertheim
^{[1]}.

## Geometric properties[edit]

The mean radius of curvature is given by
^{[2]}
^{[3]}

and the surface area is given by

where is an elliptic integral of the first kind and is an elliptic integral of the second kind, with the amplitude being

and the moduli

and

where the anisotropy parameters, and , are

and

The volume of the ellipsoid is given by the well known

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

## Maximum packing fraction[edit]

Using event-driven molecular dynamics, it has been found that the maximally random jammed (MRJ) packing fraction for hard ellipsoids is for
models whose maximal aspect ratio is greater than
^{[4]}
^{[5]}

## Equation of state[edit]

*Main article: Hard ellipsoid equation of state*

## Virial coefficients[edit]

*Main article: Hard ellipsoids: virial coefficients*

## Phase diagram[edit]

One of the first phase diagrams of the hard ellipsoid model was that of Frenkel and Mulder (Figure 6 in
^{[6]}).
Camp and Allen later studied biaxial ellipsoids
^{[7]}. It has recently been shown
^{[8]}
^{[9]}
that the SM2 structure is more stable than the face centered cubic structure for aspect ratios and densities . An updated phase diagram, encompassing the SM2 structure structure ^{[8]} ^{[9]} and the maximal packing fraction ^{[4]} ^{[5]}, can be found in ^{[10]} ^{[11]}.

## Hard ellipse model[edit]

*Main article: Hard ellipse model*(2-dimensional ellipsoids)

## References[edit]

- ↑ 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) - ↑ G. S. Singh and B. Kumar "Geometry of hard ellipsoidal fluids and their virial coefficients", Journal of Chemical Physics
**105**pp. 2429-2435 (1996) - ↑ G. S. Singh and B. Kumar "Molecular Fluids and Liquid Crystals in Convex-Body Coordinate Systems", Annals of Physics
**294**pp. 24-47 (2001) - ↑
^{4.0}^{4.1}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) - ↑
^{5.0}^{5.1}Aleksandar Donev, Frank H. Stillinger, P. M. Chaikin and Salvatore Torquato "Unusually Dense Crystal Packings of Ellipsoids", Physical Review Letters**92**255506 (2004) - ↑ D. Frenkel and B. M. Mulder "The hard ellipsoid-of-revolution fluid I. Monte Carlo simulations", Molecular Physics
**55**pp. 1171-1192 (1985) - ↑ Philip J. Camp and Michael P. Allen "Phase diagram of the hard biaxial ellipsoid fluid", Journal of Chemical Physics
**106**pp. 6681- (1997) - ↑
^{8.0}^{8.1}M. Radu, P. Pfleiderer, T. Schilling "Solid-solid phase transition in hard ellipsoids", arXiv:0908.1043v1 7 Aug (2009) - ↑
^{9.0}^{9.1}M. Radu, P. Pfleiderer, and T. Schilling "Solid-solid phase transition in hard ellipsoids", Journal of Chemical Physics**131**164513 (2009) - ↑ G. Odriozola "Revisiting the phase diagram of hard ellipsoids", Journal of Chemical Physics
**136**134505 (2012) - ↑ G. Odriozola "Further details on the phase diagram of hard ellipsoids of revolution", Journal of Chemical Physics
**138**064501 (2013)

**Related reading**

- Michael P. Allen "Computer simulation of a biaxial liquid crystal", Liquid Crystals
**8**pp. 499-511 (1990) - Carl McBride and Enrique Lomba "Hard biaxial ellipsoids revisited: Numerical results", Fluid Phase Equilibria
**255**pp. 37-45 (2007) - Wen-Sheng Xu , Yan-Wei Li , Zhao-Yan Sun and Li-Jia An "Hard ellipses: Equation of state, structure, and self-diffusion", Journal of Chemical Physics
**139**024501 (2013)