Hard ellipsoid model

A uniaxial prolate ellipsoid, a>b, b=c.

A uniaxial oblate ellipsoid, a>c, a=b.

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

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

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

Overlap algorithm

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

Geometric properties

The mean radius of curvature is given by

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 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

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 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 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 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

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

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, 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} 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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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

Maximum packing fraction

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

Equation of state

Main article: Hard ellipsoid equation of state

Virial coefficients

Main article: Hard ellipsoids: virial coefficients

Phase diagram

One of the first phase diagrams of the hard ellipsoid model was that of Frenkel and Mulder (Figure 6 in ). Camp and Allen later studied biaxial ellipsoids . It has recently been shown that the SM2 structure is more stable than the face centered cubic structure for aspect ratios 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 a/b \ge 2.0}

Hard ellipse model

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

References

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)