Hard ellipsoid model
Interaction Potential
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
The most widely used overlap algorithm is that of Perram and Wertheim:
Geometric properties
The mean radius of curvature is given by (Ref. 2)
![{\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],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e881e50476a138c1b82d0eff3fa835b1212160d)
and the surface area is given by
![{\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],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a0370210651aa0d817cf6bd9f11fc9898f4a23b)
where is an elliptic integral of the first kind and 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, 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
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.}