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)
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F(\varphi ,k)} is an elliptic integral of the first kind and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle E(\varphi ,k)} is an elliptic integral of the second kind, with the amplitude being
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \varphi =\tan ^{-1}({\sqrt {\epsilon }}_{c}),}
and the moduli
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle k_{1}={\sqrt {\frac {\epsilon _{c}-\epsilon _{b}}{\epsilon _{c}}}},}
and
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle k_{2}={\sqrt {\frac {\epsilon _{b}(1+\epsilon _{c})}{\epsilon _{c}(1+\epsilon _{b})}}},}
where the anisotropy parameters, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \epsilon _{b}} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \epsilon _{c}} , are
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \epsilon _{b}=\left({\frac {b}{a}}\right)^{2}-1,}
and
The volume of the ellipsoid is given by the well known
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V={\frac {4\pi }{3}}abc.}