Hard ellipsoid model: Difference between revisions

Revision as of 12:15, 30 January 2008

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.

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

The mean radius of curvature is given by (Refs. 5 and 6)

R = \frac{a}{2} [\sqrt{\frac{1 + ϵ_{b}}{1 + ϵ_{c}}} + {\sqrt{ϵ}}_{c} {\frac{1}{ϵ_{c}} F (φ, k_{1}) + E (φ, k_{1})}],

and the surface area is given by

S = 2 π a^{2} [1 + \sqrt{ϵ_{c} (1 + ϵ_{b})} {\frac{1}{ϵ_{c}} F (φ, k_{2}) + E (φ, k_{2})}],

where $F (φ, k)$ is an elliptic integral of the first kind and $E (φ, k)$ is an elliptic integral of the second kind, with the amplitude being

φ = \tan^{- 1} ({\sqrt{ϵ}}_{c}),

and the moduli

k_{1} = \sqrt{\frac{ϵ_{c} - ϵ_{b}}{ϵ_{c}}},

and

k_{2} = \sqrt{\frac{ϵ_{b} (1 + ϵ_{c})}{ϵ_{c} (1 + ϵ_{b})}},

where the anisotropy parameters, $ϵ_{b}$ and $ϵ_{c}$ , are

ϵ_{b} = {(\frac{b}{a})}^{2} - 1,

and

ϵ_{c} = {(\frac{c}{a})}^{2} - 1 .

The volume of the ellipsoid is given by the well known

V = \frac{4 π}{3} a b c .

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 *[http://dx.doi.org/:10.1016/0021-9991(85)90171-8  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. 2 and 3)
+The mean radius of curvature is given by (Refs. 5 and 6)
 :<math>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],
@@ Line 44: / Line 44: @@
 [http://www.qft.iqfr.csic.es/personal/carl/SR_B2_B3_ellipsoids.nb Mathematica notebook file for calculating the surface area and mean radius of curvature of an ellipsoid]
 ==See also==
 *[[Hard ellipsoid equation of state]]