5 Spheroellipsoids/Spheres

Truncated ellipsoid capped by sphere segments with continuous tangents

Advantages:

(a) convex everywhere
(b) smectic phase in pure fluid
(c) shape may be tuned between spherocylinder and ellipsoid

$z_{1}$ ... truncation half length (given)
$z_{0}=z_{1}(1-a^{2}/c^{2})$ ... center of cap
$r_{c}^{2}=a^{2}[1-(z_{1}^{2}/c^{2})(1-a^{2}/c^{2})]$ ... radius of cap

Excluded volume SE-SP:
Truncation half length

$\begin{displaymath} z_{2}=z_{1}+\frac{a^{2}}{c^{2}} \, \frac{z_{1}}{\sqrt{1-(z_{1}^{2}/c^{2})(1-(a^{2}/c^{2}))}} \end{displaymath}$

(8)

$\Longrightarrow$ Chebysheff polynomials for body:

$\displaystyle I_{body}(k)$	$\textstyle \equiv$	$\displaystyle \pi \int_{-z_{2}}^{z_{2}}dz \, \rho^{2}(z) \cos kz =C \pi \int_{-u_{2}}^{u_{2}}du \, f(u) \cos kCu$
	$\textstyle \approx$	$\displaystyle C \pi \left[ \frac{c_{0}I_{0}(u_{2})}{2} + \sum_{l=2}^{10}c_{l}I_{l}(u_{2}) \right]$	(9)

with $C=c+r_{2}$ , $u_{2}=z_{2}/C$ , and

$\begin{displaymath} I_{l}(u_{2})= \int_{-u_{2}}^{u_{2}}du \, T_{l}(u) \cos kCu =... ...{2}}du \, u^{m} \cos kCu =\sum_{m=0}^{l} d_{lm} J_{m}(u_{2};k) \end{displaymath}$

(10)

The integrals $J_{m}(u;k)$ obey the recursion

$\displaystyle J_{0}(u;k)$	$\textstyle =$	$\displaystyle \frac{2}{kC} \sin kCu$	(11)
$\displaystyle J_{m+2}(u;k)$	$\textstyle =$	$\displaystyle \frac{2}{kC} u^{m+2} \sin kCu +\frac{2(m+2)}{(kC)^{2}} u^{m+1} \cos kCu - \frac{(m+2)(m+1)}{(kC)^{2}} J_{m}(u;k)$	(12)

To this we add the caps according to $I_{12}=I_{body}+2I_{cap}$ . The cap parameters are $z_{0}$ (center), $r_{c}+r_{2}$ (radius), and $z_{2}$ (rim).

Applet Koda_se: Start

(No MC results yet)

F. J. Vesely / University of Vienna