Franz J. Vesely >   Notes > Spheroellipsoids
 


Figure 1: Definitions

Spheroellipsoids

Truncated ellipsoid capped by sphere segments; where the caps are fused to the body they have the same tangent as the ellipsoid.

Advantages:

  • (a) convex everywhere
  • (b) only one additional parameter ($z_{1}$, see below)
  • (c) shape may be tuned between spherocylinder and ellipsoid
  • (d) not homeomorphic to a sphere (like the ellipsoid); may therefore have a smectic phase
  • (e) physically more reasonable than spherocylinders, as many mesogens have a bulge around the waist

Equations:
Given long and short semiaxes $a$ and $b$ and a truncation half length $z_{1}$ we have
$z_{0}=z_{1}(1-b^{2}/a^{2})$ ... center of cap
$r_{c}^{2}=b^{2}[1-(z_{1}^{2}/a^{2})(1-b^{2}/a^{2})]$ ... radius of cap
$D=2 \, b$ ... particle width
$L=2 \, (z_{0}+r_{c})$ ... particle length



Figure 2: Hard spheroellipsoid with $a/b=6.0$, $z1/b=3.06$
In StatPhys applications, we may of course choose the unit of length; for convenience we set $b=0.5$, such that the particle width $D=1.$ The particle shape is then completely defined by the two parameters $a$ and $z_{1}$:

$z_{0}=z_{1}(1-1/4a^{2})$
$r_{c}^{2}=0.25 \,[1-(z_{1}^{2}/a^{2})(1-1/4a^{2})]$

Here is an Applet to try out various shapes:


Obviously, the shape may be continuously tuned between an ellipse (simply set $z_{1}=a$) and a spherocylinder: with a large $a$ we achieve a very small curvature of the ellipsoid near the equator, and a small truncation height $z_{1} \ll a $ produces a near-spherocylinder. Try it out with the applet!

Contact Distance between Convex Bodies

This is a fascinating topic, and I will prepare a thorough treatment in a further communication. Here, for the time being, is an Applet for determining the contact distance between two SpheroEllipses:




Here is some Fortran code (version 12.11)
[untar -> read README -> play!]
 
F. Vesely Nov-2012