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.

• (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
\$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