A3. Truncated ellipsoids

Again, let $a,c$ denote the short and long ($z$-) semiaxes of an ellipse. Truncate the ellipse at some distance $z_{1}$ from its center and complete the figure by circle segments having the same tangent as the ellipse at the point of fusion. Rotate the figure around the long axis to produce a spheroellipsoid. The pertinent equations are: $z_{0}=z_{1}(1-a^{2}/c^{2})$ (center of circle segment); $r_{c}^{2}=a^{2}[1-(z_{1}^{2}/c^{2})(1-a^{2}/c^{2})]$ (radius of circle segment). The volume of a spheroellipsoid is $v_{1}=v_{body}+2v_{cap}$, where $v_{body}=2 \pi c^{2}z_{1}(1-z{1}^{2}/3c^{2})$ and $v_{cap}=\pi\left[2r_{c}^{2}(z_{1}-z_{0})/3 + (z_{1}-z_{0})^{3}/3\right]$

Applying Koda's theory to a mixture of hard spheroellipsoids (SE) and spheres (SP) with the same diameter we make use of the relations derived earlier. For the encounter SE-SE between two parallel spheroellipsoids we find that the excluded volume is again a spheroellipsoid with the following parameters: axes $A=2 a$, $C=2 c$; truncation half-length $z_{1}$, center $z_{0}$ and radius $z_{c}$ of cap are all double the values pertaining to the individual particles. The cosine integral $I_{11}$ is then made up of a part referring to the truncated ellipsoidal body and to the spherical caps: $I_{11}(k)=I_{body}(k)+2 I_{cap}(k)$, where

$$I_{body}(k)/v_{2} = 12 \frac{C}{A} \frac{1}{kC} \left[ \left( 1+\frac{2}{(kC)^{2}}-\f... ...2}}{(kC)^{2}} \right) \sin kz_{1} -\frac{2kz_{1}}{(kC)^{2}} \cos kz_{1} \right]$$ (34)

and $I_{cap}$ is given in A1.

In the case of the SE-SP contact the truncated body of the excluded volume is again the high-order oval which we encountered before. The truncation half-length is now given by

$$z_{2}=z_{1}+\frac{a^{2}}{c^{2}} \, \frac{z_{1}}{\sqrt{1-(z_{1}^{2}/c^{2})(1-(a^{2}/c^{2}))}}$$ (35)

Again approximating this part of the excluded volume by Chebysheff polynomials we find

$$I_{body}(k) \equiv \pi \int_{-z_{2}}^{z_{2}}dz \, \rho^{2}(z) \cos kz =C \pi \int_{-u_{2}}^{u_{2}}du \, f(u) \cos kCu$$

$$\approx C \pi \left[ \frac{c_{0}I_{0}(u_{2})}{2} + \sum_{l=2}^{10}c_{l}I_{l}(u_{2}) \right]$$ (36)

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

$$I_{l}(u_{2})= \int_{-u_{2}}^{u_{2}}du \, T_{l}(u) \cos kCu =... ...{2}}du \, u^{m} \cos kCu =\sum_{m=0}^{l} d_{lm} J_{m}(u_{2};k)$$ (37)

The integrals $J_{m}(u;k)$ obey the same recursion as the $J_{m}(k)$ defined above:

$$J_{0}(u;k) = \frac{2}{kC} \sin kCu$$ (38)

$$J_{m+2}(u;k) = \frac{2}{kC} u^{m+2} \sin kCu +\frac{2(m+2)}{(kC)^{2}} u^{m+1} \cos kCu - \frac{(m+2)(m+1)}{(kC)^{2}} J_{m}(u;k)$$ (39)

Finally 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).