3 Ellipsoids/Spheres

Again, compute integrals (cosine transforms) over $V_{excl}$. Here we have a problem:

V_excl for: (a) ELL-ELL, (b) ELL-SPH, (c) SPH-SPH

The excluded volume between ellipsoid and sphere is not a simple object:

$\Longrightarrow$ Approximate $V_{excl}$ by Chebyshev polynomials. Even polynomials $T_{0}, \dots T_{10}$ suffice.

$\Longrightarrow$Integrations easy!

Details:

$\rho^{2}(z)$ ... squared radius of excluded volume
$u=z/C$ where $C=c+r_{2}$ is the long half axis of the excluded oval

$$f(u) \equiv \rho^{2}(z) \approx \frac{c_{0}}{2} + \sum_{l=2}^{L-1} c_{l} \, T_{l}(u)$$ (2)

$L-1$ in the range $10-20$
only even terms
coefficients:

$$c_{l}=\frac{2}{N} \sum_{n=1}^{N} f(u_{n}) \, T_{l}(u_{n}) \;\;\;\;\;\;l=0, \dots L-1, \dots N-1$$ (3)

$N$ ... target index $N \geq L$
$u_{n}=\cos \pi (n-1/2)/N$ ... zeros of $T_{N}(u)$

Choosing $L=11$ we have

$$I_{12}(k) \equiv \pi \int_{-C}^{C}dz \, \rho^{2}(z) \cos kz =A \pi \int_{-1}^{1}du \, f(u) \cos kCu$$

$$\approx A \pi \left[ \frac{c_{0}I_{0}}{2} + \sum_{l=2}^{10}c_{l}I_{l}\right]$$ (4)

with

$$I_{l}= \int_{-1}^{1}du \, T_{l}(u) \cos kCu =\sum_{m=0}^{l} ... ...t_{-1}^{1}du \, u^{m} \cos kCu =\sum_{m=0}^{l} d_{lm} J_{m}(k)$$ (5)

$d_{lm}$ ... $m$-th polynomial coefficient of $l-th$ Chebysheff polynomial
The integrals $J_{m}(k)$ obey the recursion

$$J_{0}=\frac{2}{kC} \, \sin kC, \;\;\;\;\;\; J_{m+2}=\frac{2... ...)}{(kC)^{2}} \, \cos kC - \frac{(m+2)(m+1)}{(kC)^{2}}\, J_{m}$$ (6)

Applet Koda_el: Start

Now: Monte Carlo simulation of Spheres and parallel ellipsoids

Overlap test for ellipses and spheres: Chebysheff again!

MC results for $c/b=12$, $conc_{sph}=0.6$:

eta=0.300		eta=0.542
Mod. squ. of smectic mode