4 Fused Spheres/Spheres

Hard spheres whose centers are fixed along a straight line segment.

Assume parallel FS particles with odd number $2n+1$ of elements. with half height $h$

Excluded volume FS/FS: again FS with radius $r=2*r_{1}$ and $4n+1$ units

Excluded volume FS/SP: also FS with radius $r=r_{1}+r_{}$ ($=2*r_{1}$) and $2n+1$ subunits.

Fused spheres: upper half of a FS particle with n=2

Cosine transforms $I_{11}$ and $I_{12}$:

$$I_{ij}(k)=2 \left( I_{0}(k) + \sum_{m=1}^{M} I_{m}(k) +I_{cap}(k) \right)$$ (7)

where
$I_{0}$ ... integral over the central truncated sphere
$I_{m}$ ... $m$-th truncated sphere on one side (above or below) center
$I_{cap}$ spherical segment terminating the particle on either end
Top index $M=2n$ for FS/FS and $M=n$ for FS/SP
Cap rim: $z_{1}= (2M+1)h$, cap radius: $r_{c}=r_{1}$, cap center: $z_{0}=2Mh$

Explicit formulae for the respective integrals are given in the appendix.



For instance:
5 units, $h=0.49$ ($L=4.92$) $\Longrightarrow$$\eta=0.448$, $\lambda=1.6625 \, L = 8.18$
9 units, $h=0.25$ ($L=5.$) $\Longrightarrow$$\eta=0.511$, $\lambda=1.61 \, L = 8.05$

Applet Koda_fs: Start

MC results for Fused Spheres/Spheres:

5 units, h=0.49 (L=4.92); eta=0.475		9 units, h=0.25 (L=5.); eta=0.51
Mod. squ. smectic mode for L=4.92