2 Spherocylinders/Spheres

Koda's Stability analysis / Second virial approximation:

$\fbox{ \parbox{450pt}{ \begin{itemize} \item Write excess free energy of mi... ...l free energy the system will prefer the smectic phase. \par \end{itemize} } }$

Central quantity: $2 \times 2$ stability matrix $\mbox{${\bf S}$}(X, \eta, k)$ with elements $s_{ij}$ that contain integrals

$\begin{displaymath} I_{ij} \equiv \int_{\textstyle V_{excl}(\vec{0})} d\vec{r} \cos kz \end{displaymath}$

Here are the matrix elements of $\mbox{${\bf S}$}_{ex}$ :

$\begin{displaymath} s_{ij}^{ex} = \frac{1}{4} \frac{\eta}{\left[ (1-X_{2})f_{1}+X_{2}\right]} X_{i} X_{j} \frac{I_{ij}(k)}{v_{2}} \end{displaymath}$

(1)

where
$\eta$ ... packing fraction (total)
$f_{1} = v_{1} / v_{2}$ ... volume ratio linear particle / sphere

This result is valid for hard particles having cylindrical and inversion symmetry ( $z \rightarrow -z$ ) mixed with hard spheres.

When $det ( \mbox{${\bf S}$})$ gets negative: $\Longrightarrow$ Smectic Demixing!

Needed: $V_{excl}$ and its cosine transform!