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KODA THEORY FOR HGO-HGS MIXTURES

Franz J. Vesely, August 2003

DC, will you please check me?

Here are some considerations on applying Koda's ideas to a mixture of Hard Gaussian Overlap particles and Hard Gaussian Spheres (HGS). By the latter we simply mean HGOs with $\sigma_{p}=\sigma_{o}$. For simplicity we also assume that the two components have the same (half) thickness $\sigma_{o}$.

Let $\sigma_{p}, \; \sigma_{o}$ be the semi(!)axes of the HGO (species 1) particles. For an encounter between two parallel HGOs the forbidden volume is then an ellipsoid with semiaxes $\sqrt{2} \sigma_{p}$ and $\sqrt{2}\sigma_{o}$. For two HGS the radius of the forbidden volume is $\sqrt{2}\sigma_{o}$, and for the unlike interaction the semiaxes are $\sqrt{\sigma_{p}^{2}+\sigma_{o}^{2}}$ and $\sqrt{2}\sigma_{o}$.

The packing density is ill defined, as our HGOs have no constant shape. Remembering that the original idea of Berne-Pechukas was

... representing, in some sense, the distribution of matter in one of our molecular ellipsoids, the spacial extent of the distribution being of order $2 \sigma$...

we may define the molecular volumes to be $v_{HGS}=4 \pi \sigma_{o}^{3}/3$ and $v_{HGO}=f_{HGO} v_{HGS}$ with $f_{HGO}=\sigma_{p}/\sigma_{o}$. Thus the packing density becomes, with the number fractions $X$ (HGO) and $1-X$ (for HGS),

$$\eta=\rho v_{HGS} \left[ X f_{HGO} + (1-X)\right]$$ (1)

If all these definitions and relations are reasonable, then the following applet should give the results of Koda's theory.

Applet Koda_hgo: Start

Following DC's suggestion, I introduced a fudge factor $ff$ allowing for a shorter excluded volume between HS and HGO. It turns out that with $ff=0.6$, a HS number fraction of $0.6$ and a long axis $L=10$ Koda predicts smectic layering at

$\eta=1.394$, with a period of $9.35$ (with larger step sizes in $\eta$ and $k$ I found $\eta=1.416$, period $9.85$.)

This is corroborated by MC, as seen from the figures. In the second image the particle dimensions are reduced by 1/2 to make the structure more transparent.