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4.1 Hard Spherocylinders

Building upon work by Somoza-Tarazona (1989) and Poniwierski-Holyst (1988, 1990) the authors Graf and L"owen (1999) determined the phase diagram of this system. They used the MWDA according to Denton and Ashcroft. Their notation is used here.

Thermodynamic State: defined by elongation $p \equiv L/D$ (where $p=0$ for spheres) and the density relative to close packing $\rho^{*} \equiv \rho/\rho_{CP}$ with $\rho_{CP} = 2/(\sqrt{2}+\sqrt{3}p)D^{3}$. The packing density is $\eta = \rho v_{0}$, where $v_{0}$ is the volume of a particle.



Approximation for the free energy:
\begin{displaymath}
\beta \Psi(\eta)=\rho B_{2}+\beta \Psi_{CS}-4\eta
\end{displaymath} (4.1)

Here, $\beta \Psi_{CS}=\eta(4-3\eta)/(1-\eta)^{2}$ is the Carnahan-Starling approximation for hard spheres at the same packing density, and $B_{2}=\pi(2D^ {3}/3+LD^{2}+L^{2}D/4)$ is the second virial coefficient for the spherocylinder fluid.

Weighting function: In keeping with the most elementary WDA a constant weighting over the particle volumes is applied:
\begin{displaymath}
w(\vec{r}_{2}-\vec{r}_{1},\vec{\omega}_{},\vec{\omega}_{2})
...
...y}{l}{\rm for}\;\;{\rm overlap} {\rm else}\end{array}\right.
\end{displaymath} (4.2)



Parametrization of the target function: Positional and orientational densities are assumed to be decoupled:
\begin{displaymath}
\rho^{(1)}(\vec{r},\vec{\omega})=\rho(\vec{r})g(\vec{\omega})
\end{displaymath} (4.3)

with the ansatz (Onsager)
\begin{displaymath}
g(\vec{\omega}) = \frac{\alpha}{4 \pi \sinh \alpha}
\cosh (\alpha \vec{\omega}\cdot \vec{\omega}_{0})
\end{displaymath} (4.4)

where $\alpha$ governs the width of the orientational distribution.

Isotropic phase: Inserting $\rho_{iso}(\vec{r},\vec{\omega})=\rho/4\pi$ in the MWDA expressions one finds
\begin{displaymath}
\frac{\beta {\rm\cal F}_{iso}^{ex}}{N}=
\frac{2/3+p+p^{2}/4}{1/6+p/4}  \eta +
\frac{5 \eta^{2}-4\eta^{3}}{(1-\eta)^{2}}
\end{displaymath} (4.5)



Nematic phase:
\begin{displaymath}
\rho_{nem}^{(1)}(\vec{r},\vec{\omega})=\rho g(\vec{\omega})
\end{displaymath} (4.6)

The free energy density is evaluated at the weighted density
$\displaystyle \hat{\rho}_{nem}$ $\textstyle =$ $\displaystyle \frac{1}{N}\int d\vec{1}\;\rho_{nem}^{(1)}(\vec{1})
\int d\vec{2} \; w(\vec{1},\vec{2})\rho_{nem}^{(1)}(\vec{2})$ (4.7)
  $\textstyle =$ $\displaystyle \rho \frac{2/3+p+p^{2}I_{2}(2\alpha)/(2 \sinh^{2}(\alpha))}{2/3+p+(1/4)p^{2}}$ (4.8)

where $I_{2}$ is a spherical Bessel function.

Thus:
\begin{displaymath}
{\rm\cal F}_{nem}^{ex}=N \Psi(\hat{\eta}_{nem}(\alpha))
\end{displaymath} (4.9)

This free energy is then numerically minimized with respect to $\alpha$.

Smectic phase: To parametrize a periodically layered structure the ansatz
\begin{displaymath}
\rho_{sm}^{(1)}(\vec{r},\vec{\omega})=
\rho c \exp\left[\sum_{n=1}^{4}\xi_{n}cos(n\cdot kz) \right]
\end{displaymath} (4.10)

is used. For the representation of the weighted density the weighting function $w(\vec{12},\omega_{1},\omega_{2})$ is expanded in spherical harmonics.
next up previous
Next: 4.2 Ellipsoids and spheres Up: 4. Application to Liquid Previous: 4. Application to Liquid
Franz J. Vesely Oct 2001
See also:
"Computational Physics - An Introduction," Kluwer-Plenum 2001