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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
(where for spheres) and the density relative to
close packing
with
. The packing density is
, where is the volume of a particle.
Approximation for the free energy:
|
(4.1) |
Here,
is the Carnahan-Starling approximation for hard spheres
at the same packing density, and
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:
|
(4.2) |
Parametrization of the target function:
Positional and orientational densities are assumed to be decoupled:
|
(4.3) |
with the ansatz (Onsager)
|
(4.4) |
where governs the width of the orientational distribution.
Isotropic phase:
Inserting
in the
MWDA expressions one finds
|
(4.5) |
Nematic phase:
|
(4.6) |
The free energy density is evaluated at the weighted density
where is a spherical Bessel function.
Thus:
|
(4.9) |
This free energy is then numerically minimized with respect to
.
Smectic phase:
To parametrize a periodically layered structure the ansatz
|
(4.10) |
is used. For the representation of the weighted density the weighting function
is expanded in spherical harmonics.
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