0.1 DFT for HSC/HS mixtures
My suggestion is to take a somewhat simplified version
of Velasco/Mederos/Sullivan (VMS) and generalize it to
mixtures.
The central equation in VMS is (their equ. 23)
|
(1) |
where I use the notation
for
the reduced excess free energy density. The meaning of the various terms is
as follows:
... excluded volume between two representative hard ellipsoids; these
are defined as ellipsoids with the same volume and aspect ratio (total)
as the given HSC; the ellipsoids may be transformed into representative
spheres with the same volume (to be used in Carnahan-Starling)
is the period of an eventual smectic density wave; since the system is
either homogeneous or periodic along we may use as the total system
length
... total length of the HSC, with the cylinder length,
and the cap diameter
... number density of the HSC fluid as a function of
... reduced excess free energy per particle
for a fluid of representative spheres, at some density , as given
by Carnahan and Starling
... weighted density of the HSC fluid; here
we will simplify matters by assuming only a basic weighting function
(see below)
... effective
potential acting on the HSC fluid at if a HSC is situated at
;
has the dimension of an area
A natural extension to mixtures of HSC (number fraction X) and HS is
|
(2) |
where the first and third terms refer to the HSC and HS components,
respectively and the cross term pertains to the interaction between
HSC and HS.
I have implicitely supposed that and are
equal. I am not sure about that yet, but if not, it may be possible to
symmetrize the various terms (see below).
For the cross-term I assume
|
(3) |
where
... excluded volume between an ellipsoid representing the HSC
and a HS; is symmetric in
... number densities of HSC and HS
... reduced excess free energy per particle
for a mixture of spheres, one species representing the HSC as above, the
other being the given HS. The formulae given by Mansoori, Carnahan,
Starling and Leland will be used
... weighted density of the mixture
... effective
potential acting on the SP component at if a HSC is situated at
; this must be symmetric, or symmetrized in some way, or else the terms
and must be treated separately
Now for the exact definitions:
Representative ellipsoid:
As in VMS, we choose the long and short axes
of
the ellipsoid such that the ellipsoidal volume equals
, and the aspect ratio
. This yields ...
Excluded volume:
|
(4) |
(continued...)
F. J. Vesely / University of Vienna