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)

$$\phi^{ex}[\rho_{1}]= \frac{1}{v_{11}} \frac{1}{d} \int_{0}^{... ...{1}(z') \tilde{v}_{11} \left(z-z';(\eta(z)+\eta(z'))/2 \right)$$ (1)

where I use the notation $\phi^{ex} \equiv \beta F^{ex}/V$ for the reduced excess free energy density. The meaning of the various terms is as follows:

$v_{11}$ ... 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)

$d$ is the period of an eventual smectic density wave; since the system is either homogeneous or periodic along $z$ we may use $d$ as the total system length

$L=L_{c}+D$ ... total length of the HSC, with $L_{c}$ the cylinder length, and $D$ the cap diameter

$\rho_{1}(z)$ ... number density of the HSC fluid as a function of $z$

$\psi_{11}^{CS}(\rho)$ ... reduced excess free energy per particle for a fluid of representative spheres, at some density $\rho$, as given by Carnahan and Starling

$\bar{\rho}_{11}(z))$ ... weighted density of the HSC fluid; here we will simplify matters by assuming only a basic weighting function $w_{0}$ (see below)

$\tilde{v}_{11} \left(z-z';(\eta(z)+\eta(z'))/2 \right)$ ... effective potential acting on the HSC fluid at $z'$ if a HSC is situated at $z$; $\tilde{v}_{11}$ has the dimension of an area



A natural extension to mixtures of HSC (number fraction X) and HS is

$$\phi^{ex}[\rho_{1},\rho_{2}]= X^{2} \phi_{11}^{ex}[\rho_{... ...2}^{ex}[\rho_{1},\rho_{2}] + (1-X)^{2}\phi_{22}^{ex}[\rho_{2}]$$ (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 $\phi_{12}$ and $\phi_{21}$ 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

$$\phi_{12}^{ex}[\rho_{1},\rho_{2}] = \frac{1}{v_{12}} \frac{1... ...L/2}dz' \rho_{2}(z') \tilde{v}_{12} \left(z-z';\eta(z) \right)$$ (3)

where

$v_{12}$ ... excluded volume between an ellipsoid representing the HSC and a HS; $v_{12}$ is symmetric in $1,2$

$\rho_{1}(z), \rho_{2}(z)$ ... number densities of HSC and HS

$\psi_{12}^{MCSL}(\rho)$ ... 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

$\bar{\rho}_{12}(z))$ ... weighted density of the mixture

$\tilde{v}_{12} \left(z-z';\eta(z) \right)$ ... effective potential acting on the SP component at $z'$ if a HSC is situated at $z$; this must be symmetric, or symmetrized in some way, or else the terms $\phi_{12}$ and $\phi_{21}$ must be treated separately

Now for the exact definitions:

Representative ellipsoid:

As in VMS, we choose the long and short axes $\sigma_{n},\sigma_{p}$ of the ellipsoid such that the ellipsoidal volume equals $L_{c}^{2} \pi + D^{2}/6 \pi$, and the aspect ratio $\sigma_{p}/\sigma_{n}=(L_{c}+D)/D$. This yields ...

Excluded volume:

$$v_{12} = \int d\vec{r} I(\vec{r})$$ (4)

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