A0. Koda's stability analysis

Here we recount the second virial approximation of Koda et al. in a format that lends itself easily to a generalisation to other mixtures than the spherocylinder/sphere system. In particular, the following relations are valid for any kind of hard linear particles with cylindrical and inversion symmetry.

Let $\rho_{1}(\vec{r})$, $\rho_{2}(\vec{r})$ denote the local number densities of two species of hard particles, with $\rho_{i}(\vec{r}) = \bar{\rho}_{i} + \delta \rho_{i}(\vec{r})$ where $\bar{\rho}_{i} \equiv N_{i}/V$. The free energy of the mixture at the level of the second virial approximation is

$$F\left\{ \rho_{1}(\vec{r}),\rho_{2}(\vec{r})\right\} = F_{id... ...\} + F_{ex}\left\{ \rho_{1}(\vec{r}),\rho_{2}(\vec{r})\right\}$$ (13)

with

$$\frac{1}{kT}F_{id}\left\{...\right\} = \sum_{i=1}^{2}\int d\... ...ambda_{i} + \rho_{i}(\vec{r}) \, \ln \rho_{i}(\vec{r})\right]$$ (14)

and

$$\frac{1}{kT}F_{ex}\left\{...\right\} = -\frac{1}{2}\sum_{i=1... ...\vec{r})\, \rho_{j}(\vec{r}\,') f_{i,j}(\vec{r}\, ,\vec{r}\,')$$ (15)

where $f_{i,j}(\vec{r}\, , \vec{r}\,') \equiv \exp(-U_{i,j}/kT)-1$ is the Mayer overlap function for a pair of particles at the given positions. Since we are dealing with hard particles, $f_{i,j}=0$ except for the overlap region of the two objects, where $f_{i,j}=-1$.

The local density variations $\delta \rho_{i}(\vec{r})$ produce a free energy difference with respect to the homogeneous density system.

Ideal free energy:
The ideal part of the free energy variation per particle is given by

$$\delta f_{id} = \frac{1}{N}\,\sum_{i=1}^{2}\int d\vec{r} \left[ \left(\bar{\rho}_... ...lta \rho_{i}(\vec{r})\right) \;\; - \bar{\rho}_{i}\, \ln \bar{\rho}_{i} \right]$$ (16)

$$\approx \frac{1}{N}\,\sum_{i=1}^{2}\int d\vec{r} \left[ \delta \rho_{i}(\... ...+\frac{ \left(\delta \rho_{i}(\vec{r})\right)^{2}}{2 \, \bar{\rho}_{i}} \right]$$ (17)

To study the possible onset of a nematic-smectic transition a periodic density variation

$$\delta \rho_{i}(\vec{r}) = \bar{\rho}_{i}\, a_{i}\, \cos kz$$ (18)

is assumed and inserted in . For the term quadratic in $a_{i}$, we have

$$\delta_{2} f_{id} = \sum_{i} \frac{X_{i}}{2 \, V} \, a_{i}^{... ...d\vec{r} \, \cos^{2}kz = \sum_{i} \frac{X_{i}}{4} \, a_{i}^{2}$$ (19)

with $X_{i}\equiv \bar{\rho}_{i}/\bar{\rho}$. Thus

$$\delta_{2}f_{id} = \vec{a}^{T} \cdot \mbox{$\left( \begin{ar... ...} \equiv \vec{a}^{T} \cdot \mbox{${\bf S}$}_{id} \cdot \vec{a}$$ (20)

where $\vec{a}^{T} \equiv (a_{1},a_{2})$, and $\mbox{${\bf S}$}_{id}$ is the ideal part of the stability matrix.

Excess free energy: From and we have

$$\delta f_{ex} = -\frac{1}{2N}\sum_{i=1}^{2}\sum_{j=1}^{2}\, a_{i} a_{j} \, \bar{\rho}_{i}\bar{\rho}_{j} \, W_{ij}(k)$$ (21)

with the overlap integral defined as

$$W_{ij}(k) \equiv \int d\vec{r} \int d\vec{r}\,' \cos kz \, ... ...vec{r} \int_{v_{ij}(\vec{r})} d\vec{r}\,' \cos kz \, \cos kz'$$ (22)

where $v_{ij}(\vec{r})$ denotes the excluded volume centered at $\vec{r}$. For linear particles symmetric with respect to $z \rightarrow -z$ the general overlap integral may be simplified as

$$W_{ij}(k) = - \int_{V} d\vec{r} \cos^{2} kz \, \int_{v_{ij}... ...j}(0)}d\vec{r}\,'' \cos kz'' \equiv - \frac{V}{2} \, I_{ij}(k)$$ (23)

where $v_{ij}(0)$ is now centered at the origin.

Thus the central task in this approach is the calculation of the cosine transform of the three kinds of excluded volumes,

$$I_{ij}(k) = \int_{v_{ij}(0)} d\vec{r} \cos kz$$ (24)

For the quadratic term in the expression for the excess free energy density we have $\delta_{2} f_{ex} = \vec{a}^{T} \cdot \mbox{${\bf S}$}_{ex} \cdot \vec{a}$, where the matrix elements of $\mbox{${\bf S}$}_{ex}$ are

$$s_{ij}^{ex} = \frac{\bar{\rho}}{4} X_{i} X_{j} I_{ij}(k)$$ (25)

It is convenient here to introduce the volume of the linear particles and relate it to the volume of the hard spheres, thus: $v_{1}=f_{1}\, v_{2}$. Using the total packing fraction $\eta$ in place of $\bar{\rho}$ we have

$$s_{ij}^{ex} = \frac{1}{4} \frac{\eta}{\left[ (1-X_{sp})f_{lb}+X_{sp}\right]} X_{i} X_{j} \frac{I_{ij}(k)}{v_{2}}$$ (26)

This is a general result valid for hard particles having cylindrical and inversion symmetry ( $z \rightarrow -z$) mixed with hard spheres. Specific systems made up of spherocylinders/spheres, ellipsoids/spheres, and fused spheres/spheres will be considered below. Also, a mixture of ``spheroellipsoids'' and spheres is studied.

The total stability matrix is just $\mbox{${\bf S}$} =\mbox{${\bf S}$}_{id}+\mbox{${\bf S}$}_{ex}$. The system will become unstable towards smectic demixing if at the given densities there exists a wave number $k$ for which the determinant $\vert S\vert$ becomes zero. An eventual smectic phase will be stable if $-s_{11}/s_{22}<0$.

Applet Koda_sc: Start