Here are results of a more extensive sweep through the region
$l_{1}=10.$, $l_{2}=1.0-4.5$, with
$d_{1}=d_{2}=1.$ The concentration $X_{2}$ was scanned in steps of 0.01.
For each length $l_{2}$ and each $X_{2}$ the packing density $\eta$ and the
wave number $k$ were scanned until the determinant $ S(\eta,k)$ crossed zero.
Starting at this point the nearest minimum of $S(\eta,k)$ with $S=0$ was
found using a 2D Newton-Raphson strategy.
Figure 1: "Phase surface" of the $l_{1}=10$, $d_{1}=d_{2}=1$
BC (bare cylinders) system.
Figure 2: Transition density $\eta(l_{2},X)$. Theory will produce
numbers for $\eta$, regardless of physical meaning; we have
arbitrarily cut away values of $\eta$ larger than 0.96 (white
area on top.) The region in the right upper corner,
$l_{2} \approx 1$, $X_{2} \approx 0.99$, will be shown in more
detail below.
Figure 3: Smectic period $\lambda$ as a function of $(l_{2},X)$.
Values higher than 45 are regarded as indicative of global
demixing and are cut off.
Let us now look at some details. In Figure 1 the region
$l_{2} \approx 1.-2.$, $X_{2} \approx 0.9-0.99$ has an interesting
structure which we magnify in Figure 4. The gap beginning at
$l_{2}=1.67$ and coming to a dead end at $l_{2}=2.06$ denotes the
following situation: for given $l_{2}$ (1.7, say) addition of the
short particles first destabilizes the S2 (smectically demixed)
structure, then stabilizes it once more. Inspection of the smectic
layer spacing shows what happens: the smectic structure that was
dominated by the size of the longer particles ($l_{1}=10$) is
destroyed and upon further addition of short particles is replaced
a smectic dominated by $l_{2}$. Indeed, the volume fraction of the
short species is near 0.9, meaning that we have a liquid of
short cylinders with a dilute admixture of long particles.
Figure 4: Detail in phase surface of Fig. 1.
Figure 5: Smectic wave length, detail.
The other interesting region is $l_{2}=2.17-3.12$. Above and below,
the system may attain a smectically demixed structure at all
concentrations $X_{2}$. Within the
given region, high concentrations of species 2 favour a mixed
but smectic state.
Figure 6: Phase surface, intermediate range of $l_{2}$.
Figure 7: Smectic wave length, detail.
It should be noted that some of the bifurcation points in this region
actually refer to unphysical states: they are lying in the white
area of Figure 8, where the packing density at transition is beyond
0.96. Inspecting this environment more closely, and for the moment
permitting even higher $\eta$, we see that formally the jump from
large to small smectic layer periods may occur in two steps.
Starting at $l_{2}=3.05$ we find that the period first
drops from $\approx 16$ (phase a) to $\approx 5.2$ (phase b),
then - at the remixing concentration - to $\approx 3.9$ (phase c).
While the packing density at the a-b transition is unphysically high,
the b phase may extend to reasonable densities before phase c
takes over.
Figure 8: Packing density at the smectic transition.
Table: Quantitative description of the steps in Fig. 9
Finally, the region $l_{2}= 3.99-4.50$; see Figure 1.
Below that 3.99 (but above 3.12) a smectically
demixed phase exists at all concentrations, and the transition
occurs at reasonable densities. All of a sudden,
at $l_{2} = 3.99$, a concentration
cutoff $X_{2}$ near 0.6 appears, with a smectic S1 phase at lower
concentrations and a demixed S2 phase above. The smectic period
in the demixed, low-$X_{2}$ phase corresponds to the long particle
length, while the mixed S1 phase is dominated by the short
cylinders.
Will look at the region $l_{2} \geq 4.5$ next.
Actually this region spans $l_{2}=3.99-5.56$.
Above that, there is a smectic but non-demixed phase. Figure 11
shows that the smectic period decreases smoothly from 14 to 8, in
accordance with Van Roij and Mulder's $1.4 l_{1,2}$.
The transition densities (Fig. 12) are
unspectacular, extending only near $l_{2}=5.3$ above the
arbitrary limit of 0.96.
Figure 10: Phase surface
Figure 11: $\lambda_{bif}$ surface
Figure 12: $\eta_{bif}$ surface
Addenda, Feb-26:
The entire phase surface between $l_{2}/l_{1}=0.08$ and $0.6$ is
shown in Fig. A1.
Figure A2 shows $\eta$ and $p$ at the S2 transition for $l_{2}/l_{1}=0.1$
Figure A1: Figures 1 & 10 joined together
Figure A1red: As A1, reduced width
Figure A2: l1=10.0, l2=1.0: $\eta_{bif}$ and $p_{bif}$
Figure A3: As A2 but with phase boundary of smectic S2
Figure A4: k-dependence of determinant $S(\eta_{bif},k)$. $l_{2}=1$,
$X_{2}=0.889-0.892$.
