Synclinic-anticlinic symmetry in the structure of multilayer polar liquid crystals

A large number of smectic phases can have the same energy when short-range interlayer interactions (interaction of nearest layers and frustrating next-nearest layer interaction) are taken into account. In this situation even weak long-range interactions become important. They lift the degeneracy and lead to the appearance of a manifold of polar structures with multilayer periodicity. We analyze the reasons for the appearance of six-layer structures observed in experiments. Due to specific symmetry of interlayer interactions each antiferroelectric structure corresponds to a ferrielectric one in which synclinic orientations of molecules in nearest layers are replaced by anticlinic and vice versa. It is shown how different long-range interlayer interactions transform the degeneracy and induce different stable and metastable structures.

Figure 1

(a) Two-component vector ${\mathbf{\xi }}_i$ characterizes the polar (angle ${\theta }_i$) and azimuthal (angle ${\phi }_i$) orientation of molecules in $i\mathrm{th}$ smectic layer. (b) Multilayer polar phases are formed by change of the orientation of ${\mathbf{\xi }}_i$ from layer to layer.

Figure 2

Orientations of ${\mathbf{\xi }}_i$ in polar phases $\mathrm{Sm}{C}^{*}$, $\mathrm{Sm}C{}_{\mathrm{A}}^{*}$, $\mathrm{Sm}C{}_{\mathrm{d}4}^{*}$, $\mathrm{Sm}C{}_{\mathrm{d}5}^{*}$, $\mathrm{Sm}C{}_{\mathrm{d}6/4\mathrm{A}}^{*}$, $\mathrm{Sm}C{}_{\mathrm{d}3}^{*}$, $\mathrm{Sm}C{}_{\mathrm{d}6/4\mathrm{F}}^{*}$, and $\mathrm{Sm}C{}_{\mathrm{d}6/2\mathrm{A}}^{*}$. $\mathrm{Sm}C{}_{\mathrm{d}3}^{*}$ and $\mathrm{Sm}C{}_{\mathrm{d}6/2\mathrm{A}}^{*}$ structures can be obtained from $\mathrm{Sm}C{}_{\mathrm{d}6/4\mathrm{A}}^{*}$ and $\mathrm{Sm}C{}_{\mathrm{d}6/4\mathrm{F}}^{*}$ by changing synclinic orientations in nearest layers to anticlinic and vice versa. The rectangle in Fig. 2 shows the period of the $\mathrm{Sm}C{}_{\mathrm{d}3}^{*}$ phase.

Figure 3

Dependencies of the energies per layer $F_0$ on $a_1/a_2$ for $\mathrm{Sm}{C}^{*}$, $\mathrm{Sm}C{}_{\mathrm{A}}^{*}$, and $\mathrm{Sm}C{}_{\mathrm{d}4}^{*}$ phases (solid curves). $|a_1/a_2|=0.5$ are the global degeneracy points in which energies of a large number of phases coincide. The dependencies of energy of several phases on $a_1/a_2$ passing through the degeneracy points $|a_1/a_2|=0.5$ are given in the figure. Model parameters: $\alpha =0.01{\mathrm{K}}^{-1},T-T^{*}=-10\mathrm{K},b_0=1,a_2=0.01$.

Figure 4

Stability regions of $\mathrm{Sm}C{}_{\mathrm{d}6/4\mathrm{A}}^{*}$ (a), $\mathrm{Sm}C{}_{\mathrm{d}3}^{*}$ (b), $\mathrm{Sm}C{}_{\mathrm{d}6/4\mathrm{F}}^{*}$ (c), and $\mathrm{Sm}C{}_{\mathrm{d}6/2\mathrm{A}}^{*}$ (d) phases in the parameter space of interlayer interactions ($a_3$, $a_1/a_2$). Model parameters: $\alpha =0.01{\mathrm{K}}^{-1},T-T^{*}=-10\mathrm{K},b_0=1$, $a_2=0.01$, $a_4=0$ (a), (b); $a_4=2\times {10}^{-5}$ (c), (d). Solid points show the results of calculations for uniplanar structures. Open points in (a),(b) illustrate the influence of chiral interaction on the phase diagram. Calculations for nonplanar structures were performed with parameters $f=0.002$, $b=0.09$. The upper horizontal axis stands for (a),(b), the lower axis for (c), (d).

Figure 5

Dependence of the energies per layer $F_0$ on $a_1/a_2$ for $\mathrm{Sm}{C}^{*}$ (solid squares), $\mathrm{Sm}C{}_{\mathrm{d}6/4\mathrm{A}}^{*}$ (solid circles), $\mathrm{Sm}C{}_{\mathrm{d}4}^{*}$ (solid triangles), and $\mathrm{Sm}C{}_{\mathrm{d}5}^{*}$ (open diamonds). Vertical dashed lines border stability ranges of the $\mathrm{Sm}{C}^{*}$, $\mathrm{Sm}C{}_{\mathrm{d}6/4\mathrm{A}}^{*}$, and $\mathrm{Sm}C{}_{\mathrm{d}4}^{*}$ phases. Model parameters:$\alpha =0.01{\mathrm{K}}^{-1},T-T^{*}=-10\mathrm{K},b_0=1,a_2=0.01$, $a_3=3\times {10}^{-5}$, $a_4=0$. The short dotted lines show the stability range of the $\mathrm{Sm}C{}_{\mathrm{d}5}^{*}$ phase with the fourth-nearest layer interaction $a_5=-3\times {10}^{-6}$.

Figure 6

Stability regions of $\mathrm{Sm}C{}_{\mathrm{d}6/4\mathrm{F}}^{*}$ (a), $\mathrm{Sm}C{}_{\mathrm{d}6/4\mathrm{A}}^{*}$ (b), $\mathrm{Sm}C{}_{\mathrm{d}3}^{*}$ (c), $\mathrm{Sm}C{}_{\mathrm{d}6/2\mathrm{A}}^{*}$ (d) phases (uniplanar model) in the parameter space of long-range interactions ($a_3$, $a_4$). $\alpha =0.01{\mathrm{K}}^{-1},T-T^{*}=-10\mathrm{K},b_0=1,a_2=0.01$. $a_1=-0.005$ (a), (b), $a_1=+0.005$ (c), (d).