Phase transitions of heliconical $\text{smectic-}C$ and heliconical nematic phases in banana-shaped liquid crystals

A mean-field theory is introduced to describe heliconical nematic ($N_{\text{TB}}$), heliconical $\text{smectic-}C$ ($S_m{C}_{\text{TB}}$), and biaxial heliconical $\text{smectic-}C$ ($S_m{C}_{\text{TB},b}$) phases with mirror symmetry breaking. We extend our previous theories of the $N_{TB}$ phase to the heliconical $\text{smectic-}C$ phases, by taking into account one-dimensional spatial ordering of smectic layers. The calculated phase diagrams on the temperature–alkyl chain length plane show a rich variety of phase transitions: first- and second-order $N_{\text{TB}}\text{−}{S}_m{C}_{\text{TB}}$ transitions, etc., including tricritical, tetracritical, and multicritical points. Our theory is qualitatively consistent with an experimental phase diagram.

Figure 1

(a) A schematic representation of a bent-core molecule with the bend angle ${\theta }_B$ between the arm vectors ${\mathbf{u}}_A$ and ${\mathbf{u}}_B$ on the orthonormal molecular axes $\{\mathbf{a},\mathbf{b},\mathbf{c}\}$. The vectors ${\mathbf{u}}_A$ and ${\mathbf{u}}_B$ are on the same plane $\mathbf{b}\text{−}\mathbf{c}$. The angle ${\chi }_B$ between the vectors $\mathbf{c}$ and ${\mathbf{u}}_A$ is given by ${\chi }_B=(\pi -{\theta }_B)/2$. (b) A schematic representation of the $S_m{C}_{\text{TB}}$ phase with the one-dimensional spatial order. The director $\mathbf{n}$ is uniformly twisted along the pitch axis $\mathbf{p}$ (parallel to the $z$ axis) with a cone angle $\epsilon$.

Figure 2

Order parameters plotted against the temperature $T/T_{NI}$ for $\alpha =0.4$ (a) and $\alpha =1.0$ (b) with ${\gamma }_L=0.02$.

Figure 3

Phase diagrams on the $(\alpha ,T/T_{NI})$ plane for ${\gamma }_L=0.02$ (a), 0.012 (b), and 0.002 (c). The solid lines represent the first-order phase transitions while the dotted lines show the second-order phase transitions. The symbols $▵, \square , •, \circ$, and $\diamond$ show the TP, TeCP, TCP, MCP, and CEP, respectively.