Biaxial smectic-$A^{*}$ phase and its possible misidentification as a smectic-$C_{\alpha }^{*}$ phase

The biaxial smectic-$A^{*}$ (Sm-$A_B^{*}$) phase, appearing in the phase sequence Sm-$A^{*}$–Sm-$A_B^{*}$–Sm-$C^{*}$, is analyzed using Landau theory. It is found to possess a helical superstructure with a pitch that is significantly shorter than the pitch of the Sm-$C^{*}$ helical superstructure. The Sm-$A_B^{*}$–Sm-$C^{*}$ transition can be either first or second order, and correspondingly there will be either a jump or continuous variation in the pitch. The behaviors of the birefringence and electroclinic effect are analyzed and found to be similar to those of a Sm-$C_{\alpha }^{*}$ phase. As such, it is possible that the Sm-$A_B^{*}$ phase could be misidentified as a Sm-$C_{\alpha }^{*}$ phase. Ways to distinguish the two phases are discussed.

Figure 1

Schematics of (a) Sm-$A^{*}$, (b) Sm-$A_B^{*}$, and (c) Sm-$C^{*}$ phases. In each case a single layer is shown. ${\widehat{\mathbf{e}}}_{\mathbf{1}}$, ${\widehat{\mathbf{e}}}_{\mathbf{2}}$, and $\widehat{\mathbf{n}}$ are the eigenvectors of the orientational order tensor.

Figure 2

The phase diagram in $r_{\alpha }$-$r_{\theta }$ space for the Sm-$A^{*}$, Sm-$A_B^{*}$, Sm-$C^{*}$ phases. First- and second-order phase boundaries are shown as solid and dashed lines, respectively. Four decreasing temperature paths from the Sm-$A^{*}$ to Sm-$C^{*}$ phase are shown. In the region between the (black) solid and (orange) dot-dashed lines the system will exhibit a particularly dramatic electroclinic effect (see Fig. 3). (Top inset) The expected behavior of the helical pitch across the first- and second-order Sm-$A_B^{*}$–Sm-$C^{*}$ transitions. (Bottom inset) Schematic location of the normal or oblique incidence Bragg scattering wave-vector peaks associated with the helical superstructure of the Sm-$A_B^{*}$ or Sm-$C^{*}$ phases, with $K_b/K_t\approx 2$. In the Sm-$A_B^{*}$ phase there are only peaks at even multiples of $q_{0B}$ for both normal and oblique incidence. In the Sm-$C^{*}$ phase the peaks are located at even multiples of $q_{0C}$ for normal incidence but at integer multiples of $q_{0C}$ for oblique incidence.

Figure 3

(Top inset) $\theta \left(E\right)$ in Sm-$A^{*}$ or Sm-$A_B^{*}$ phases above a continuous transition to the Sm-$C^{*}$ phase. The susceptibility ${\chi }_0$ is the slope of the $\theta \left(E\right)$ curve at $E=0$. (Bottom inset) ${\chi }_0\left(T\right)$ for phase sequences Sm-$A^{*}$–Sm-$C^{*}$ (red dotted line) or Sm-$A^{*}$–Sm-$A_B^{*}$–Sm-$C^{*}$ (green solid line). Paths (1) and (4) refer to Fig. 2. The continuous transition to the Sm-$C^{*}$ phase is at $T_C$. For a first-order transition the divergence of ${\chi }_0\left(T\right)$ is cut off at $T>T_C$. (Main panel) $\theta \left(E\right)$ curves (i) and (ii) are in the Sm-$A_B^{*}$ phase above a first-order Sm-$A_B^{*}$–Sm-$C^{*}$ transition at $T_{AB}$. (i) is above the critical temperature $T_E>T_{AB}$ and (ii) is at $T_{AB}<T<T_E$ (see Fig. 2 for the corresponding region in the phase diagram). Curve (iii) is in the Sm-$C^{*}$ phase, below a second- or first-order transition.