Type-I and type-II smectic-$C^{*}$ systems: A twist on the electroclinic critical point

We conduct an in-depth analysis of the electroclinic effect in chiral, ferroelectric liquid crystal systems that have a first-order smectic-$A^{*}$–smectic-$C^{*}$ (Sm-$A^{*}$–Sm-$C^{*}$) transition, and show that such systems can be either type I or type II. In temperature-field parameter space type-I systems exhibit a macroscopically achiral (in which the Sm-$C_M^{*}$ helical superstructure is expelled) low-tilt (LT) Sm-$C_U^{*}$–high-tilt (HT) Sm-$C_U^{*}$ critical point, which terminates a LT Sm-$C_U^{*}$–HT Sm-${}^{*}{C}_U$ first-order boundary. Notationally, Sm-$C_M^{*}$ or Sm-$C_U^{*}$ denotes the Sm-$C^{*}$ phase with or without a modulated superstructure. This boundary extends to an achiral-chiral triple point at which the macroscopically achiral LT Sm-$C_U^{*}$ and HT Sm-$C_U^{*}$ phases coexist along with the chiral Sm-$C_M^{*}$ phase. In type-II systems the critical point, triple point, and first-order boundary are replaced by a Sm-$C_M^{*}$ region, sandwiched between LT and HT Sm-$C_U^{*}$ phases, at low and high fields, respectively. Correspondingly, as the field is ramped up, the type-II system will display a reentrant Sm-$C_U^{*}$–Sm-$C_M^{*}$–Sm-$C_U^{*}$ phase sequence. Moreover, discontinuity in the tilt of the optical axis at each of the two phase transitions means the type-II system is tristable, in contrast to the bistable nature of the LT Sm-$C_U^{*}$–HT Sm-$C_U^{*}$ transition in type-I systems. Whether the system is type I or type II is determined by the ratio of two length scales, one of which is the zero-field Sm-$C^{*}$ helical pitch. The other length scale depends on the size of the discontinuity (and thus the latent heat) at the zero-field first-order Sm-$A^{*}$–Sm-$C^{*}$ transition. We note that this type-I vs type-II behavior in this ferroelectric smectic is the Ising universality class analog of type-I vs type-II behavior in XY universality class systems. Lastly, we make a complete mapping of the phase boundaries in all regions of temperature–field–enantiomeric-excess parameter space (not just near the critical point) and show that various interesting features are possible, including a multicritical point, tricritical points, and a doubly reentrant Sm-$C_U^{*}$–Sm-$C_M^{*}$–Sm-$C_U^{*}$–Sm-$C_M^{*}$ phase sequence.

Figure 1

A schematic representation of (a) the Sm-$C_U^{*}$ phase and (b) the Sm-$C_M^{*}$ phase. The inset shows the tilt angle $\theta$ of the director $\widehat{\mathbf{n}}$ relative to the layer normal $\widehat{\mathbf{z}}$, and also $\mathbf{c}$, the Sm-$C_{U/M}^{*}$ order parameter. In the Sm-$C_M^{*}$ phase the average molecular direction precesses from layer to layer. This modulation along $z$ has period $P$. For zero field the modulation is perfectly helical, as shown here. For sufficiently large electric field the modulation is expelled, and the average molecular direction is uniform, as shown for the Sm-$C_U^{*}$ phase.

Figure 2

(a) The electroclinic effect, i.e., average tilt ($c_{x_{\mathrm{av}}}$) vs electric field ($E$), for fixed temperatures for a material with a first-order Sm-$A^{*}$–Sm-$C^{*}$ transition. For $T\lesssim T_c$ there is a discontinuous jump between low-tilt and high-tilt states, and the system is bistable. (b) The first-order phase boundary separating uniform low-tilt from uniform high-tilt Sm-$C_U^{*}$ phases. The phase boundary terminates at an Ising-like critical point at $T_c$, as found in [12]. The phase diagram assumes zero modulation.

Figure 3

$T\text{−}E$ phase diagrams for (a) type-I systems and (b) type-II systems. The smectic-$A$ phase exists for $E=0$ and $T>T_{A{C}^{*}}$. CP denotes critical point, TP denotes triple point, MCP denotes multicritical point, and TCP denote tricritical points. Each panel shows a fixed temperature path that will exhibit the reentrant phase sequence Sm-$A^{*}$–Sm-$C_U^{*}$–Sm-$C_M^{*}$–Sm-$C_U^{*}$, as $E$ is ramped up. The inset shows the corresponding average tilt vs $E$, which is tristable. The fixed $E$ path shown on the type-II phase diagram will exhibit the doubly reentrant phase sequence Sm-$C_U^{*}$–Sm-$C_M^{*}$–Sm-$C_U^{*}$–Sm-$C_M^{*}$.

Figure 4

(a) The uniform-modulated phase boundary for two type-II systems, near the unstable critical point at $\rho =h=0$. As type-I behavior is approached, the modulated region (below the phase boundary) shrinks. Between the two tricritical points (TCP) the uniform-modulated transition is continuous. The parabolic shape of the phase boundaries, i.e., $\rho -{\rho }_v\propto -{(h-h_v)}^2$, is only valid between the two TCPs. The narrower modulated region corresponds to smaller $m$. Note that for smaller $m$ the tricritical points are closer together. (b) The type-I phase diagram near the high-tilt–low-tilt–modulated triple point. Note the different slopes of the phase boundary between the low-tilt–modulated states and the phase boundary between the high-tilt–modulated states.

Figure 5

(a) The dashed curve shows $\varphi \left(z\right)$ for the modulated phase at the triple point, i.e., $\rho ={\rho }_{TP}$ and $h=0$. The low and high tilt domains are the same size ($\approx P/2$). The solid curve shows $\varphi \left(z\right)$ for the modulated phase below the triple point, $\rho \lesssim {\rho }_{TP}$ and $h\gtrsim 0$. The high-tilt domains are larger than the low-tilt domains. The inset shows the variation of fractional low and high tilt domain sizes, $L_L/P$ and $L_H/P$, as well as $L_{H,M-L}/P$ and $L_{L,M-H}/P$. (b) Schematic illustration of the modulated Sm-$C_M^{*}$ structure in a type-I system close to the triple point. The structure consists of long domains ($L_L\approx P/2$ and $L_H\approx P/2$) of achiral low tilt and high tilt, with the domains separated by short $\ll P$ chiral domain walls in which the system rapidly twists from low to high tilt (or vice versa).

Figure 6

The phase diagram in $\rho \text{−}h\text{−}m$ space near the critical triple point. The modulated phase lies below the curved surface, while the uniform phase lies above the curved surface. The green region of the surface (lower region of surface, beneath solid line) and gold region of the surface (upper region of surface, above solid line) correspond to first-order and continuous uniform-modulated phase boundaries, respectively. The gold (upper) surface is bounded by a line of tricritical points. The vertical $h=0$ plane separates the low and high tilt uniform phases, and crossing the plane corresponds to the first-order uniform-low–uniform-high phase transition. The dotted line at the top of the vertical plane is a line of uniform-low–uniform-high critical points, while the dashed line at the bottom of the plane is a line of uniform-low–uniform-high–modulated triple points. These lines, and the line of tricritical points, all meet at $\rho =0$, $h=0$, $m=m_c$ at what we term a critical triple point. For $m<m_c$ the system is type I and and for $m>m_c$ it is type II.

Figure 7

The modulated phase boundary for systems with different $m$ values (rightmost being largest). For $m>1.27$ each phase boundary has two tricritical points, shown as diamonds. For $m<1.27$ the tip of each “nose” is a triple point. There is a uniform low-tilt–high-tilt first-order boundary (not shown) that extends from each triple point to a critical point located at $R=\epsilon =1$.

Figure 8

The modulated phase boundary for systems with different $m$ values (topmost being largest). For $m>1.27$ each phase boundary has two tricritical points, shown as diamonds. For $m<1.27$ the tip of each “nose” is a triple point. There is a uniform low-tilt–high-tilt first-order boundary (only shown in inset for $m=1.02$) that extends from each triple point to a critical point located at $\rho =h=0$.