Ferrielectricity in smectic-$C^{*}$ dechiralization-line lattices

Recent experiments probing a new ferroelectric liquid crystal (CLF08) confined in cells with planar alignment have shown dielectric and optic anomalies suggesting the onset of ferrielectric ordering within the surface lattice of dechiralization lines. We present a phenomenological theory describing the corresponding phase transition sequence $\mathrm{Sm}A\to \mathrm{Sm}{C}^{*}\to \mathrm{Ferri}$. Phase diagrams and thermodynamic, dielectric, and optic properties are worked out and compared with experiments. The anomalies are related to the predicted tristability of the experimental cells under applied electric field. The order parameters of Landau theory are reinterpreted in terms of line positions, allowing description of the entrance and exit line behavior, and yielding the prediction and identification of new limit phases within a nonconventional Landau approach.

Figure 1

Scheme of the polarization field in a $\mathrm{Sm}C*$ cell. Black thin arrows and large open circles represent the local in-plane $\left(zy\right)$ polarization and large open circles represent the local out-of-plane ($x$) polarization. Small solid circles represent cuts of the dechiralization lines (parallel to Ox) in the $zy$ plane. Close to the surfaces the field is homogeneous. Between the lines the helix is wound. Thick arrows symbolize the antiparallel polar vectors defining the globally antiferroelectric character of the cell.

Figure 2

Hysteresis cycle of the dielectric permittivity ${\varepsilon }^{\prime }$ vs dc bias voltage $E_{\mathrm{bias}}$ in a 7-μm-thick cell at 800 Hz. The phases ($\mathrm{Sm}C*$), ferrielectric ($f$), and ferroelectric ($F$) are indicated. $E1–E11$ represent field values of the main characteristic points of the curve. They are interpreted at the dechiralization line scale in Fig. 9. The figure shows the difference between the first polarization curve ($E1-E3$) and the low-field part of the hysteresis curve ($E7$ to 0). The main hysteresis appears between $-E8$ and $-E11$, whereas the residual hysteresis is present between 0 and $-E8$. $-E10$ shows the anomalous maximum of permittivity.

Figure 3

Polarization versus electric field for different frequencies $f$.

Figure 4

(a) Group-subgroup relationships between the $\mathrm{Sm}A$, $\mathrm{Sm}C*\left(A\right)$, ferroelectric ($F$), and ferrielectric ($f$) phases, in the bulk and in the cell, with and without field. (b) Symmetry breakdown pattern in a cell at zero field. (c) Symmetry breakdown pattern in a cell under electric field.

Figure 5

Zero-field phase diagram of the ferrielectric model. Dotted lines represent first-order transitions. Thick solid lines represent second-order transitions. Thin lines represent the stability limits of the stable phases: $\mathrm{Sm}A$, $A$ (SmC*), ferrielectric $f$, and ferroelectric $F$. The dotted line indicates the thermodynamic path followed by the system when temperature is varied. At low temperature (below $T_1$) the $A$ phase remains stable but coexists with the metastable $f$ phase.

Figure 6

(a) Temperature-field phase diagram along the thermodynamic path. The thin lines indicate the limits of stability of the various stable or metastable phases. (b) Polarization ($P$) vs applied field ($E$) hysteresis cycles at different fixed temperatures.

Figure 7

(a) Polarization-field $P\left(E\right)$ cycles inn the first scenario: transition $f^{+}\to A$ at $-E2$. (b) Permittivity-field ${\varepsilon }^{\prime }\left(E\right)$ cycle in the first scenario. (c) ${\varepsilon }^{\prime }\left(E\right)$ in the second scenario: transition $f^{+}\to f^{–}$ at $-E2$.

Figure 8

Sample surface under crossed analyzer and polarizer in a 15-μm-thick sample. The dechiralization lines appear as horizontal broken parallel dark stripes. The end point of one line (where it leaves the liquid crystal normal to the figure) is indicated.

Figure 9

Schemes of the migration of the dechiralization lines during field cycling, as observed at $T_C-T=5{}^{\circ }\mathrm{C}$ in a 15-μm-thick sample from $E=0$ to $E_{\max }=1\mathrm{V}/\mu \mathrm{m}$. Schemes in the first column represent the cell during the first polarization curve. The second and third rows represent one-half of one cycle. The field values $E1$, $E2$, … are defined in Fig. 2. Small arrows represent the polarization field. Large arrows show the displacements of the lines submitted to Coulomb forces. A wrapped line indicates the presence of the helix. $H$ and $D\left(E=0\right)$ represent the upper and lower line lattices, respectively.

Figure 10

(a) Order-parameter space. The parent phase (PP) lies at the center of the square, with $A$ and $F$ on the diagonals. The gray area corresponds to the normal ferrielectric f phase. The limit phase $f_L$ lies on the square boundaries. $A_L$ and $F_L$ are represented by dots located on the boundaries. (b) Dechiralization line lattices in normal and limit phases. Parts of the sample where the helix is unwound are represented by dark arrows indicating the polarization direction; wrapped thin solid lines represent the wound helix.

Figure 11

Zero-field phase diagrams. (a) $K^2<cd$ and (b) $K^2>cd$. Arrows indicate possible thermodynamic paths.