Simple model for the stripe phase in compounds with bent-core molecules which exhibit a lower-temperature ferroelectric smectic-$A$ phase

Bent-core (BC) molecules usually exhibit polar packing in smectic layers in which the long or bow axes tilt with respect to the layer normal. In many compounds, the tilt angle goes to zero, and typically the polarization (P) of neighboring layers has an antiferroelectric order ($\mathrm{Sm}A{P}_{\mathrm{AF}}$). A careful molecular engineering has led to the discovery of the ferroelectric $\mathrm{Sm}A{P}_{\mathrm{F}}$ phase in a few BC compounds. Detailed experimental studies [Zhu et al., J. Am. Chem. Soc. 134, 9681 (2012)] have shown that one of the compounds undergoes a weak first order transition to a striped phase $\left(\mathrm{Sm}A{P}_{\mathrm{Fmod}}\right)$, in which the repolarization switching occurs at a threshold electric field, the latter increasing with temperature. The main optical birefringence of the $\mathrm{Sm}A{P}_{\mathrm{Fmod}}$ phase increases rapidly with the field at lower temperatures of its range. A small $(<10\%)$ fraction of the BC molecules can be expected to have less bent conformations in excited states (ESs), and we argue that the ES conformers rotate freely about the bow axes, and aggregate to form regions without any polar order to gain rotational entropy. In turn, a favorable divP term leads to the formation of stripes in the polarized regions made of ground state conformers. Based on this physical picture, we develop a simple phenomenological model to reproduce all the experimental trends qualitatively. In sample cells with insulating layers neighboring stripes prefer to have an antiparallel orientation of the mean P direction. The dielectric constant of the stripe phase increases with a dc bias electric field, a trend which is opposite to that in other ferroelectric liquid crystals.

Figure 1

Schematic diagram of a cell filled with liquid crystal in the $\mathrm{Sm}A{P}_{\mathrm{Fmod}}$ (stripe) phase. The outer indium-tin-oxide (ITO) coated plates have a thickness of $\sim 1\mathrm{mm}$, the insulating layer a thickness of $d_{il}\sim 10\mathrm{nm}$, and the cell gap a thickness of $d_{lc}\sim 10\mu \mathrm{m}$. The smectic-$A$ layers are stacked parallel to the plane of the page, with the layer normal along the $z$ axis. The splayed structures of the polar order $\mathit{m}$ exhibited by the more strongly bent ground state (GS, lower left) BC molecules, and described by the angle φ between $\mathit{m}$ and the $Y$ axis, have opposite configurations in adjacent stripes of a rested sample, with the applied electric field = 0, implying that the polar anchoring of $\mathit{m}$ at the insulating layer is negligible. The less bent excited state (ES) conformers indicated by circles can rotate freely, and aggregate to form walls of width $v$ ($\sim 1$ or 2 nm) between the polar stripes of width $w$ (approximately tens of nanometers). $\mathit{m}$ is orthogonal to the interface with the walls made of ES molecules, implying a strong anchoring at that interface. The small angle x-ray Bragg peak corresponds to $v+w$. The molecular frame is described by the arrow (−a) axis, the bow (−b) axis, and the $o$ axis orthogonal to the molecular plane.

Figure 2

Calculated variations of the order parameter $m$ (topmost curve near relative temperature $T-T^{*}=-65\mathrm{K}$) and spatially averaged projection of $\mathit{m}$ along the $y$ axis ($\langle \mathit{m}cos\phi \rangle$, bottom-most curve near $T-T^{*}=-65\mathrm{K}$) across the $\mathrm{Sm}A{P}_{\mathrm{Fmod}}$ (stripe) to $\mathrm{Sm}A{P}_{\mathrm{F}}$ (uniform) phase transition at $T_{tr}-T^{*}=-85.3\mathrm{K}$. The order parameter $m$ exhibits a small jump in a thermodynamically weak transition, but the polarization P ($=p\langle \mathit{m}cos\phi \rangle$, $p=3.5\times {10}^{-3}\mathrm{C}/{\mathrm{m}}^2$) exhibits a large jump due to the change in the spatial distribution of $\mathit{m}$. The reduced stripe width $[w\left(T\right)/w\left(T_{tr}\right)]$ is shown in the middle curve, where $w$ ($T_{tr}$) (the stripe width at $T_{tr}$) = 56 nm.

Figure 3

Variation of $\phi$, the angle made by $\mathit{m}$ with the $y$ axis (in the symmetric configuration of a stripe in the left half of Fig. 1) with the reduced position $2x/w\left(T\right)$. The relative temperatures $(T-T^{*})$ are $-65\mathrm{K}$ (topmost curve, $w=25.7\mathrm{nm}$), $-75\mathrm{K}$ (middle curve, $w=33.8\mathrm{nm}$), and $-85\mathrm{K}$ (bottom-most curve, $w=54.1\mathrm{nm}$). The φ angle lingers close to 0 over a larger fraction of the stripe at a lower temperature.

Figure 4

Variations of different parameters with the dc electric field ($E_{lc}$) acting on a sample poled along the positive $y$ axis (see Fig. 1). In each figure, the curves correspond, from bottom to top, to the following relative temperatures $(T-T^{*})$: −65, −70, −75, −80, −83, and −85 K, respectively. The jumps in the curves correspond to the $\mathrm{Sm}A{P}_{\mathrm{Fmod}}$ to $\mathrm{Sm}A{P}_{\mathrm{F}}$ transition temperature ($T_{tr}$) which shifts to a higher temperature with $E_{lc}$, the relative change in $T_{tr}$ decreasing at higher temperatures. The parameters are (a) order parameter $m$, (b) polarization $P$, (c) biaxiality $\delta n$, (d) birefringence $\mathrm{\Delta }n$, and (e) stripe width $w$, all of which increase with electric field at a given temperature in the stripe phase. The relative jumps at the fields corresponding to the stripe to uniform transition in the order parameter $\mathrm{\Delta }m\left(E\right)/\mathrm{\Delta }m\left(0\right)$ with $\mathrm{\Delta }m\left(0\right)=0.019$, which is the upper curve in (f), increase significantly with field, whereas that in polarization $\mathrm{\Delta }P\left(E\right)/\mathrm{\Delta }P\left(0\right)$ with $\mathrm{\Delta }P\left(0\right)=64\mathrm{nC}/\mathrm{c}{\mathrm{m}}^2$, which is the lower curve in (f), shows a small decrease.

Figure 5

Electric field dependence of free energy density minimum ${\langle F\rangle }_m$ in the $\mathrm{Sm}A{P}_{\mathrm{Fmod}}$ and $\mathrm{Sm}A{P}_{\mathrm{F}}$ phases at the relative temperature $T-T^{*}=-75\mathrm{K}$. At zero field, ${\langle F\rangle }_m$ is lower in the $\mathrm{Sm}A{P}_{\mathrm{Fmod}}$ phase, and has a negative slope as a function of $E_{lc}$ for both $\mathrm{Sm}A{P}_{\mathrm{F}}$ (topmost curve at $E_{lc}=0.2\mathrm{V}/\mu \mathrm{m}$) and the stripe phase with $\mathit{m}$ parallel to ${\mathbf{E}}_{\mathbf{lc}}$ on average (bottom-most curve at $E_{lc}=0.2\mathrm{V}/\mu \mathrm{m}$), as in the left half of Fig. 1. It has a positive slope (middle curve at $E_{lc}=0.2\mathrm{V}/\mu \mathrm{m}$) when the two vectors point in opposite directions (as in the right half of Fig. 1).

Figure 6

Threshold electric field $E_{th}$ at which the stripe structure overcomes a barrier to switch from the unfavorable orientation, with average $\mathit{m}$ oriented opposite to ${\mathbf{E}}_{\mathbf{lc}}$, to the favorable orientation (upper curve). For the parameters used in the calculations, it switches to the $\mathrm{Sm}A{P}_{\mathrm{F}}$ phase, which has the lowest ${\langle F\rangle }_m$ value at the threshold field. For comparison, the temperature dependence of the field $E_{tr}$ required for the transition from the stripe to the uniform phase of a poled sample is shown in the lower curve.

Figure 7

Birefringence variations shown in Fig. 4 redrawn as functions of the applied (external) field $E_{apl}$, which is higher than the field $E_{lc}$ experienced by the medium, due to an opposing field generated by the polarization, as described in the text. Note that the transition between the two phases has a finite slope, signifying a coexistence of the phases.

Figure 8

The $\mathrm{Sm}A{P}_{\mathrm{Fmod}}$ phase can be expected to have an antiparallel orientation of neighboring stripes when $E_{apl}$ is reduced to 0, and the cell rested for a minute or so, to avoid the positive self-energy due to polarization of a poled structure. This configuration is maintained until the external field reaches the threshold value for the switching of the unfavorably oriented stripes. (a) $E_{apl}$ dependences of the order parameter $m$ averaged over the widths of both types of stripes are shown at $T-T^{*}=-65\mathrm{K}$ (bottom-most curve), −75 K (middle curve), and −85 K (topmost curve). The variations with field are smaller than those in a poled sample, shown in Fig. 4. (b) Applied field dependence of the polarization at three temperatures. Note the relatively small value of $P$, which rapidly rises with $E_{apl}$. The polarization contribution to the dielectric constant which is ∝ to the slope of the curve increases with $E_{apl}$, unlike in the $\mathrm{Sm}A{P}_{\mathrm{F}}$ phase.

Figure 9

(a) Schematic diagram showing the likely collection of the ES molecules (which do not exhibit a spontaneous polarization) near the negative electrode when the GS conformers exhibit the $\mathrm{Sm}A{P}_{\mathrm{F}}$ phase, stabilized by a large enough $E_{apl}$. As the field is reduced towards the value at which the $\mathrm{Sm}A{P}_{\mathrm{Fmod}}$ phase has the lower free energy, divP fluctuations involving a redistribution of the ES molecules as shown in (b) can be triggered. The process of covering the entire cell with the stripe structure with optimum stripe width can take about 0.1 s, and a faster field cycling can lead to a broad coexistence range as seen in experiments [21].