Two-state model for the $\mathrm{Sm}A{P}_R$ phase with randomized layer polarization exhibited by some compounds with bent-core molecules

One of the unusual liquid crystalline phases exhibited by some compounds with bent-core (BC) molecules is designated as $\mathrm{Sm}A{P}_R$, in which transverse polarization (P) of smectic layers with upright molecules has a random orientational distribution. Most of such compounds undergo a transition to the $\mathrm{Sm}A{P}_A$ phase with antiferroelectric order of adjacent layers as the temperature is lowered. Second harmonic studies have shown that the medium consists of polarized domains with only a few hundred molecules, the number increasing at lower temperatures. This is in contrast to the random orientations of entire layers predicted by a few phenomenological models which have been proposed for the $\mathrm{Sm}A{P}_R$ phase. In this paper we show that the two-state model, developed earlier by us to describe modulated phases found in compounds with BC molecules, can successfully account for all the experimental results on the $\mathrm{Sm}A{P}_R$ phase. It is proposed that polarized domains which are made of ground state conformers, with a large bend of the aromatic cores, nucleate as circular disks in a first order transition from the isotropic phase. Excited state conformers (with a smaller bend) freely rotate about their long axes and surround the disks. The polarization in the disk has a spontaneous splay distortion which generates a texture with an expelled center of a partial disclination. The interdisk electrostatic energy is lower than the thermal energy and the disks are subject to significant rotational fluctuations, resulting in the uniaxial symmetry of the medium. Calculated equilibrium properties of the medium as functions of temperature and electric field reflect the trends seen in experiments. The model overestimates properties such as the radius of the disks as the transition to a lower temperature uniform phase is approached, and physical arguments are given to show that interlayer interactions ignored in the model become important in that regime, and should be included for an improved description of the medium.

Figure 1

Schematic figure showing the more bent average shape of ground state (GS) conformers (left) and less-bent average shape of the excited state (ES) conformers (right).

Figure 2

Schematic diagram showing a circular domain with radius $R$ made of a splay distorted polarization (P, indicated by the radial arrows) field exhibited by the ground state conformers. The freely rotating ES conformers are symbolized by the filled circles surrounding the disk, as well as in the core of radius ξ around the melted +1 disclination of the P field at the center of the disk.

Figure 3

Schematic diagram showing the texture with the center of the +1 disclination expelled from the disk to the virtual location $O$ with $x$ = −ρ from the center of the disk at $C$ (a). The $m$ vector corresponding to a point $B$ in the disk is oriented along OB making an azimuthal angle θ with respect to the $X$ axis (b). Both div$m$ (= 1/$r_e$) and $m_r$ needed in calculating the free energy density averaged over the area of the disk can be expressed in terms of the radius vector $r$, its azimuthal angle φ, and ρ.

Figure 4

Dependence of the radius of the disk $R$ on the relative temperature. $R$ increases at lower temperatures, with an accelerating growth below dT ≈ −60 K.

Figure 5

Temperature dependences of the ratio $A=R/\rho$ (upper curve) and the order parameter σ (lower curve). σ has an extremely small jump at the transition to the uniform phase at dT≈ −67.3 K, indicating an almost second order transition.

Figure 6

Theoretically calculated temperature variations of the dipole moment of the disk (upper curve) and the polarization contribution to the low frequency relative dielectric constant (lower curve) in the $\mathrm{Sm}A{P}_R$ phase.

Figure 7

Electric field dependences of the induced polarization P_E (C/${\mathrm{m}}^2$, lower curve beyond 2 × ${10}^6$ V/m) and field-induced enhancement in birefringence $\delta \mathrm{\Delta }{\mathrm{n}}_E$ (≈half of biaxiality $\delta n_E$, upper curve) at dT = −50 K.

Figure 8

Electric field dependence of the polarization contribution to the dielectric constant ε_P at dT = −50 K.

Figure 9

Rotational time constant τ_R of the disks as a function of relative temperature, with the estimated value of $C$ = ${10}^{28}$ sec $\mathrm{K}/{\mathrm{m}}^4$ .

Figure 10

Temperature variations of the electric- field-induced polarization ${\mathrm{P}}_{\mathrm{E}}$ of the medium. The lower curve corresponds to $E$ = 4 × ${10}^6$ V/m and the upper one to $E$ = 2 × ${10}^7$ V/m.

Figure 11

Temperature variations of the field-induced biaxiality $\delta n_E$. The lower curve corresponds to $E=4\times {10}^6\mathrm{V}/\mathrm{m}$ and the upper one to $E=2\times {10}^7\mathrm{V}/\mathrm{m}$.

Figure 12

Temperature variations of the contribution of the field-induced polarization to the relative dielectric constant ε_P at two applied fields. The upper curve corresponds to $E=4\times {10}^6\mathrm{V}/\mathrm{m}$ and the lower one to $E=2\times {10}^7\mathrm{V}/\mathrm{m}$.

Figure 13

Calculated dependence of the relative temperature ${dT}_{\mathrm{tr}}$ of the transition between the $\mathrm{Sm}A{P}_R$ and uniform phases on $p$.

Figure 14

Stacking of three disks with splayed polarization texture in neighboring layers with antiparallel orientation of adjacent disks. Along the layer normal, which has the shortest separation equal to the layer spacing $l$, the P vectors of nearest neighbor disks have perfect antiparallel orientation only along the central line, whereas the parallel orientations of the P vectors in next neighbor disks are in perfect registry.