Effective long-range interlayer interactions and electric-field-induced subphases in ferrielectric liquid crystals

Microbeam resonant x-ray scattering experiments recently revealed the sequential emergence of electric-field-induced subphases (stable states) with exceptionally large unit cells consisting of 12 and 15 smectic layers. We explain the emergence of the field-induced subphases by the quasimolecular model based on the Emelyanenko-Osipov long-range interlayer interactions (LRILIs) together with our primitive way of understanding the frustration in clinicity using the $q_E$ number defined as $q_E=|\left[R\right]-\left[L\right]|/(\left[R\right]+\left[L\right])$; here $\left[R\right]$ and $\left[L\right]$ refer to the numbers of smectic layers with directors tilted to the right and to the left, respectively, in the unit cell of a field-induced subphase. We show that the model actually stabilizes the field-induced subphases with characteristic composite unit cells consisting of several blocks, each of which is originally a ferrielectric three-layer unit cell stabilized by the LRILIs, but some of which would be modified to become ferroelectric by an applied electric field. In a similar line of thought, we also try to understand the puzzling electric-field-induced birefringence data in terms of the LRILIs.

Figure 1

Electric-field effects in (a) $\text{Sm-}{C}^{*}$, (b) $\text{Sm-}{C}_A^{*}$, (c) $q_T=1/3$, and (d) $q_T=1/2$. The averaged tilt plane direction is perpendicular to the applied electric field in the (a) ferroelectric and (c) ferrielectric phases. It may become parallel to the electric field in the (b) and (d) antiferroelectric phases; at the same time, the directors in the unit cell may show some small asymmetric change to produce induced polarization in the initial averaged tilt direction, which is observed as the pretransitional effect in the antiferroelectric phases [38, 39, 40, 48, 49, 50]. The electric field is applied along the $y$ direction.

Figure 2

Nine composite planar superlattice structures used in the free-energy calculation. We only need to calculate the free energy given in Eq. (17) using ${\alpha }_k^{\left(0\right)}$ in Eq. (29) for these nine, if we assume that any field-induced subphase in the temperature region of $q_T=1/3$ consists of an orderly array of the 3-layer ferrielectric and ferroelectric blocks and that the largest unit cells are of 15-layer periodicity (5 blocks) with simple $q_E$ in lower terms up to 7/9; notice that we have disregarded $q_E$ in terms higher than 7/9 (either in numerator or denominator), i.e., 7/15, 11/15, and 13/15. Hereafter we designate these nine as $q_{E}n=1,...,9$.

Figure 3

Reproduced $g-\tilde{T}$ phase diagram with parameter values of $\chi c_{\mathrm{p}}{c}_{\mathrm{f}}/B=-0.12$ and $c_{\mathrm{f}}/c_{\mathrm{p}}=-1.0$ [36, 51], and nine chosen points in red for studying the electric field effect: $\tilde{T}=-0.24, -0.21$, and $-0.18$ at $g=0.1$; $\tilde{T}=-0.21, -0.16$, and $-0.09$ at $g=0.2$; and $\tilde{T}=-0.09, -0.02$, and 0.05 at $g=0.3$.

Figure 4

Calculated ${\phi }_i$ for $q_{E}n=4$ at $\tilde{E}=0.573$ and for $q_{E}n=6$ at $\tilde{E}=0.574$. Other parameters are $g=0.2,\tilde{T}=-0.09, \chi c_{\mathrm{p}}{c}_{\mathrm{f}}/B=-0.12$, and $c_{\mathrm{p}}/c_{\mathrm{f}}=-1.0$. The relevant dimensionless free energies are $-1.5113$ for $q_{E}n=1$ at $\tilde{E}=0.572, -1.51166$ for $q_{E}n=4$ at $\tilde{E}=0.573$, and $-1.51228$ for $q_{E}n=6$ at $\tilde{E}=0.574$. The electric field is applied along the $y$ axis and it is assumed that $c_{\mathrm{p}}>0$ and hence that the spontaneous polarization is positive and the microscopic short-pitch distorted helix is right handed. The tilt directions in different layers are symmetrical with respect to the middle of the period indicated by closed red circles; this property defines the chirality of these subphases.

Figure 5

The compound Iida et al. used in the recent microbeam RXRS experiments [33]. The two chiral centers are $(S,S)$. They noticed the emergence of the field-induced subphases that have unbelievably large 12- and 15-layer unit cells.