Anisotropy of electrostatic interaction in smectic-$C^{*}$ liquid crystals

The contribution to the free energy of distortion of the ferroelectric smectic-$C^{*}$ due to the electrostatic interaction of polarization charges is calculated. These calculations are performed by accounting for the anisotropy of the permittivity, which is essential for smectic-$C^{*}$. Fluctuations of the $\mathbf{c}$ director in an external electric field are considered. It is shown that the anisotropy of the permittivity strongly affects the interaction of the polarization charges, the spectrum orientation fluctuations, and the angular dependence of the light scattering intensity.

Figure 1

Positions of the vectors $\mathbf{n},\mathbf{c}$, and $\mathbf{P}$ in smectic-$C^{*}$. The external electric field $\mathbf{E}$ is directed along the $y$ axis.

Figure 2

Positions of the vectors $\mathbf{n},\mathbf{c}$, and $\mathbf{P}$ in smectic-$C^{*}$. External electric field $\mathbf{E}$ is directed along the $y$ axis.

Figure 3

Angular dependence of the light scattering intensity. (a) and (b) show the dependencies of the scattered light intensity on the polar angle ${\theta }_{\text{sc}}$ for fixed azimuth angles ${\varphi }_{\text{sc}}$. (c) and (d) show the dependence of the scattered light intensity on the azimuth angle ${\varphi }_{\text{sc}}$ for the fixed polar angle ${\theta }_{\text{sc}}={20}^{\circ }$. In (a) and (c) the polarization $P=5\text{statcoulomb}/{\mathrm{cm}}^2=1.67\times {10}^4\mathrm{nC}/{\mathrm{m}}^2$. In (b) and (d) the polarization $P=20\mathrm{statcoulomb}/{\mathrm{cm}}^2=6.67\times {10}^4\mathrm{nC}/{\mathrm{m}}^2$. In all panels the scattering intensity is presented for three different combinations of the principal values of the static permittivity tensor. The solid red lines correspond to ${\varepsilon }_1={\varepsilon }_2={\varepsilon }_3=5$, the dashed black lines are for the uniaxial system with ${\varepsilon }_1={\varepsilon }_2=4$ and ${\varepsilon }_3=7$, and the dotted blue lines are for the biaxial system with ${\varepsilon }_1=3,{\varepsilon }_2=7$, and ${\varepsilon }_3=5$.

Figure 4

Angular dependence of the Fourier image of the $\mathbf{c}$-director fluctuation correlation function. The parameters used are the same as in Fig. 3.