Theoretical model of the frequency and temperature dependence of the complex dielectric constant of ferroelectric liquid crystals near the smectic-C*–smectic-Aphase transition

Using the generalized Landau model [Phys. Rev. A 36, 1484 (1987)], the temperature and frequency dependence of the complex dielectric constant of the ferroelectric smectic-${\mathit{C}}^{\mathrm{*}}$ (Sm-${\mathit{C}}^{\mathrm{*}}$) phase and the corresponding smectic-A (Sm-A) phase is calculated. It is demonstrated how the dielectric response of the Sm-${\mathit{C}}^{\mathrm{*}}$ phase generally consists of four modes—two high-frequency polarization modes and two modes of lower frequency that are connected to the reorientation of the director, commonly denoted the soft mode and the Goldstone mode. In the Sm-A phase only two modes are present—one doubly degenerate soft mode and one doubly degenerate polarization mode. The temperature dependences of the dielectric strengths and relaxation frequencies of the modes in question are calculated, and simplified expressions of these quantities are given. The most important feature of the generalized Landau model is the presence of a biquadratic coupling between tilt and polarization in the free-energy density of the system and we show how the general thermodynamic and dielectric properties of the system depend on the strength of this coupling. Comparing the results of the calculations with existing data, we finally conclude that the model provides a description of the Sm-${\mathit{C}}^{\mathrm{*}}$–Sm-A transition that takes all experimentally known features of the dielectric properties of the system into account in a qualitatively correct way.