Landau theory of the helicoidalC/emph>phase in smectic liquid crystals: Re-entrance of the smectic-C/emph>phase and order of the smectic-C–smectic-C/emph>transition

We present a Landau theory for smectic liquid crystals in a magnetic field, and apply it to systems exhibiting the smectic-A (Sm-A), smectic-C (Sm-C), and smectic-$C^{\mathrm{*}}$ (Sm-$C^{\mathrm{*}}$) phases; our theory provides a unified explanation of two phenomena, the temperature dependence of the pitch in the Sm-$C^{\mathrm{*}}$ phase and the re-entrance of the Sm-$C^{\mathrm{*}}$ phase. We find that the Sm-$C^{\mathrm{*}}$ phase is always re-entrant. Two types of phase diagrams are found, depending on the relative signs of two of the Landau parameters; in one case, the Sm-$C^{\mathrm{*}}$ phase is re-entrant only for fields greater than some value, while in the other case it is re-entrant for all field values (but the helicity has different sign above and below the Sm-C phase). The results are in qualitative agreement with experiment. The order parameter in the Sm-$C^{\mathrm{*}}$ phase has a well-developed domain structure only near the Sm-C–Sm-$C^{\mathrm{*}}$ boundary; a new feature is that the order parameter is almost sinusoidal at low temperatures, well below the boundary. We provide a complete analytical treatment of the Sm-C–Sm-$C^{\mathrm{*}}$ transition, extending previous results in the theory of commensurate-incommensurate transitions.