Molecular orientation arrangements in the smectic-$C^{*}$ variant liquid-crystal phases

Recent experiments have identified three-layer and four-layer distorted helical structures in the smectic liquid-crystal phases Sm $C_{\mathrm{FI}1}^{*}$ and Sm $C_{\mathrm{FI}2}^{*},$ respectively. However, no theories have explained the existence of all these phases. A discrete phenomenological model of the free-energy is analyzed and found to predict the stability of distorted three-layer and four-layer structures, as well as simple helical solutions in smectic liquid crystals. A simple physical picture is provided to explain the stability of the phases exhibiting distorted helical structures.