Critical point pairs for smectic-A* - smectic-C* phase transitions.

Liquid crystals is a class of materials possessing properties from both solids and fluids. Similar to solids the molecules arrange themselves in some sort of order. In the liquid crystal state there are multiple phases, smectic being one of them. In a smectic liquid crystal the molecules are aranged (along $z$) in layers. Of the smectic liquid crystals there exists different phases. In the smectic-A (Sm-A) phase the avarage tilt is $0$ relative to $z$ and in the Smectic-C (Sm-A) phase the avarage tilt is non-zero relative to $z$. Normally the liquid crystal will transition between the two phases by altering the temperature. In chiral smectics (i.e. Sm-A*, Sm-C*) it is possible to induce director tilting (i.e. the Sm-C* phase) from the Sm-A* phase via the application of an electric field. This is known as the bulk electroclinic effect (BECE). Also possible to have high-tilt - low-tilt Sm-C* transition, analogous to liquid-gas transition. Like liquid-gas transition, there is a critical point. We investigate possibility of high-tilt - low-tilt phase boundary with \textbf{two} critical points, one at each end of the boundary