Landau-Ginzburg mean-field theory for the nematic to smectic- C and nematic to smectic- A phase transitions

A Landau-Ginzburg free energy is presented that yields a nematic to smectic-$A$ (NA) transition, a nematic to smectic-$C$ (NC) transition, and a Lifshitz point (at the boundary between the NA and NC transitions). Fluctuation enhancements of the Frank elastic constants are calculated. For the NC transition, all three elastic constants ${K}_{1}$, ${K}_{2}$, and ${K}_{3}$ diverge as ${\ensuremath{\xi}}^{2}$, where $\ensuremath{\xi}$ is the correlation length for fluctuations of the smectic order parameter. At the Lifshitz point, ${K}_{1}$ and ${K}_{2}$ diverge as $\mathrm{ln}\ensuremath{\xi}$ whereas ${K}_{3}$ diverges as $\ensuremath{\xi}$.