A modified Landau–de Gennes theory for smectic liquid crystals: phase transitions and structural transitions

We mathematically model smectic-A (SmA) phases with a modified Landau–de Gennes (mLdG) model as proposed in Xia et al. [Phys. Rev. Lett., 126 (2021), 177801]. The orientational order of the SmA phase is described by a tensor-order parameter , and the positional order is described by a real scalar , which models the deviation from the average density of liquid crystal molecules. First, we prove the existence and regularity of global minimizers of the mLdG free energy in three-dimensional settings. Then we analytically prove that the mLdG model can capture the isotropic-nematic-smectic phase transition as a function of temperature, under some assumptions. Further, we explore stable smectic phases on a square domain with edge length and tangent boundary conditions. We use heuristic arguments to show that defects repel smectic layers and that nematic ordering promotes layer formation. We use asymptotic arguments in the and limits which reveal the correlation between the number and thickness of smectic layers, the amplitude of density fluctuations with the phenomenological parameters in the mLdG energy. For finite values of , we numerically recover BD-like and D-like stable smectic states observed in experiments. We also study the frustrated mLdG energy landscape and give numerical examples of transition pathways between distinct mLdG energy minimizers.