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. The orientational order of the SmA phase is described by a tensor-order parameter $\mathbf{Q}$, and the positional order is described by a real scalar $u$, which models the deviation from the average density of liquid crystal molecules. Firstly, we prove the existence and regularity of global minimisers 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 $\lambda$, and tangent boundary conditions. We use heuristic arguments to show that defects repel smectic layers and strong nematic ordering promotes layer formation. We use asymptotic arguments in the $\lambda\to 0$ and $\lambda\to\infty$ 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 $\lambda$, 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 minimisers.