# Transition from nematic phase to smectic phase in calamitic liquid crystals

I am a math student and not very good at physics. I am so confused when I read the paper about the mathematical problems in liquid crystals.

My question is about the nematic phase and smectic phases. I think that for a specified material of liquid crystals, they will transit from nematic phase to smectic phase when the temperature decreases. In the mathematical study, I have seen some studies using the Landau-de-Gennes energy (the simplified form) to describe the equilibrium state, that is, $$\int_{B_r} |\nabla Q|^2 - a\text{tr}(Q^2) + b \text{tr}(Q^3) + c\text{tr}(Q^4)dx.$$ Here $$a$$ is the reduced temperature. In the study "On the local instability of radial hedgehog configurations in nematic liquid crystals under Landau-de Gennes free-energy models", they consider $$a$$ very large and still call it nematic liquid crystals. I am so confused by that since I think it will be smectic when temperature is low.

I think the main thing to note is that the free energy functional you refer to is only intended to describe the normal-nematic transition. $$Q_{ij}\sim n_i n_j -\delta_{ij}/3$$ is an order parameter constructed from the molecular orientation $$n_i$$. If $$Q_{ij}$$ is zero we have a normal liquid, and $$Q_{ij}\neq 0$$ corresponds to the nematic phase. This transition is driven by the parameter $$a$$, which depends on temperature, but possibly also on other quantities, like the concentration.
Some (but not all) nematic liquid crystals also exhbit phases with smectic order. This is a phase that also shows a density modulation, which can be described by an additional order parameter $$\psi$$ which encodes the amplitude of a density modulation with some wavenumber $$k$$. A free energy functional for the nematic-smectic transition can also be found in the literature, for example here (also see the text book by Chaikin and Lubensky).