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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.

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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).

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  • $\begingroup$ Thank you so much! I misunderstand that all LC will be smectics when temperature is suitably low. But it is not true, right? $\endgroup$
    – mnmn1993
    Aug 7, 2022 at 9:15
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    $\begingroup$ @mnmn1993 Indeed, the phase diagram can be quite complex, and there are regions of pressure or concentration where the sequence of phases as a fct of T is isotropic-nematic-solid. $\endgroup$
    – Thomas
    Aug 8, 2022 at 2:09
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    $\begingroup$ But, maybe more importantly, because the phase diagram is complex, people will typically not try to construct a single functional that describes all of it. If you want to study the normal-nematic transition, then you need a functional for Q. $\endgroup$
    – Thomas
    Aug 8, 2022 at 2:11

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