Landau–de Gennes theory

In physics, Landau–de Gennes theory describes the NI transition, i.e., phase transition between nematic liquid crystals and isotropic liquids, which is based on the classical Landau's theory and was developed by Pierre-Gilles de Gennes in 1969. The phenomonological theory uses the $\mathbf{Q}$ tensor as an order parameter in expandiing the free energy density.

Mathematical description
The NI transition is a first-order phase transition, albeit it is very weak. The order parameter is the $$\mathbf{Q}$$ tensor, which is symmetric, traceless, second-order tensor and vasbishes in the isotripic liquid phase. We shall consider a uniaxial $$\mathbf Q$$ tensor, which is defined by


 * $$\mathbf Q = S\left(\mathbf n\mathbf n - \frac{1}{3}\mathbf I\right)$$

where $$S=S(T)$$ is the scalar order parameter. The $$\mathbf Q$$ tensor is zero in the isotropic liquid phase since the scalar order parameter $$S$$ is zero, but becomes non-zero in the nematic phase.

Near the NI transition, the (Helmholtz or Gibbs) free energy density $$\mathcal{F}$$ is expanded about as


 * $$\mathcal{F} = \mathcal{F}_0 + \frac{1}{2}A (T) Q_{ij}Q_{ji} - \frac{1}{3}B(T) Q_{ij}Q_{jk}Q_{ki} + \frac{1}{4}C(T) (Q_{ij}Q_{ij})^2 $$

or more compactly


 * $$\mathcal{F} = \mathcal{F}_0 + \frac{1}{2}A(T)\mathrm{Tr}(\mathbf{Q}^2) - \frac{1}{3}B(T)\mathrm{Tr}(\mathbf{Q}^3) + + \frac{1}{4}C(T)(\mathrm{Tr}(\mathbf{Q}^2))^2.$$

Further, we can expand $$A(T)=a (T-T_*)+\cdots$$, $$B(T) = b + \cdots$$ and $$C(T)=c + \cdots$$ with $$(a,b,c)$$ being three positive constants. Now substituting the $$\mathbf Q$$ tensor results in


 * $$\mathcal{F} - \mathcal{F}_0 = \frac{1}{3}a(T-T_*)S^2 - \frac{2}{27} bS^3 + \frac{1}{9}cS^4.$$

This is minimized when


 * $$3a(T-T_*) - b S^2 + 2c S^3=0.$$

The two required solutions of this equation are


 * $$\begin{align}\text{Isotropic:} & \,\,S_I = 0,\\

\text{Nematic:} & \,\,S_N = \frac{b}{4c} [1+\sqrt{1-24a(T-T_*)c/b^2}]>0. \end{align}$$

The NI transition temperature $$T_{NI}$$ is not simply equal to $$T_*$$ (which would be the case in second-order phase transition), but is given by


 * $$T_{NI} = T_* + \frac{b^2}{27ac}, \quad S_{NI} = \frac{b}{3c}$$

$$S_{NI}$$ is the scalar order parameter at the transition.