Landau–de Gennes theory

Mathematical description

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

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

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

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

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

or more compactly

${\displaystyle \mathcal{F}={\mathcal{F}}_0+\frac{1}{2}A(T)\mathrm{t}\mathrm{r}({\mathbf{Q}}^2)-\frac{1}{3}B(T)\mathrm{t}\mathrm{r}({\mathbf{Q}}^3)+\frac{1}{4}C(T)[\mathrm{t}\mathrm{r}({\mathbf{Q}}^2){]}^2.}$

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

${\displaystyle \mathcal{F}-{\mathcal{F}}_0=\frac{a}{3}(T-T_{\ast })S^2-\frac{2b}{27}{S}^3+\frac{c}{9}{S}^4.}$

This is minimized when

${\displaystyle 3a(T-T_{\ast })S-b{S}^2+2c{S}^3=0.}$

The two required solutions of this equation are

${\displaystyle \begin{array}{cc}\text{Isotropic:}& S_I=0,\\ \text{Nematic:}& S_N=\frac{b}{4c}\left[1+\sqrt{1-\frac{24ac}{b^2}(T-T_{\ast })}\right]>0.\end{array}}$

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

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

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