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.^[1][2] The phenomonological theory uses the ${\displaystyle \mathbf {Q} }$ tensor as an order parameter in expanding the free energy density.^[3][4]

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

${\displaystyle {\mathcal {F}}={\mathcal {F}}_{0}+{\frac {1}{2}}A(T)\mathrm {tr} (\mathbf {Q} ^{2})-{\frac {1}{3}}B(T)\mathrm {tr} (\mathbf {Q} ^{3})+{\frac {1}{4}}C(T)[\mathrm {tr} (\mathbf {Q} ^{2})]^{2}.}$

Further, we can expand ${\displaystyle A(T)=a(T-T_{*})+\cdots }$, ${\displaystyle B(T)=b+\cdots }$ and ${\displaystyle C(T)=c+\cdots }$ with ${\displaystyle (a,b,c)}$ being three positive constants. Now substituting the ${\displaystyle \mathbf {Q} }$ tensor results in^[5]

${\displaystyle {\mathcal {F}}-{\mathcal {F}}_{0}={\frac {a}{3}}(T-T_{*})S^{2}-{\frac {2b}{27}}S^{3}+{\frac {c}{9}}S^{4}.}$

This is minimized when

${\displaystyle 3a(T-T_{*})S-bS^{2}+2cS^{3}=0.}$

The two required solutions of this equation are

${\displaystyle {\begin{aligned}{\text{Isotropic:}}&\,\,S_{I}=0,\\{\text{Nematic:}}&\,\,S_{N}={\frac {b}{4c}}\left[1+{\sqrt {1-{\frac {24ac}{b^{2}}}(T-T_{*})}}\,\right]>0.\end{aligned}}}$

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

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

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