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

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

${\displaystyle \mathbf {Q} =S(\mathbf {n} \otimes \mathbf {n} -{\tfrac {1}{3}}\mathbf {I} )}$

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

${\displaystyle {\mathcal {F}}={\mathcal {F}}_{0}+{\frac {A}{2}}Q_{ij}Q_{ji}-{\frac {B}{3}}Q_{ij}Q_{jk}Q_{ki}+{\frac {C}{4}}(Q_{ij}Q_{ij})^{2}}$

or more compactly

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

where ${\displaystyle (A,B,C)}$ are functions of temperature. Near the phase transition, 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.