Anisotropic critical properties of the de Gennes model for the nematic to smectic-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>A</mml:mi></mml:math>phase transition

Abstract

The generalized de Gennes model for the nematic to smectic-$A$ phase transition in liquid crystals with an order parameter with $\frac{n}{2}$ complex components is investigated using the $\ensuremath{\epsilon}$ expansion in the vicinity of four dimensions. It is found that anisotropic critical behavior with correlation lengths in directions parallel and perpendicular to the direction of molecular alignment diverging at different rates described by exponents ${\ensuremath{\nu}}_{\ensuremath{\parallel}}$ and ${\ensuremath{\nu}}_{\ensuremath{\perp}}$ is possible. When the splay elastic constant ${K}_{1}=\ensuremath{\infty}$ and $n&gt;{n}_{c2}=238.17$, there is an accessible fixed point with ${\ensuremath{\nu}}_{\ensuremath{\parallel}}&gt;{\ensuremath{\nu}}_{\ensuremath{\perp}}$. If ${K}_{1}$ is large but not infinite and $n=2$, it is expected that there will be a series of crossovers from anisotropic quasicritical behavior to isotropic heliumlike quasicritical behavior and finally to a first-order transition. Present theory and current experiments are compared briefly and found to be in incomplete agreement.