Orientational phase transition of tilted molecules moving in a planar layer—tentative model for the smectic-Cphase

A recent x-ray study, by De Jeu and De Poorter, of the smectic phases of heptyloxybenzylidene-pentylaniline shows that the individual molecules are tilted with respect to the layer normal in both the smectic-$A$ and the smectic-$C$ phases. We demonstrate here that a system of long molecules, moving within a planar layer, and making a fixed angle $\epsilon >0\mathrm{}$ with the layer normal, has a second-order transition from a phase with disordered tilt directions to a phase with the tilt directions aligned along some axis. It is suggested that this model phase transition underlies the smectic-$A$—smectic-$C$ transition, at least in those cases where the smectic-$A$ layer thickness is significantly smaller than the molecular length. In this model the effective tilt angle $\alpha$, as measured in optical and magnetic resonance experiments, grows continuously with increasing order of the tilt directions, and saturates at the value $\alpha =\epsilon$. On the other hand, the biaxiality ($\Delta$) in the molecular orientational order passes through a maximum and remains small at all temperatures; e.g., we find $\left|\Delta \right|<\mathrm{}6\mathrm{}\times {10}^{-2\mathrm{}}$ for $\epsilon =30\mathrm{}°$. The phase transition is driven by the gains in translational entropy and total attractive energy which accompany an increase in order of the tilt directions. (At the transition temperature these effects balance the loss in orientational entropy.)