Phase-Diagram of a Frustrated Smectic Liquid-Crystal System and the Absence of the Incommensurate Smectic-A(I2) Phase

The phase diagram and the structure of smectic-${\mathit{A}}_{\mathit{d}}$ (Sm-${\mathit{A}}_{\mathit{d}}$) and Sm-${\mathit{A}}_{2}$ phases of binary mixtures of 4-octyloxy-4'-cyanobiphenyl (8OCB) and 4-n-heptyloxyphenyl-4'-cyanobenzoyloxybenzoate (DB7OCN) have been determined by high-resolution x-ray diffraction. This system had previously been reported by Ratna, Shashidhar, and Raja [Phys. Rev. Lett. 55, 1476 (1985)] to exhibit an incommensurate Sm-${\mathit{A}}_{\mathit{i}2}$ phase for concentrations of 8OCB ranging from 25 to 40 mol %. The phase diagram obtained from the present study is in excellent agreement with the mean-field theory of Prost and co-workers [J. Phys. (Paris) 46, 391 (1985)]. The results of the x-ray study establish that the Sm-${\mathit{A}}_{\mathit{i}2}$ phase is, in fact, a coexistence of the Sm-${\mathit{A}}_{\mathit{d}}$ and Sm-${\mathit{A}}_{2}$ phases at the first-order phase transition between them. Coexistence of the two phases is observed at all concentrations of 8OCB ranging from 35.1 to 0 mol %. At a fixed temperature in the coexistence region, the system evolves towards one phase at a very slow pace. The coexistence region becomes narrower at lower concentrations. Temperature dependence of the smectic wave vectors indicates the presence of a critical point at (physically inaccessible) negative concentrations of 8OCB. The coexistence behavior at the Sm-${\mathit{A}}_{\mathit{d}}$ to Sm-${\mathit{A}}_{2}$ transition also conforms to the predictions of the theory and resembles the coexistence curve in the proximity of the liquid-gas critical point.