Devil’s staircase formed by competing interactions stabilizing the ferroelectric smectic-${\mathit{C}}^{\mathrm{*}}$ phase and the antiferroelectric smectic-${\mathit{C}}_{\mathit{A}}^{\mathrm{*}}$ phase in liquid crystalline binary mixtures

In order to demonstrate experimentally that the competition between the antiferroelectric and ferroelectric interactions stabilizing Sm${\mathit{C}}_{\mathit{A}}^{\mathrm{*}}$ and Sm${\mathit{C}}^{\mathrm{*}}$ (the pairing energy of the transverse dipole moments in adjacent smectic layers and the packing entropy resulting from the excluded volume effect) produces a variety of antiferroelectric and ferroelectric subphases in the temperature region between antiferroelectric Sm${\mathit{C}}_{\mathit{A}}^{\mathrm{*}}$ and ferroelectric Sm${\mathit{C}}^{\mathrm{*}}$, we have tried to observe the subphases in several binary mixtures of chiral smectic liquid crystals. The subphases were identified by the electric field dependence of cono- scopic figures and the apparent tilt angle as a function of field strength. Systematically changing the mixing ratio of two properly chosen compounds and studying the stability of each subphase, we have substantiated that the subphases between Sm${\mathit{C}}_{\mathit{A}}^{\mathrm{*}}$ and Sm${\mathit{C}}^{\mathrm{*}}$ form a Devil’s staircase, Sm${\mathit{C}}_{\mathit{A}}^{\mathrm{*}}$(${\mathit{q}}_{\mathit{T}}$). Here, ${\mathit{q}}_{\mathit{T}}$ is an irreducible rational number, which specifies a fraction of the ferroelectric ordering in the antiferroelectric ordering. We have also discussed the relationship of Sm${\mathit{C}}_{\mathit{A}}^{\mathrm{*}}$(${\mathit{q}}_{\mathit{T}}$) to the other staircase, Sm${\mathit{C}}_{\mathrm{\alpha }}^{\mathrm{*}}$(${\mathit{q}}_{\mathit{T}}$), that was proposed to exist as a result of the rather macroscopic, electrostatic interaction among two-dimensional spontaneous polarizations in smectic layers.