Type-I and type-II smectic-C^{*} systems: A twist on the electroclinic critical point

We conduct an in-depth analysis of the electroclinic effect in chiral, ferroelectric liquid crystal systems that have a first-order smectic-A^{*}-smectic-C^{*} (Sm-A^{*}-Sm-C^{*}) transition, and show that such systems can be either type I or type II. In temperature-field parameter space type-I systems exhibit a macroscopically achiral (in which the Sm-C_{M}^{*} helical superstructure is expelled) low-tilt (LT) Sm-C_{U}^{*}-high-tilt (HT) Sm-C_{U}^{*} critical point, which terminates a LT Sm-C_{U}^{*}-HT Sm-^{*}C_{U} first-order boundary. Notationally, Sm-C_{M}^{*} or Sm-C_{U}^{*} denotes the Sm-C^{*} phase with or without a modulated superstructure. This boundary extends to an achiral-chiral triple point at which the macroscopically achiral LT Sm-C_{U}^{*} and HT Sm-C_{U}^{*} phases coexist along with the chiral Sm-C_{M}^{*} phase. In type-II systems the critical point, triple point, and first-order boundary are replaced by a Sm-C_{M}^{*} region, sandwiched between LT and HT Sm-C_{U}^{*} phases, at low and high fields, respectively. Correspondingly, as the field is ramped up, the type-II system will display a reentrant Sm-C_{U}^{*}-Sm-C_{M}^{*}-Sm-C_{U}^{*} phase sequence. Moreover, discontinuity in the tilt of the optical axis at each of the two phase transitions means the type-II system is tristable, in contrast to the bistable nature of the LT Sm-C_{U}^{*}-HT Sm-C_{U}^{*} transition in type-I systems. Whether the system is type I or type II is determined by the ratio of two length scales, one of which is the zero-field Sm-C^{*} helical pitch. The other length scale depends on the size of the discontinuity (and thus the latent heat) at the zero-field first-order Sm-A^{*}-Sm-C^{*} transition. We note that this type-I vs type-II behavior in this ferroelectric smectic is the Ising universality class analog of type-I vs type-II behavior in XY universality class systems. Lastly, we make a complete mapping of the phase boundaries in all regions of temperature-field-enantiomeric-excess parameter space (not just near the critical point) and show that various interesting features are possible, including a multicritical point, tricritical points, and a doubly reentrant Sm-C_{U}^{*}-Sm-C_{M}^{*}-Sm-C_{U}^{*}-Sm-C_{M}^{*} phase sequence.