Critical points in layered systems

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.

A temperature-concentration (T-X) phase diagram calculated using the mean field theory for liquid crystals

Phase-line equations for smectic-hexatic phase transitions in liquid crystals were derived using the Landau phenomenological theory. In particular, second-order transitions for the smectic-A-smectic-C (SmA-SmC) and hexatic-B-hexatic-F (or HexI) transitions were studied and the tricritical points for these transitions were located. The calculated phase-line equations were fitted (using experimental data for various liquid crystals) to construct a generalized T-X phase diagram. It was shown that the T-X phase diagram calculated from the free energy adequately describes the observed behavior of liquid crystals during smectic-hexatic transitions.

de Vries behavior of the electroclinic effect in the smectic-A* phase near a biaxiality-induced smectic-A*-smectic-C* tricritical point

Using a generalized Landau theory involving orientational, layering, tilt, and biaxial order parameters we analyze the smectic-A* and smectic-C* (Sm-A*-Sm-C*) transitions, showing that a combination of small orientational order and large layering order leads to Sm-A*-Sm-C* transitions that are either continuous and close to tricriticality or first order. The model predicts that in such systems the increase in birefringence upon entry to the Sm-C* phase will be especially rapid. It also predicts that the change in layer spacing at the Sm-A*-Sm-C* transition will be proportional to the orientational order. These are two hallmarks of Sm-A*-Sm-C* transitions in de Vries materials. We analyze the electroclinic effect in the Sm-A* phase and show that as a result of the zero-field Sm-A*-Sm-C* transition being either continuous and close to tricriticality or first order (i.e., for systems with a combination of weak orientational order and strong layering order), the electroclinic response of the tilt will be unusually strong. Additionally, we investigate the associated electrically induced change in birefringence and layer spacing, demonstrating de Vries behavior for each, i.e., an unusually large increase in birefringence and an unusually small layer contraction. Both the induced changes in birefringence and layer spacing are shown to scale quadratically with the induced tilt angle.

Field-temperature phase diagrams in chiral tilted smectics, evidencing ferroelectric and ferrielectric phases

Usual ferroelectric compounds undergo a paraelectric-to-ferroelectric phase transition when the susceptibility of the electric polarization density changes its sign. The temperature is the only thermodynamic field that governs the phase transition. Chiral tilted smectics may also present an improper ferroelectricity when there is a tilt angle between the average long axis direction and the layer normal. The tilt angle is the order parameter of the phase transition which is governed by the temperature. Although the electric susceptibility remains positive, a polarization proportional to the tilt appears due to their linear coupling allowed by the chiral symmetry. Further complications come in when the chirality increases, as new phases are encountered with the same tilt inside the layers but a distribution of the azimuthal direction which is periodic with a unit cell of two (SmC(A)*, three (SmC(Fi1)*, four (SmC(Fi2)* or more (SmC(alpha)* layers. In most of these phases, the layer normal is a symmetry axis so there is no macroscopic polarization except for the SmC(Fi1)* in which the average long axis is tilted so the phase is ferrielectric. By studying a particular compound with only a SmC(Fi2)* and a SmC(alpha)* phase, we show that we recover the uniformly tilted ferroelectric SmC* when applying an electric field. We are thus led to build field-temperature phase diagrams for this class of compounds by combining different experimental techniques described here.

Emergence of hexatic and threefold hidden order in two-dimensional smectic liquid crystals: A Monte Carlo study

Using a high resolution Monte Carlo simulation technique based on a multihistogram method and cluster algorithm, we have investigated the critical properties of a coupled XY model, consisting of a sixfold symmetric hexatic and a hidden order parameter of threefold symmetry in two dimensions. The simulation results demonstrate a series of continuous transitions in which both kinds of orderings are established simultaneously. It is found that the specific-heat anomaly exponents for some regions in coupling constants space are in excellent agreement with the experimentally measured exponents extracted from heat-capacity data near the smectic- A -hexatic- B transition of two-layer free standing films.

Behavior of the layer compression elastic modulus near, above, and below a smectic C-hexatic I critical point in binary mixtures

We present the first study of the layer compression modulus B carried out near, above and below the Smectic C-Hexatic I critical point in racemic mixtures of methylbutyl phenyl octylbiphenyl-carboxylate (8SI) and the octyloxy biphenyl analog (8OSI), at frequencies ranging from 0.2 Hz to 2 x 10(3) Hz. The behavior of B as a function of temperature shows a progressive evolution from a first order transition in 8SI to a continuous supercritical behavior in 8OSI. The latter is characterized by an increase in B, which appears above the transition, and which is followed by a leveling off when the temperature is decreased towards the transition. It is proposed that this behavior stems from the relaxation of the hexatic domains which are frozen in the frequency range studied. For the supercritical and near-critical compounds, B exhibits a small dip near the transition temperature, which is visible in the low frequency range only, indicating that the dynamics associated with the critical point is very slow. We also report measurements in the Crystal-J phase of the pure compounds, and show that 8SI behaves mechanically as a hexatic phase and 8OSI as a soft crystal phase.

Hexatic-herringbone coupling at the hexatic transition in smectic liquid crystals: 4-epsilon renormalization group calculations revisited

Simple symmetry considerations would suggest that the transition from the smectic-A phase to the long-range bond-orientationally ordered hexatic smectic-B phase should belong to the XY universality class. However, a number of experimental studies have reported over the past twenty years "novel" critical behavior with non-XY critical exponents for this transition. Bruinsma and Aeppli argued [Phys. Rev. Lett. 48, 1625 (1982)], using a 4-epsilon renormalization-group calculation, that short-range molecular herringbone correlations coupled to the hexatic ordering drive this transition first order via thermal fluctuations, and that the critical behavior observed in real systems is controlled by a "nearby" tricritical point. We have revisited the model of Bruinsma and Aeppli and present here the results of our study. We have found two nontrivial strongly coupled herringbone-hexatic fixed points apparently missed by these authors. Yet, these two nontrivial fixed points are unstable, and we obtain the same final conclusion as the one reached by Bruinsma and Aeppli, namely that of a fluctuation-driven first-order transition. We also discuss the effect of local twofold distortion of the bond order as a possible "extra" order parameter in the Hamiltonian.

Critical fluctuations near the smectic-hexatic phase transition with anticlinic structure

Layer compression modulus B measurements have been conducted near the transitions between smectic and hexatic phases with synclinic and anticlinic structures. In the synclinic structure, B shows no pretransitional softening near the phase transition. However, in the anticlinic structure, we observed evident critical softening of B near the smectic-hexatic phase transition. These results clearly reveal that the introduction of the in-plane hexatic order in the anticlinic structure is different from the usual smectic-hexatic phase transition.