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Mean-Field Behavior of the Paraelectric-Ferroelectric Phase Transition of a Fluorinated Compound
Abstract
The partially fluorinated MHPOBC analogue (MHPO(F)PBC) shows an attractive phase sequence: SmA*, SmC f *, SmC* and SmC A *. The SmC f * phase was only observed by reversal current switching method. Measurements of spontaneous polarization P s (T) reveal the existence of the SmC f * sub-phase between the SmA* and SmC* phases. Linear dielectric spectroscopy studies corroborate almost model behavior of the soft mode relaxation frequency - obtained under bias field of 20V/20 w m -in the vicinity of the SmA*-SmC* phase transition.
The dielectric properties of six symmetrical azomethines with various number of benzene rings were studied in the isotropic phase. The permittivity as well as the impedance spectra were analyzed because the only one relaxation process was visible in the permittivity spectra while two relaxation processes appear in the impedance presentation. It was found that the process at low frequency range, characterized by a very high dielectric increment and parameter α ~ 0.1, is attributed to the ionic motion, while the high frequency Debye-type process with the relatively small increment is connected with the dipolar rotation of the molecules (displacement current). The appearance of an abnormally high value of the static permittivity is discussed, which may be useful for biological and/or other molecular systems where the ionic conductivity is usually very high.
In this section a historical overview will be presented of the development of phenomenological theoretical models of ferroelectric and antiferroelectric liquid crystals. In Sect. 5.1.2 the two order parameters of ferroelectric smectic C* (Sm C*) liquid crystals will be introduced, which are the tilt of long axes of molecules from the normal to the smectic layers and the in-plane electric polarization, that corresponds to the ordering of molecules transverse to their long axes. The phenomenological models of ferroelectric liquid crystals (FLC) will be presented, that correspond to the Landau expansion of the free-energy density in the tilt and the polarization order parameters. An extension of the phenomenological model of FLC will also be demonstrated, that includes as the additional order parameter the transverse quadrupolar ordering and that describes some anomalies of FLC’s, that cannot be accounted by the model with only two order parameters. In these models the layered structure of smectic liquid crystals is not explicitly taken into account and the same is true for the phenomenological models of antiferroelectric liquid crystals (AFLC), that are presented and described in Sect. 5.1.3. Such models cannot describe all existing phases in AFLC and therefore discrete models of AFLC’s were introduced, which take explicitly smectic layers into account and assume some inter-layer interactions, which must have longer range then only between nearest neighboring layers. Discrete models are described in Sect. 5.1.4, while in Sect. 5.1.5 the nature and the structure of the inter-layer interactions is presented together with possible phases predicted by such models.
We have investigated the orthoconic antiferroelectric liquid crystal mixture W107 by means of optical, X-ray and calorimetry measurements in order to assess the origin of the unusally high tilt angle between the optic axis and the smectic layer normal in this material. The optical birefringence increases strongly below the transition to the tilted phases, showing that the onset of tilt is coupled with a considerable increase in orientational order. The layer spacing in the smectic A* (SmA*) phase is notably smaller than the extended length of the molecules constituting the mixture, and the shrinkage in smectic C* (SmC*) and smectic Ca* (SmCa*) is much less than the optical tilt angle would predict. These observations indicate that the tilting transition in W107 to a large extent follows the asymmetric de Vries diffuse cone model. The molecules are on average considerably tilted with respect to the layer normal already in the SmA* phase but the tilting directions are there randomly distributed, giving the phase its uniaxial characteristics. At the transition to the SmC* phase, the distribution is biased such that the molecular tilt already present in SmA* now gives a contribution to the macroscopic tilt angle. In addition, there is a certain increase of the average tilt angle, leading to a slightly smaller layer thickness in the tilted phases. Analysis of the wide angle scattering data show that the molecular tilt in SmCa* is about 20° larger than in SmA*. The large optical tilt (45°) in the SmCa* phase thus results from a combination of an increased average molecule tilt and a biasing of tilt direction fluctuations.
The general shape of the temperature dependence of the static susceptibility in a biasing field conjugated to the order parameter is analysed with the use of the simplest equation of state compatible with the Widom and Griffiths scaling hypothesis. The corresponding curves are demonstrated to show from two to four inflection points, from which a discontinuous inflection point is found to occur exactly at the critical point whenever the critical exponent of susceptibility differs from one: . The unique inflection point occurring below the temperature of the maximum of the susceptibility in the case of the classical critical exponents, i.e. in the mean field theory, is also shown to be strictly independent of the biasing field. New scaling invariants related to the inflection points are found and their analytical expressions are given for the considered equation of state. The usefulness of the theoretical results to the analysis of experimental data is discussed.
Antiferroelectric liquid crystals with a fluorinated terminal chain are reviewed. Their mesomorphism, interlayer spacing, spontaneous polarization and AF-F switching properties are described. Mixtures with an induced antiferroelectric phase are shown.
Soft mode and Goldstone mode properties have been studied for a fluorinated substance. The dielectric spectra have been measured on the planar oriented samples, in the frequency range from 10 Hz to 10 MHz. The thickness of the sample was 10 μm and two kinds of capacitors were used: (i) a low resistance EHC cell and (ii) gold coated electrodes. A bias field up to 10kV cm has been applied to show the existence of both the soft mode and domain mode in the S*c phase below Tc. In the S*c phase a strong Goldstone mode has been observed with a low critical frequency (vC = 15 Hz). The high frequency side at the Goldstone mode spectrum is accompanied by a shoulder which consists of the soft mode and domain mode as well. In the vicinity of the S*A-S*C transition the dielectric parameters of the soft mode obey a Curie-Weiss law. The slope ratio is equal to −1·62 for the inverse of dielectric increments (Δ∈) and −1·73 for the critical frequencies (vc) obtained by using gold electrodes. The respective values received for the EHC cell are −4·14 and −2·1. The dielectric parameters of the domain mode have been obtained versus temperature and bias field. We can qualitatively show that a high frequency molecular relaxation is present in the S*A and S*C phases as the high frequency limit of dielectric permittivity is distinctly higher than the refractive index squared.
The frequency and temperature dependence of the complex dielectric constant was measured near the smectic-C*-smectic-A liquid-crystal phase (Sm-C*-Sm-A) transition for two ferroelectric liquid crystals with a very large spontaneous polarization. The dielectric strengths as well as the corresponding relaxation frequencies of the dielectric modes were determined. Due to the large polarization we were able to resolve the contributions from the soft mode and the Goldstone mode in a comparatively large temperature interval in the Sm-C* phase close to the transition to the Sm-A phase. In the Sm-A phase, the relaxation frequency and the inverse dielectric strength of the soft mode decrease linearly when approaching Tc. Comparing the experimental data with a recent theoretical calculation of the complex dielectric constant, we conclude that we have a fairly good understanding of the behavior of the complex dielectric constant in the whole Sm-C*-Sm-A transition range.