Frustrated smectics

The frustration effects are implicit in many physical systems, as different as spin glass magnets, adsorbed monomolecular films and liquid crystals [32, 54, 55], In the case of polar mesogens the dipolar frustrations may be modelled by a spin system on a triangular lattice (Fig, 5), The corresponding Hamiltonian consists of a two particle dipolar potential that has competing parallel dipole and antiparallel dipole interactions [321, The system is analyzed in terms of dimers and trimers of dipoles. When the dipolar forces between two of them cancel, the third dipole experiences no overall interaction. It is free to permeate out of the layer, thus frustrating smectic order. 

Figure C2.2.5. Frustrated smectic phases. Here the arrows denote longitudinal molecular dipoles. 

Another class of frustrated smectic phases are the incommensurate smectic A phases in which the competing periodicities coexist along the layer normal. The first example of such a phase was observed for a non-symmetric dimer, KI5 [144]... 

Aj-Nr-Aj point show that the hicritical point has, in fact, split into a tricritical point (for the Aj-N boundary) and a critical end point (for the Afl-N boundary, fig. 5.6.5). This new result, though predicted theoretically for magnetic systems, has not been envisaged in any of the theories of frustrated smectics. 

J. Wang, The Frustrated Smectic Phases in Liquid Crystals , thesis. University of Pennsylvania, 1985. 

Another class of frustrated phase results from the frustration between bend or twist deformations in smectic phases (Section 5.6) and the tendency to form a layered structure. Twisted grain boundary phases are frustrated smectic phases and both SmA and SmC versions have been observed. The phases are denoted TGBA and TGBC respectively and are formed by chiral mesogens. The phases are macroscopically chiral and result from arrays of screw dislocations (i.e. defects in lattice order) which lead to a twist in the director between grains of layers, i.e. to a helical rotation of layers. 

Frost showed that the properties and structures of frustrated smectics can be described by two order parameters [72, 82]. The first p(r) measures mass density modulation familiar in SmA phases [1 ]. The second (r), often referred to as a polarization wave, describes long range head-to-tail correlations of asymmetric molecules along the z axis... 

Figure 6. Mean field phase diagram obtained from the model of frustrated smectics for different values of the incommensurability parameter (a) Very weak incommensurability N, SmA, and SmA2 form the generic phase diagram of frustrated smectics. First calculated by Frost [82] this diagram is similar to the experimental one observed on the mixture DB5-TBBA [67].

The mean field analysis of the Frost s model of frustrated smectics describes most of the reported experimental observations on polar smectics. Situations where fluctuations are important are however not correctly described the understanding of multiply re-entrant behavior, the appearance of nematic bubbles and the scaling properties of the SmA2-SmAcritical point for instance require more elaborate analysis. Some of these points will be discussed in the following sections. 

Although not specific of liquid crystals, the re-entrant phenomenon (i.e. re-appearance of the phase of higher symmetry upon cooling) is often observed in frustrated smectics [80,106]. Theoretical analyses suggest that there is no universal explanation for re-en-trance in the N-SmA problem. 

Figure 15. Heat capacity of 80PCB0B near the nematic to smectic A1 phase transition. The smooth curve represents a fit to the data with Eq. (6) based on critical parameters in agreement with the three-dimensional XY model. The index 1 in SmA refers to the monolayer structure of this frustrated smectic compound [65].

A phenomenological model for frustrated smectics proposed that 2-D modulated smectic phases might arise as an escape from underlying incommensurability in the system [22,161,162]. The role of polar and steric molecular assymetry in smectic polymorphism has been comprehensively reviewed [23, 163]. The forms of the X-ray diffraction patterns from modulated and incommensurate smectic phases have been extensively discussed [20-23] and these papers should be consulted for earlier references and for details which will not be covered here. 

Modulated smectic phases are just one of a number of classes of frustrated smectic phases, all of which arise from the competition between different characteristic length scales but differ in the manner by which this frustration is relieved. In the incommensurate SmA phases the competing periodicities coexist along the layer normal and the first example of such a phase was observed for a nonsymmetric dimer, KI5 (see Fig. 1 g). In this phase the larger periodicity appears to correspond to the molec-... 

Fig. 9.22 (a) X-ray pattern of oriented smectic phase observed in BPCO3-011 fibers at 200 °C. The fiber specimen was prepared by pulling up the isotropic melt and its axis is placed in the vertical direction. It includes several other inner reflections except for the usual layer reflections, indicating the frustrated phase, (b) Tentative structural model of frustrated smectic phase. For convenience, the polymer chains in an all-trans conformation are illustrated... 

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