Tilt plane

Figure 8.6 Three-dimensional slice of C2 symmetrical SmC phase, showing tilt cone, polar axis (congruent with twofold symmetry axis), smectic layer planes, tilt plane, and polar plane.

By our definition, the tilt plane is normal to the polarization in the ferroelectric state in the illustration in Figure 8.13 this is a vertical plane normal to the plane of the page. Since there is no tilt of the director projected onto this plane, the phase should be considered a type of SmA. We name this structure SmAPp (an untilted polar smectic the subscript F referring to a ferroelectric structure, in this case a ferroelectric state of an antiferroelectric phase). The antiferroelectric phase is therefore also an SmA denoted SmAPA (the subscript A for antiferroelectric). While this idea is certainly intriguing, no such antiferroelectric has yet been discovered. 

Figure 8.24 Illustration of layer structure and symmetries observed for NOBOW thermodynamic phase (majority domains) in freely suspended films, (a) Films of even-layer number have achiral, nonpolar C symmetry, (b) Films of odd-layer number have chiral and polar C2 symmetry, with net polarization normal to tilt plane (lateral polarization).

The layer stacking is synclinic in the tilt plane and antiferroelectric in the polar plane. The phase composed of an infinite number of SmCP layers stacked in this way is termed SmCsP, where the subscripts S and A each refer a structural feature of the layer interfaces between adjacent pairs of layers. If two adjacent layers are tilted in the same direction, the interface is synclinic (subscript S) in the tilt plane. If two adjacent layers have antiparallel orientation of their polar axes, then the layer interface is said to be antiferroelectric (A) in the polar plane. 

As can be easily seen by inspection of these illustrations of the SuiCsPa and ShiCsPf phases, while the director tilt in the tilt plane is synclinic for both, the layer interfaces have a different character when observed in projection in the bow plane. The antiferroelectric diastereomer has synclinic character at the layer interfaces, while the ferroelectric diastereomer is anticlinic in the bow plane. This suggests a very simple reason for the tendency toward antiferroelectric bananas, this being basically the same as the tendency toward ferroelectric calamitic smectics preference for synclinic layer interfaces. 

A simple consideration of the synclinic banana phases in the context of the prior discovery of the Soto Bustamante-Blinov achiral antiferroelectric bilayer is illuminating. In Figure 8.28, the achiral antiferroelectric SmAPA bilayer structure is illustrated on the left. The layers are horizontal and normal to the plane of the page, and the tilt plane is vertical and normal to the plane... 

Figure 8.27 Illustrations showing SmCsPA and SniCsP. supermolecular structures projected onto tilt plane and onto bow plane are given. Projections onto bow plane are meant to illustrate effective synclinic layer interface present in SmCsPA phase and corresponding anticlinic layer interface in SmCsPF phase, though both are synclinic in tilt plane.

From this discussion the clear similarity between the SmAPA and SmCsPA structures is easily seen. In addition, the suggestion of Brand et al.29 that a bilayer smectic with all anticlinic layer interfaces (the SmAPF) would produce an achiral ferroelectric smectic follows directly. The unanticipated tilt of the director in the tilt plane, leading to a chiral layer structure, seems to be a general response of the bent-core mesogens to the spontaneous nonpolar symmetry breaking occurring in these rigid dimer structures. 

FIG. 6.9 A drop on a tilted plane, showing advancing and receding contact angles. (Adapted from Johnson and Dettree 1969.)... 

The existence of the layers and director tilt in the achiral smectic C liquid crystal phase are experimental facts. Given these, the maximum possible symmetry of the phase would be Ci, with a C2 axis normal to the tilt plane, and a a plane congruent with the tilt plane. In fact, there is no fundamental reason why a given C phase must possess either of these symmetry elements. But, breaking of either of the symmetry elements would afford polar symmetry, and no C phase has ever been shown to possess any property associated with polar symmetry (e.g. pyroelectricity). Therefore, we can say that all known C phases indeed possess the maximum possible symmetry consistent with the layers and tilt. 

The macroscopic polarization of the phase is given by equations 1 and 2, where Di is the number density of the ith conformation, jlj is the component of the molecular dipole normal to the tilt plane when the ith conformation of the molecule is oriented in the rotational minimum in the binding site, ROFj is the "rotational orientation factor", a number from zero to one reflecting the degree of rotational order for the ith conformation, and e is a complex and unmeasured dielectric constant of the medium (local field correction). 

Figure 2. Preferred conformational and rotational orientation relative to the tilt plane for compound 3 in the C phase according to the Boulder Model.

The experimental observations could be consistently explained if the general tilt structure (SmCo) inside the layers is assumed. For most bent-core smectics the polar vector is perpendicular to the tilt plane, defined by the layer normal and averaged long axis direction, just as polarization in the ferroelectric rod-like liquid crystalline systems. However, since in the bent-core liquid crystals the polar order is decoupled from the tilt order, the polar director can in general have any direction in space thus it can also have a non-zero component along the layer normal. This can be achieved by a combination of tilting (rotation around the polar director) and leaning (rotation around the direction perpendicular to the polar director) of... 

The first term in (7) describes the coupling between the polarization splay and tilt of the molecules with respect to the smectic layer normal. This coupling is responsible for the chiral symmetry breakdown in phases where bent-core molecules are tilted with respect to the smectic layer normal [32, 36]. The second term in (7) stabilizes a finite polarization splay. The third term with positive parameter Knp describes the preferred orientation of the molecular tips in the direction perpendicular to the tilt plane (the plane defined by the nematic director and the smectic layer normal). However, if Knp is negative, this term prefers the molecular tips to lie in the tilt plane. The last term in (7) stabilizes some general orientation (a) of the polar director (see Fig. 7) which leads to a general tilt (SmCo) structure. 

Let us first consider the case where the preferred orientation of the polar director is perpendicular to the tilt plane (K > 0). The spatial variation of the layer normal and the nematic and polar directors is shown in Fig. 12. We see that regions of favorable splay (called blocks or layer fragments in Sect. 2) are intersected by regions of unfavorable splay (defects, walls). In the region of favorable splay the smectic layer is flat. In the defects regions the tilt angle decreases to reduce energy... 

Let us now consider the case of in (7) being negative. The term with K p alone would now require the molecular tips to lie in the tilt plane. So the last term in (7) is needed to stabilize a finite angle between the polar director and the tilt plane. As shown in Fig. 10 the system can profit by the orientation of the polar director towards the tilt plane since this can lead to a better packing of the bulky molecular cores. 

Numerical results show that the reduction of the cone angle in the region of the unfavorable splay is smaller in the general tilt structure than in the structure with polarization being perpendicular to the tilt plane. The cone angle reduction is coupled to the undulation of layers. So in the general tilt structure the undulation is less pronounced than in the regular Blrev phase (Fig. 13), as was indeed observed experimentally [25]. 

We have also shown the existence of 2D phases due to more subtle electron density changes. In some cases additional peaks are observed in the XRD pattern, signifying a double layer periodicity in the system, which can be accounted for if a general orientation of the polar director is allowed. If the polar director is not perpendicular to the tilt plane there exists a component of polarization in the direction of the smectic layer normal (longitudinal polarization). By double layer periodicity the system escapes from the polar structure and in addition achieves better packing of the molecular cores and molecular tails. 

Miyachi K, Matsushima J, Ishikawa K, Takezoe H, Fukuda A (1995) Spontaneous polarization parallel to the tilt plane in the antiferroelectric chiral smectic-CA phase of liquid-crystals as observed by polarized infrared-spectroscopy. Phys Rev E 52 R2153-R2156... 

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