Tilted cones

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.

In a bulk SmC material the tilt angle 0 is fixed, but the director is free to take up any azimuthal orientation, defining a tilt cone of degenerate azimuthal... 

Since P must remain normal to z and n, the polarization vector forms a helix, where P is everywhere normal to the helix axis. While locally a macroscopic dipole is present, globally this polarization averages to zero due to the presence of the SmC helix. Such a structure is sometimes termed a helical antiferroelectric. But, even with a helix of infinite pitch (i.e., no helix), which can happen in the SmC phase, bulk samples of SmC material still are not ferroelectric. A ferroelectric material must possess at least two degenerate states, or orientations of the polarization, which exist in distinct free-energy wells, and which can be interconverted by application of an electric field. In the case of a bulk SmC material with infinite pitch, all orientations of the director on the tilt cone are degenerate. In this case the polarization would simply line up parallel to an applied field oriented along any axis in the smectic layer plane, with no wells or barriers (and no hysteresis) associated with the reorientation of the polarization. While interesting, such behavior is not that of a true ferroelectric. 

Apparently this switching mode is disfavored since, in fact, the chirality of the layers does not change upon switching to the ferroelectric state rather the layer interface clinicity changes. This occurs when the molecules in alternate layers simply precess about the tilt cone in a manner exactly analogous to antiferroelectric to ferroelectric switching in the chiral SmC phase. As shown in Figure 8.25, the ferroelectric state obtained from the ShiCsPa antiferroelectric phase is a ShiCaPf structure, an achiral macroscopic racemate with anticlinic layer interfaces. 

Application of a field to the ShiCaPa phase causes switching by precession of the director around the tilt cone in alternate layers, to give a ferroelectric ShiCsPf state with uniform tilt. In this case, there can be no domains of opposite tilt since such domains would necessarily have their polarization opposing the applied field. This leads to a uniform SmC-like texture with a green birefringence color. The extinction brushes in the cylindrical focal conic rotate counterclockwise when the net tilt rotates clockwise, as indicated in Figure 8.25. As anticipated, the chiral rotation of the brushes is a direct manifestation of the chirality of the phase. Elsewhere in the sample there must be ShiCaPa domains of opposite handedness, which would possess the opposite sense of tilt for the same sign of the applied field. 

In addition, in the absence of the bounding plates, a spontaneous helix develops in the director field about the layer normal along the tilt cone (indicated in the Figure)... 

In this section we present a model [32, 33] which predicts that regions of favorable splay are connected by regions of unfavorable splay in which layers are inclined to reduce the molecular tilt. Unfavorable splay occurs on a smaller tilt cone and as a result the energy penalty in the unfavorable splay regions is reduced. The local increase of the layer thickness due to the smaller tilt angle leads to the bending of layers and as a result to the formation of undulated layers or fragmented layer structure. 

Figure 8.7. Idealized configuration of the SSFLC cell. Depending on the polarity of the applied dc field E, the director n switches between two different positions on the tilt cone, corresponding to two different positions of the optical axis in a plane parallel to the cell plates. In real devices the smectic layer planes are not perfectly perpendicular to the cell plates but show a chevron structure with a kink inside the cell.

Another quantity which has been studied in chiral-racemic systems [75], and which is important for applications, is the rotational viscosity corresponding to the motion of the director n on the tilt cone at constant tilt magnitude 6. On first approximation, the switching time t in response to the reversal of a field E is given as... 

If we keep the tilt angle 9 in Fig. 48 fixed (just by keeping the temperature constant), we can move the molecule around the tilt cone, and we see that the rotational bias stays sterically fixed to the molecule and the tilt plane. The direction of the resulting P lies in the direction zxn. If we change the azimuthal angle by 180°, this corresponds to a tilt -9 (along the -x direction), and results in a change of P direction from y to -y. This means that we could tentatively write the relation between the secondary and pri-... 

Figure 9.10 Pumping performance as a function of Sp. (a) Tilted cone (6 = n/4) each curve corresponds to a different opening angle (() with changes of A<() = 0.1 (in radians) between curves up to a maximum <() = 0.7...

Soft-mode FLC (SMFLC) was introduced as an alternative way to switch FLC. In this case, instead of vaiying the azimuthal angle ( ) around the tilt cone, i.e., 6 is constant as in SSFLC, SMFLCs use changes in the tilt (9) while (f) remains constant. As a result, SSFLCs exhibit bistability, but SMFLCs are capable of continuous intensity change. The SMFLCs employ smectic-A phase, and uniform alignment is much easier to obtain for SMFLCs than for SSFLCs, which employ smectic-C. ... 

The topography of the cone affects the system s dynamics. Simple classic arguments can rationalize the way topography affects a trajectory Vertical cones facilitate transitions from the upper surface to the lower surface whereas tilted cones are less efficient. Actual quantum mechanical calculations have confirmed these generalizations. " The efficacy of a conical intersection in promoting a nonadiabatic transition reflects the topography in the vicinity of a conical intersection. ... 

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