# Helix-Free Ferroelectric Liquid Crystals: Electro Optics and Possible Applications
## 4 Ferroelectric Liquid Crystals Director Reorientation by an Electric Field in the Soliton Mode  
As shown in [15; 16], the periodic deformation of smectic layers means that for FLC molecules initially inclined at the angle \(\Theta_{0}\) with respect to the normal to the FLC layer at a given point, it is energetically preferable to additionally deflect by some angle \(\pm\Psi\) with respect to the direction \(z\) of substrate rubbing (Figure 3). Because of this, the position of the FLC main optical axis along the smectic layers changes, and the birefringence depends on the electric field frequency. The electric field \(E\) applied along the coordinate \(y\) interacts with the spontaneous polarization and changes the director distribution (the angle \(\Psi\)) in each smectic layer. The reorientation of the FLC director can occur both with the change of the azimuthal angle \(\varphi\) of the director orientation by 180\({}^{\circ}\), when the director is reoriented along the cone generatrix with its 2\(\Theta_{0}\) turn, and with the change in the distribution of the angle \(\psi\) characterizing the deformation of smectic layers. In the first case, the  
Figure 3: An electro-optical cell with helix-free FLC (**a**) and a fragment of a deformed smectic layer (**b**) 1—glass substrates with conductive covers; 2—smectic layers; \(\Theta_{0}\)—the angle of molecule tilt in smectic layers; \(\psi\)—angle of tilt of a smectic layer; \(P_{S}\)—vector of spontaneous polarization; \(d\)—FLC cell thickness; \(l\)—smectic layer thickness. dissipative coefficient is the rotational viscosity \(\gamma_{\varphi}\), and in the second case, it is the viscosity for the shear deformation \(\gamma_{\psi}\). When after the electric field is turned off the director relaxation time begins to depend on the electric field frequency, the transition to the Maxwellian mechanism of energy dissipation takes place--that is, the transition from the rotation viscosity \(\gamma_{\varphi}\) to the shear viscosity \(\gamma_{\psi}\). To describe the nonlinear dynamic process of FLC director reorientation in an external electric field, we used the soliton approach [15; 16; 17; 18] by which many nonlinear effects and processes have been described before, including those in ferroelectric liquid crystals [17; 19; 20; 21]. The soliton mode manifests in the new FLC compositions possessing coefficients of rotational viscosity in the range from 0.15 to 1 P, when the electric field frequency or, simultaneously, the frequency and the strength increase. At frequencies corresponding to the transition to the Maxwellian mechanism of energy dissipation, the viscosity \(\gamma_{\psi}\) for the shear strain becomes a dissipative coefficient, and the dynamic solitons move along the smectic layers. The director of the FLC is reoriented by the motion of soliton waves. The appearance of the soliton waves is due to the joint influence of the medium nonlinearity and the presence of a dispersion of velocities of deformation waves (or displacements). They sharply increase the steepness of the wave front and cause the growth of gradients of the field variables. This leads to a spatial redistribution of the excitation energy and its localization (the wave period tends to infinity). The shape of dynamic solitons is uniquely related to the independent parameters--in particular, to the velocity of the soliton center [18]:  
\[V=\frac{\Theta_{0}}{\gamma_{\psi}}\Bigg{(}2K(P_{S}E\cos\varphi_{0}+M)-\left( \frac{2K}{d\Theta_{0}}\right)^{2}\Bigg{)}^{1/2}, \tag{2}\]  
where \(K\) is the FLC elastic modulus describing the deformation of the director with respect to the angle \(\Psi\); \(\gamma_{\psi}\) is the shear viscosity; \(M\) is the bending energy of smectic layers; and \(\varphi_{0}\) is the initial azimuth angle of the director orientation. The transition to the Maxwellian mechanism of energy dissipation and the soliton mechanism of director reorientation is accompanied by a strong frequency dependence of the electro-optical response time \(\tau_{0.1\text{-}0.9}\).