# Helix-Free Ferroelectric Liquid Crystals: Electro Optics and Possible Applications
## 5 Theoretical Model of the Ferroelectric Liquid Crystal Deformation and Director Reorientation  
### Deformation of Ferroelectric Liquid Crystal Smectic Layers  
The bulk density of free energy associated with the periodic deformation of smectic layers can be written as [16; 18; 22]  
Figure 5: Frequency dependences (**a**) of the optical response time at \(\pm\)1.5 V (curve 1) and \(\pm\)12 V (curve 2) pulses acting and (**b**) of the director relaxation time after \(\pm\)1.5 V pulses turning off. The FLC composition is HF32E (\(\gamma_{\varphi}\) = 1.5 P). A bipolar voltage of rectangular shape (meander) was applied. The thickness of the electro-optical cell is 1.7 \(\upmu\)m.  
Figure 6: Frequency dependencies of the electro-optical response time for the HF32D composition (\(\gamma_{\varphi}\) = 0.15 Poise). 1—amplitude of the bipolar voltage (meander) \(\pm\)1.5 V; 2—amplitude \(\pm\)9 V. The thickness of the electro-optical cell is 1.7 \(\upmu\)m.  
\[F=\frac{1}{2}K\bigg{(}\frac{d\psi}{dy}\bigg{)}^{2}+\frac{1}{2}M\bigg{(}\frac{l-l_{0} }{l_{0}}\bigg{)}^{2}. \tag{3}\]  
Here, \(K\) is the elasticity coefficient describing the director's deformation over the angle \(\psi\) (Figure 3) and \(M\) is the bending energy of smectic layers. We also take into account the smallness of the angles \(\psi\) and \(\Theta_{0}\) (\(\psi\), \(\Theta_{0}<<1\)), and the relative change of the smectic layer thickness (\(l-l_{0}\))/\(l_{0}\cong(\psi^{2}-{\Theta_{0}}^{2})/2\)[23].  
If to minimize Equation (3) we can find the distribution of the angle \(\psi\) along the coordinate \(\mathbf{y}\) (\(0\leq y\leq d\), where \(d\) is the thickness of the electro-optical cell), then  
\[\frac{d^{2}\psi}{dy^{\prime 2}}+\frac{M\[PERSON]_{0}^{2}d^{2}}{2K}\psi-\frac{ Md^{2}}{2K}\psi^{3}=0, \tag{4}\]  
where \(y^{\prime}=y/d\) and \(0\leq y^{\prime}\leq 1\).  
The exact solution of Equation (4) is written in terms of the Jacobi elliptic sine:  
\[\psi=\sqrt{2}\[PERSON]_{0}\frac{k}{\sqrt{1+k^{2}}}sn\bigg{(}\[PERSON]_{0}d\sqrt{ \frac{M}{2K}\big{(}\frac{y^{\prime}+C_{1}}{\sqrt{1+k^{2}}}},k\bigg{)} \tag{5}\]  
where \(k=\sqrt{\frac{M\Theta_{0}^{4}d^{2}}{8 KC_{2}}}-\sqrt{\frac{M\Theta_{0}^{4}d^{2 }}{8 KC_{2}}-1}\), \(0<C_{2}<\frac{M\Theta_{0}^{4}d^{2}}{8K}\), and \(0<k<1\) is the modulus of Jacobi elliptic sine.  
Using the [PERSON] method [24] and substituting the boundary conditions \(\psi(y=0)=0\) and \(d\psi/dy=0\) at \(\psi=\Theta_{0}\), we obtain an approximate solution to Equation (4):  
\[\psi=\Theta_{0}\sin\left(\frac{5}{8}\Theta_{0}\sqrt{\frac{M}{2K}y}\right). \tag{6}\]  
This analytical solution given by Equation (6) describes the structure of smectic layers deformed periodically in the direction \(\mathbf{y}\). Maxima and minima in the dependence \(\psi(y)\) correspond to light and dark stripes in Figure 2a for a cell with FLC layer thickness of 5 \(\upmu\)m, \(\Theta_{0}=23^{\circ}\), \(M=4\times 10^{3}\) erg/cm\({}^{3}\), and \(K=5\times 10^{-12}\) N.  
Reorientation of the FLC director due to the interaction of the alternating electric field \(\mathbf{E}\) applied along the coordinate \(\mathbf{y}\) (Figure 2b) with the spontaneous polarization \(\mathbf{P}_{S}\) can occur both by changing the azimuth angle \(\varphi\) of the director orientation by 180\({}^{\circ}\) (if the director is reoriented along the generating lines of a cone with \(\backslash\) apex angle 2\(\Theta_{0}\)) and by changing the distribution of \(\psi\) (deformation of smectic layers). In the first case, the dissipative coefficient is the rotational viscosity \(\gamma_{\phi}\), and in the second one, the viscosity \(\gamma_{\psi}\) of the shear deformation predominates [16; 18].