Fredericks transition

Table 7. Measurements of the Fredericks transition in the magnetic field 71) (H = critical field, r = relaxation time, kn = splay elastic constant, yl = twist viscosity coefficient...

Such an antisymmetric distortion differs from the symmetric distortion characteristic of the Frederiks transition. It is instructive to compare these two cases. In Fig. 11.28 the space distributions of the director n and its x-projection rix = sinfl 9 are pictured for the Fredericks transition (a) and flexoelectric effect (b) the anchoring energy at both surfaces is infinitely strong in case (a) and finite in case (b). 

In both the cases considered, an optical contrast of the patterns observed in isotropic liquids is very small. Certainly, the anisotropy of Uquid crystals brings new features in. For instance, the anisotropy of (helectric or diamagnetic susceptibility causes the Fredericks transition in nematics and wave like instabilities in cholesterics (see next Section), and the flexoelectric polarizaticm results in the field-controllable domain patterns. In turn, the anisotropy of electric conductivity is responsible for instability in the form of rolls to be discussed below. All these instabilities are not observed in the isotropic liquids and have an electric field threshold controlled by the corresponding parameters of anisotropy. In addition, due to the optical anisotropy, the contrast of the patterns that are driven by isotropic mechanisms , i.e. only indirectly dependent on anisotropy parameters, increases dramatically. Thanks to this, one can easily study specific features and mechanisms of different instability modes, both isotropic and anisotropic. The characteristic pattern formation is a special branch of physics dealing with a nonlinear response of dissipative media to external fields, and liquid crystals are suitable model objects for investigation of the relevant phenomena [39]. 

The sketch of the experimental set-up is shown in Figure 1. A Q-switched Nd-YAG laser, operating at 1.06 ixm and a pulse repetition 2-12.5 Hz was used to provide the fundamental (pump) beam. The peak power was 200-300 kW. The beam was focused with a 43 cm lens so that the power density on the sample placed in a thermostate was about 100-200 MW-cm. " For investigation the field-induced SHG, short pulses (tp = 20 fxs) of high voltage Up = 4kV) were provided by an electrical generator. The pulse duration was chosen from the condition Trelaxation time for dipolar (Debye) polarization, and T is the director reorientation time. Under such a condition, molecular dipoles are oriented by the field but the Fredericks transition does not take place. The sensitivity of our set-up was about 30 photons of the optical second harmonic per single laser pulse. The cell temperature was stabilized with an accuracy of 0.1° K. 

In our view, all of the results of this experiment can be explained by the reorientation of the director in the electric field of the light wave, in analogy with the Fredericks transition. Fredericks transitions are observed in uniform, constant magnetic and electric fields. Our case is more complex, but the basic features of the Fredericks effect can be clearly seen. 

The reorienution of the director of a nematic liquid crystal induced by the field of a light wave is considered. An oblique (with respect to the director) extraordinary wave of low intensity yields the predicted and previously observed giant optical nonlinearity in a nematic liquid crystal. For normal incidence of the light wave on the cuvette with a homeotropic orientation of the nematic liquid crystal, the reorientation appears only at light intensities above a certain threshold, and the process itself is similar to the Fredericks transition. The spatial distribution of the director direction is calculated for intensities above and below threshold. Hysteresis of the Fredericks transition in a light field, which has no analog in the case of static fields, is predicted. 

In contrast to the simplest model of the Fredericks transition in static fields (see Refs. 12 and 13), here we take into account the following two facts 1) the amplitude for the deformation of the director above threshold must be found taking into account the distortion of the longitudinal profile of the light wave itself as it propagates in an inhomogeneous anisotropic medium ... 

Let us first give a qualitative picture of the Fredericks transition. When a plane light wave falls at normal incidence on a homeotropic cell the electric field of the wave is exactly perpendicular to the director. At a positive value of it would be energetically favorable to orient the director in the direction of the field. However, this is prevented by the homeotropic orientation of the director by the curvette walls. Furthermore, in the first approximation in the light intensity the orientational effect of the field on the unperturbed director is absent, in other words, for E=e,E,j the function exact solution to equation (5). 

Equation (6) is analogous to the linearized equation for the behavior of the director near the Fredericks transition in a static field E,, .. (cf. Ref. 13,... 

In the case where the transverse dimension of the beam is smaller than or on the order of the cuvette thickness X, the Frank energy due to the transverse gradients of the director becomes dominant. Let us first estimate the order of magnitude of the threshold power of the Fredericks transition. The energy of the perturbed state is... 

In order to exactly determine the threshold of the Fredericks transition, it is necessary to solve the three-dimensional problem of the stability of the solution ( =0. We shall consider this problem for several particular cases. 

In the case of more than a single constant the threshold of the Fredericks transition can be determined analytically for a ribbon beam of the form P t ) =P(y) polarized along the xaxis. The equation linearized in (p r) in this case has the form... 

To study the Fredericks transition for B 0 we integrate Eq. (10) with respect to z after multiplying it by 2d(p/dz. After determining the integration constant in terms of the maximum angle of deviation of the director from the unperturbed direction (p we obtain... 

Assuming that the power density P of the radition falling on a cell of the NLC is close to the threshold value for the Fredericks transition, that is, (p 1, let us compute the integral in (33) up to terms The solution of the resulting biquadratic equation for cp has the form... 

FIG. 4. Ifysteresis of the Fredericks transition. The arrows indicate the direction of variation of the power of the light field P. 

The behavior of the cell in the oblique-incidence case discussed here is analogous to the case of a cell in an oblique magnetic field-in both cases there is no rigorous Fredericks transition (cf. Ref. 13, 4.2.3). 

This value is 3.1 times larger than that for the Fredericks transition in a cuvette of thickness L = 150 pm, which is also in very good agreement with the results of Ref. 11. 

As was shown in Sec. 5, for certain types of liquid crystals the Fredericks transition can be accompanied by hysteresis. Let us calculate for the case of the well studied liquid crystal PAA the main characteristics of this phenomenon. As seen from formula (37) with B = -0.03 and G =0.06, the powers at which the Fredericks effect in PAA is switched on and off differ little Pn 0.992P,.. The angle corresponding to the appearance of a local minimum of the free-energy function [that is, (p corresponding to the inflection point of the function is [Pg.123]

Note added in proof. We have been kindly informed by the authors of a paper submitted to this journal that they have also studied the theory of light-induced Fredericks transitions (A. S. Zolot ko, V. F. Kitaeva,... 

In the paper we described a simple theoretical model of the Fredericks transition induced by optical fields. This model predicts the threshold intensity as a function of the spot size of the laser beam. The experimental data fit well the theoretical curve. From the fitting parameter, Po, and effective elastic con-... 

In conclusion, the theoretical model explains the main features of the experimental observations and provides a reliable numerical value for the threshold. This proves the underlying idea, i.e. that the observed phenomena is due to a Fredericks transition in optical field, which can be described by the continuum theory of nematics, taking into account the finite size of the deforming light beam. 

FIGURE 3.18. Local Fredericks transition. The sample (1) mica cleavage (2) a stair of Langmuir-Blodgett bilayers (3) a drop of MBBA. Photos are taken for various thicknesses of a Langmuir-Blodgett film (a) 5 = 0 (no film), (b) 100 A (2 bilayers), (c) 150 A (3 bilayers). 

This equation has already been encountered when discussing the Fredericks transition in nematics. Its solution may be expressed in terms of the elliptic function... 

During demonstrations with our 16 pixel display, it is interesting to show that its optical action is reversed when the polarizers are parallel instead of being crossed (Figure 2.5). Another interesting feature of the display is its relaxation time which is of the order of 15 s. Such a long relaxation time (much longer than in real TN displays) is due to the thickness of the cell. Indeed, the relaxation time in the Fredericks transition varies as so that when the thickness of the cell is 20 pm instead of 200 pm, the relaxation time becomes 0.15 s instead of 15 s. 

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