Polarization order parameter

Two of the most important nonlinear optical (NLO) processess, electro-optic switching and second harmonic generation, are second order effects. As such, they occur in materials consisting of noncentrosymmetrically arranged molecular subunits whose polarizability contains a second order dependence on electric fields. Excluding the special cases of noncentrosymmetric but nonpolar crystals, which would be nearly impossible to design from first principles, the rational fabrication of an optimal material would result from the simultaneous maximization of the molecular second order coefficients (first hyperpolarizabilities, p) and the polar order parameters of the assembly of subunits. (1)... 

Fig. 5.2. Definitions of the tilt- and polarization-order parameters. The z axis coincides with the smectic layer normal. The x and y directions are chosen arbitrarily. For the tilt-order parameter the nail notation is used.

The polar order parameter is the first Legendre polynomial P = (Lh) = cos 0 where L is a new polar vector parallel to n and called polar director. The polar order contributes to the elastic surface energy linearly while the quadrupolar (nematic) order contributes quadratically with a sign dependent on surface treatment ... 

Stable ferroelectric nematic phases have been found in high density simulations of DHS [134,135], DSS [101-103] and Stockmayer particles [127,136]. So far the transition densities at fixed X or temperatures (at fixed density) have been estimated only quahtatively by the value at which the polarization order parameter (rotmded by finite size effects) Pi = fi-i d)/N) pa o.5... 

This transition is usually second order [18,19 and 20]. The SmC phase differs from the SmA phase by a tilt of the director with respect to the layers. Thus, an appropriate order parameter contains the polar (0) and azimuthal ((]i) angles of the director ... 

Fig. 3. Order parameter as a function of temperature for -methoxybeiizylidene-/) - -butylariiline (MBBA), a room temperature nematic Hquid crystal. S(T) is determined from tbe polarization of tbe absorption (dicbroism) of small quantities of a dye molecule of similar stmcture (/n / .f-dimetby1aminonitrosti1bene) wbicb bas been dissolved in tbe Hquid crystal bost (1).

In order to develop the dyes for these fields, characteristics of known dyes have been re-examined, and some anthraquinone dyes have been found usable. One example of use is in thermal-transfer recording where the sublimation properties of disperse dyes are appHed. Anthraquinone compounds have also been found to be usehil dichroic dyes for guest-host Hquid crystal displays when the substituents are properly selected to have high order parameters. These dichroic dyes can be used for polarizer films of LCD systems as well. Anthraquinone derivatives that absorb in the near-infrared region have also been discovered, which may be appHcable in semiconductor laser recording. 

The classical scheme for dichroism measurements implies measuring absorbances (optical densities) for light electric vector parallel and perpendicular to the orientation of director of a planarly oriented nematic or smectic sample. This approach requires high quality polarizers and planarly oriented samples. The alternative technique [50, 53] utilizes a comparison of the absorbance in the isotropic phase (Dj) with that of a homeotropically oriented smectic phase (Dh). In this case, the apparent order parameter for each vibrational oscillator of interest S (related to a certain molecular fragment) may be calculated as S = l-(Dh/Di) (l/f), where / is the thermal correction factor. The angles of orientation of vibrational oscillators (0) with respect to the normal to the smectic layers may be determined according to the equation... 

Normal incidence transmission IRLD measurements are used to study thin films (typically 100 pm thickness and less, depending on the molar extinction coefficient of the bands) with in-plane uniaxial orientation. Two spectra are recorded sequentially with the radiation polarized parallel (p) and perpendicular (s) to the principal (machine) direction of the sample. The order parameter of the transition moment of the studied vibration is calculated from either the dichroic ratio (R — Ap/As) or the dichroic difference (AA = Ap—As) as ... 

Ap and As are the absorbances measured with p- and s-polarization, respectively, and A0 — (Ap + 2As)/3 is the structural absorbance spectrum that would be measured for an isotropic sample. The order parameter of the main chain can be determined using the Legendre addition theorem (Equation (24)). 

Like Raman scattering, fluorescence spectroscopy involves a two-photon process so that it can be used to determine the second and the fourth rank order parameters. In this technique, a chromophore, either covalently linked to the polymer chain or a probe incorporated at small concentrations, absorbs incident light and emits fluorescence. If the incident electric field is linearly polarized in the e direction and the fluorescent light is collected through an analyzer in the es direction, the fluorescence intensity is given by... 

Recently, a formalism has been developed to determine the second and the fourth order parameters of films using polarized total internal reflection fluorescence (TIRF) [71]. Similarly to IR-ATR spectroscopy (Section 4), the experiment makes use of p- and s-polarized excitation, but the fluorescence emission (analyzed either in p- or s-direction) is detected normal to the substrate. Two approaches are developed based on the measurements of two intensity ratios. In the first one, the S angle has to be known experimentally or theoretically, and the order parameters (P2) and (P4) can be determined. In the second one, the order parameter (R ) is obtained by another technique, for instance IR-ATR spectroscopy, which allows deducing the order parameter (P4) and (cos2<5). 

To calculate d4/d , we need to evaluate partial derivatives, such as U->4/7) , which measures the rate of change in energy with the order parameter. To do so we need to define generalized coordinates of the form ( , qi, , qN-1). Classical examples are spherical coordinates (r, 6, o), cylindrical coordinates (r, 0, z) or polar coordinates in 2D. Those coordinates are necessary to form a full set that determines... 

In this simple model characterized by a single scalar order parameter, the structures with periodic surfaces are metastable. It simply means that we need a more complex model including the surfactant degrees of freedom (its polar nature) in order to stabilize structures with P, D, and G surfaces. In the Ciach model [120-122] indeed the introduction of additional degrees of freedom stabilizes such structures. 

B. Schartel, Y. Wachtendorf, M. Grell, D.D.C. Bradley, and M. Hennecke, Polarized fluorescence and orientational order parameters of a liquid-crystalline conjugated polymer, Phys. Rev. B, 60 277-283, 1999. 

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