Layer periodicity

Detailed x-ray diffraction studies on polar liquid crystals have demonstrated tire existence of multiple smectic A and smectic C phases [M, 15 and 16]. The first evidence for a smectic A-smectic A phase transition was provided by tire optical microscopy observations of Sigaud etal [17] on binary mixtures of two smectogens. Different stmctures exist due to tire competing effects of dipolar interactions (which can lead to alternating head-tail or interdigitated stmctures) and steric effects (which lead to a layer period equal to tire molecular lengtli). These... 

Undoubtedly the most successful model of the nematic-smectic A phase transition is the Landau-de Gennes model [201. It is applied in the case of a second-order phase transition by combining a Landau expansion for the free energy in tenns of an order parameter for smectic layering with the elastic energy of the nematic phase [20]. It is first convenient to introduce an order parameter for the smectic stmcture, which allows both for the layer periodicity (at the first hannonic level, cf equation (C2.2A)) and the fluctuations of layer position ur [20] ... 

The smectic phases Ai, A2 and A have the same macroscopic symmetry, differing from each other in the wavelength of spacing. Hence it is possible to go from Ai to Aa or from Aa to A2 by varying only the layer periodicity in a continuous or discontinuous way(with the jump in the layer spacing d). Smectic-smectic transition lines of first order may terminate at a critical point, where the differences between the periodicities of the smectic A phases vanish, providing a continuous evolution from an Aa to bilayer A2 phase [12]. 

Samples Process and Layers Periodic Number Film Thickness (A)... 

The devolatilization of a component in an internal mixer can be described by a model based on the penetration theory [27,28]. The main characteristic of this model is the separation of the bulk of material into two parts A layer periodically wiped onto the wall of the mixing chamber, and a pool of material rotating in front of the rotor flights, as shown in Figure 29.15. This flow pattern results in a constant exposure time of the interface between the material and the vapor phase in the void space of the internal mixer. Devolatilization occurs according to two different mechanisms Molecular diffusion between the fluid elements in the surface layer of the wall film and the pool, and mass transport between the rubber phase and the vapor phase due to evaporation of the volatile component. As the diffusion rate of a liquid or a gas in a polymeric matrix is rather low, the main contribution to devolatilization is based on the mass transport between the surface layer of the polymeric material and the vapor phase. 

A section in the full-bed models was isolated that was comparable to the WS model. The layout of these different sections was identical, except that the WS model had a two-layer periodicity and the full-bed models had a six-layer periodicity. To be able to make direct comparisons of velocity profiles, several sample-points needed to be defined. In the three different models seven tangential planes were defined and on each plane three axial positions were defined. This reduced the data to single radial velocity profiles at corresponding positions in all three models, as shown in Fig. 10, for the WS model. Identical planes were defined in the full-bed models. Some spheres and sample planes 4 and 5 are not displayed to improve the visibility of the sample planes and lines. In the right-hand part of the figure, plane 4 is shown with the axial positions at which data were taken and compared. 

The data from the WS model in some cases deviated slightly from the full-bed models. This could be explained by the slightly different layout of the WS model. Some spheres had to be relocated in the WS model to create a two-layer periodicity from the six-layer periodicity in the full-bed models. The differences in velocity magnitudes were mainly found in the transition area between the wall layers and the center layers. The effect of slightly larger gaps between spheres from the nine-sphere wall layers and the three-sphere central layers, due to the sphere relocations, had a noticeable effect on the velocity profile. Differences were also found in the central layer area where the sphere positions were not identical. 

Bragg mirrors on periodic stacks of layers Periodic stacks of metal nanoparticles or dielectric layers with alternating high and low refractive index produce a desired reflectance of the mirror that depends on the thickness and the refractive index of the layers in the stack 16,17... 

A standard method for confirming coherence of the layers is the study of x-ray diffraction spectra. If the layers are coherent and there are enough of them to provide a relatively strong Bragg diffraction pattern, satelhtes due to superlattice (see Chapter 16) formation should appear on each side of the Bragg diffraction peak. Although detailed treatments can be found in the literature, we present below a simplified but rather useful formula for the determination of layer periodicity. 

Cr Cub, Cubv d E G HT Iso Isore l LamN LaniSm/col Lamsm/dis LC LT M N/N Rp Rh Rsi SmA Crystalline solid Spheroidic (micellar) cubic phase Bicontinuous cubic phase Layer periodicity Crystalline E phase Glassy state High temperature phase Isotropic liquid Re-entrant isotropic phase Molecular length Laminated nematic phase Correlated laminated smectic phase Non-correlated laminated smectic phase Liquid crystal/Liquid crystalline Low temperature phase Unknown mesophase Nematic phase/Chiral nematic Phase Perfluoroalkyl chain Alkyl chain Carbosilane chain Smectic A phase (nontilted smectic phase)... 

So far we have discussed 2D density modulated phases that are formed by deformation or breaking of the layers. However, there are also 2D phases with more subtle electron density modulations. In some cases additional peaks observed in the XRD pattern (Fig. 10) are related to a double layer periodicity in the structure. As double layer periodicity was observed in the bent-core liquid crystals formed by the asymmetric as well as symmetric molecules [22-25] it should be assumed that the mechanism leading to bilayers must be different from that of the pairing of longitudinal dipole moments of molecules from the neighboring layers, which is valid for smectic antiphases made by asymmetric rod-like molecules. 

We have also shown the existence of 2D phases due to more subtle electron density changes. In some cases additional peaks are observed in the XRD pattern, signifying a double layer periodicity in the system, which can be accounted for if a general orientation of the polar director is allowed. If the polar director is not perpendicular to the tilt plane there exists a component of polarization in the direction of the smectic layer normal (longitudinal polarization). By double layer periodicity the system escapes from the polar structure and in addition achieves better packing of the molecular cores and molecular tails. 

Clintonite. Clintonite is the trioctahedral brittle mica with ideal composition of Ca(Mg2Al)(SiAl3)Oio(OH)2. This structure violates the Al-avoidance principle of Loew-enstein (1954). It crystallizes in H20-saturated Ca-, Al-rich, Si-poor systems under wide P-T conditions. Clintonite, usually found in metasomatic aureoles of carbonate rocks, is rare in nature because crystallization is limited to environments characterized by both alumina-rich and silica-poor bulk-rock chemistry and very low CO2 and K activities (Bucher-Nurminen 1976 Olesch and Seifert 1976 Kato et al. 1997 Grew et al. 1999). The IM polytype and IMj sequences are the most common forms. The 2Mi form is rare (Akhundov et al. 1961) and no 3T forms have been reported. Many IM crystals are twinned by 120° rotation about the normal to the 001 cleavage. Such twinning causes extra spots on precession photographs that simulate an apparent three-layer periodicity (MacKinney et al. 1988). 

Correct analytical expressions are obtained for reflection and transmission coefficients of a finite three-layered periodic structure. An enhancement of light energy localization inside the structure simultaneously at all transmission resonance frequencies, is shown. 

Well investigated stratified periodical structures (SPSs) are widely used as dielectric mirrors, polarization devices for integrated optics, tunable narrow band filters, time delay devices, nonreciprocal elements, devices of parametric and non-linear optics [1], The two-layer SPSs are a particular case of more general class of three-layer structures. For example, the three-layered structures (ABB) A with layers A and B and number of periods N can be considered as a two-layer SPS. In the paper we present features of optical wave transformation in three-layer periodical structures. 

Consider a normal light transmission through a finite isotropic three-layer periodical structure, made of identical layers indicated by numbers 1, 2 and 3 with a thickness c/, and and refraction indexes , and, respectively. Let this SPS with N periods to be surrounded by isotropic dielectric media with refraction indexes and (Fig. 1). 

Figure 1. A three-layer periodical dielectric structure.

In conclusion, the correct analytical expressions have been derived for transmission and reflection coefficients of a three-layer periodic structure surrounded by two different isotropic dielectric media. On the basis of the given formulas it was shown that for three-layer SPS it is possible to realize the enhancement of resonant interaction of light with the structure. By this way one can achieve an increased light energy localization inside the structure and decreased group velocity of light inside SPS. It is necessary to point out that the enhancement of these phenomena is put into practice by using quasi-periodic structures. The possibility to enhance the resonant phenomena by the use of finite periodic structure, optical properties of which are predictable, easily calculated and practically realizable, opens wide perspectives for their application. 

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