Spatially modulated structures

Selke, W. (1992). Spatially modulated structures in systems with compeing interactions. In Phase transtitions and critical phenomena, (ed. C. Domb and J. L. Lebowitz), p. 1. Academic, London. 

It is interesting that the relaxation of the field-induced, i.e., untwisted, state under these conditions is accompanied by the formation of spatially modulated structures in the form of strips or grids. 

In this book the authors present a complete and readily understood treatment of virtually all known phenomena occurring in liquid crystals under the influence of an electric field. In the first three chapters (Chapters 1-3) bulk and surface properties of liquid crystalline materials are discussed. The next two chapters (4, 5) are devoted to consideration of the electrooptical effects due to the formation of uniform and spatially modulated structures in nematics. In Chapters 6 and 7 the electrooptical properties of the cholesteric and smectic mesophases are presented, including a discussion of ferroelectric materials. Major emphasis is given to explaining the qualitative aspects of the phenomena and to portraying their physical basis. The prospects for the practical application of electrooptical effects are also discussed (Chapter 8). 

For a second-order transition, a, b, and c are positive constants and Tc is the phase transition temperature. The term is necessary for a correct description of the temperature-dependence of the tilt angle and heat capacity [10]. If a spatially modulated structure is to be described, it is convenient to use a two-component tilt vector y), corresponding to the projection of ft... 

Field untwisting of a helix for thin cells with planar boundary conditions occurs differently from the case of an infinite cholesteric medium. For a field perpendicular to the axis of a cholesteric helix in the planar texture, a stepwise change in the pitch with increase in the field is predicted [128]. The size of the step increases with a decrease in the ratio 2electric field perpendicular to the helix axis has been observed for the case of planar cholesteric texture with a>0 and with strong anchoring of the molecules to the limiting surfaces [129]. Under these conditions the relaxation of the field-induced (i.e. untwisted) state is accompanied by the formation of spatially modulated structures in the form of strips or grids. 

Lincot, Froment, and Cachet review the chemical and mechanistic aspects of chemical bath deposition of chalcogenide compounds with special emphasis on structural properties associated with epitaxial growth. Present applications of chemically deposited films are reviewed and several characteristic advantages are identified that may be exploited in the future for applications such as small band gap semiconductors, large area electrochromic devices, electroluminescence, quantum sized films, and films with spatially modulated composition and structure. 

It is very peculiar that the spatial distribution of the director field of the modulated structure forms a chiral structure. This became evident much later [36] when the numerical calculations had been made in the same geometry with director components n K., rPy = co qy, = sinf/y. Thus the projections of the director on the zy plane, i.e. = Uy, rotate about the x-axis upon translation along the y-axis. The corresponding picture is demonstrated in Fig. 11.31. The calculations show that the chirality changes its handedness when the sign of the electric field applied in the z-direction inverses. Therefore, we again see the field induced break of the mirror symmetry. 

Due to low symmetry (C2) of the chiral smectic C phase, its theoretical description is very complicated. Even description of the achiral smectic C phase is not at all simple. In the chiral SmC phase two new aspects are very important, the spatially modulated (helical) structure and the presence of spontaneous polarisation. The strict theory of the SmA -SmC transition developed by Pikin [10] is based on consideration of the two-component order parameter, represented by the c-director whose projections ( 1, 2) = are combinations of the director compo-... 

Photoetching has been also applied to fabricate microstructures on the surface of semiconductor. Because the photoetching reaction occurs selectively at the irradiated part of the surface, one can etch the semiconductor to form three-dimensional structures by spatial modulation of the... 

The physical origin of these phenomena could be explained by faster response times of the modulated structures, as compared with uniform ones. When the equilibrium director distribution is approached, i.e., a relaxation process is over, the transient structures disappear. The emergence and subsequent evolution of the spatial periodicity of the transient structures were considered theoretically in [26] for different domain orientations with respect to the initial homogeneous and twisted director structure. 

References [59] investigate the static and dynamic behavior of the electrohydrodynamic instability in fireely suspended layers of nematic liquid crystals. The existence of a domain mode was shown, which consists of adjacent elongated domains with a spatial period proportional to the thickness of the layer. This mode occurs only if the thickness of the layer exceeds a critical value 7 /x), and can be understood in terms of the same anisotropic mechanism as the Carr-Helfrich-type, as in the case of the Kapustin-WiUiams modulated structure. 

To develop a surface relief structure, the UV-curable polymer was spin coated on the substrates at a rate of 4000 rpm for 30 sec. The resultant thickness of the spin coated polymer film was about 3 pm. After a polymer film formed on the glass substrate, spatially modulated UV light was irradiated onto the film through a photomask. The photomask was designed to generate the intensity modulation of the UV light to form a pattern of microlens arrays. The irradiation led to the diffusion of monomers in the polymer composites from the unexposed to exposed regions. The result was the formation of a lens-shaped surface relief structure. 

The kink d5mamics of the sine-Gordon equation is studied in the model of the loealized spatial modulation of the periodic potential. A case of two identical areas (or impurities) of the spatial modulation of the periodic potential is considered. It is shown that observing the collective effects of impurity influence is possible and depends on the distance between the impurities. A definite critical value of impurity distances causing two quite different ways of the d5mamic kink behavior is demonstrated. The structure and properties of three-kink solutions of the sine-Gordon equation in the impurity area are studied. 

---