Lamellar structure/thickness

Dye structures of passive tracers placed in time-periodic chaotic flows evolve in an iterative fashion an entire structure is mapped into a new structure with persistent large-scale features, but finer and finer scale features are revealed at each period of the flow. After a few periods, strategically placed blobs of passive tracer reveal patterns that serve as templates for subsequent stretching and folding. Repeated action by the flow generates a lamellar structure consisting of stretched and folded striations, with thicknesses s(r), characterized by a probability density function, f(s,t), whose... 

Nylon crystallites consist of sheets of chains that are hydrogen-bonded to their neighbors. On a supermolecular scale, crystallites have a lamellar structure, that is they are many times longer and broader than they are thick. When nylon crystallizes from an isotropic molten state, it generally forms spherulites, which consist of ribbon-like lamellae radiating in all directions... 

Lamellar structures. In these structures, octahedra form layers which are electrically neutral or charged and cohesion of the layers is through van der Waals interaction, hydrogen bonding or ionic bonding. Two cases must be considered in this category of oxides. When the octahedra share only their corners, the layers are sufficiently thick to avoid distortion. When thin layers are formed, the octahedra share both edges and corners. 

Fig. 16a-c. Schematic model of the lamellar structure of the copolymer in the, a. high temperature range (paraelectric phase) b. Curie transition region and c. low temperature region L and I denote respectively the long period and the average crystal thickness comprising a mixture of non ferroelectric and ferroelectric domains... 

Lattice Model Carlo simulations of a block copolymer confined between parallel hard walls by Kikuchi and Binder (1993,1994) revealed a complex interplay between film thickness and lamellar period. In the case of commensurate length-scales (f an integral multiple of d), parallel ordering of lamellae was observed. On the other hand, tilted or deformed lamellar structures, or even coexistence of lamellae in different orientations, were found in the case of large incommensurability. Even at temperatures above the bulk ODT, weak order was observed parallel to the surface and the transition from surface-induced order to bulk ordering was found to be gradual. The latter observations are in agreement with the experimental work of Russell and co-workers (Anastasiadis et al. 1989 Menelle et al. 1992) and Foster et al. (1992). 

Liquid crystals stabilize in several ways. The lamellar structure leads to a strong reduction of the van der Waals forces during the coalescence step. The mathematical treatment of this problem is fairly complex (28). A diagram of the van der Waals potential (Fig. 15) illustrates the phenomenon (29). Without the liquid crystalline phase, coalescence takes place over a thin liquid film in a distance range, where the slope of the van der Waals potential is steep, ie, there is a large van der Waals force. With the liquid crystal present, coalescence takes place over a thick film and the slope of the van der Waals potential is small. In addition, the liquid crystal is highly viscous, and two droplets separated by a viscous film of liquid crystal with only a small compressive force exhibit stability against coalescence. Finally, the network of liquid crystalline leaflets (30) hinders the free mobility of the emulsion droplets. 

The three fundamental lyotropic liquid crystal structures are depicted in Figure 1. The lamellar structure with bimolecular lipid layers separated by water layers (Figure 1, center) is a relevant model for many biological interfaces. Despite the disorder in the polar region and in the hydrocarbon chain layers, which spectroscopy reveals are close to the liquid states, there is a perfect repetition in the direction perpendicular to the layers. Because of this one-dimensional periodicity, the thicknesses of the lipid and water layers and the cross-section area per lipid molecule can be derived directly from x-ray diffraction data. 

Provided the crystalline stem length t,c is known and a stacked lamellar structure is assumed, the thicknesses of the interphase and amorphous phase can be evaluated from these data by the Eqs. (6) and (7) ... 

A few words of explanation are not useless in order to understand this formalism. As a consequence of mixing, the medium is assumed to have a lamellar structure and n is a unit vector which remains normal to the material slices undergoing deformations in the velocity field, n n denotes a dyadic product (the dyadic product of vectors a and b is the tensor a.jbj) and 13 n n denotes the scalar product of the two tensors (the scalar product of tensors i = Tij and W = is the scalar quantity T W = E Z T j wji)- Assume that we start with two miscible fluids A J and B (having for instance different colors). Upon mixing, we obtain a lamellar marbled structure characterized by a striation thickness 6 and a specific "interfacial" area av. If the fluid is incompressible, avS = 1. Then,by application of (7-1)... 

Mixing by molecular diffusion. This is the ultimate and finally the only process really able to mix the components of a fluid to the molecular scale. The time constant for this process is the diffusion time t = yLy is a shape factor and L is the ratio of the volume to the external surface area of the particle. For instance let us consider various shapes slabs (thickness 2R, case of lamellar structure with striation thickness 6 = 2R), long cylinders (diameter 2R, case of filamentous structure), and spheres (diameter 2R, case of spherical aggregates). 

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