Long range order lamellar structure

Moreau, J.J.E., Vellutini, L., Wong Chi Man, M., Bied, C, Bantignies, J.L, Dieudonnee, P., and Sauvajol, L.J. (2001) Self-organized hybrid silica with long-range ordered lamellar structure. /. Am. Chem. Soc., 123, 7957-7958. 

Well-ordered lamellar structures. The lamellae are arranged in parallel, giving rise to long-range order. Examples are soaps, phosphoUpids and clays. 

The important conclusion to be drawn is that the process has a clear two-state character, i.e. the structures go directly from one into the other, without involving noticeable amounts of irregular intermediary structures once the local concentration of the choline lipid exceeds a critical value, the entire domain transforms from a well-ordered hexagonal to the lamellar state, without loosing long-range order as evident from the relatively constant width of the lamellar (002) reflection which is rather sensitive to lattice distortions. 

These conclusions have been strengthened by an analysis of suitable correlation functions and structure factors [99]. These results show (Fig. 31) that a cylindrical bottle brush is a quasi-lD object and, as expected for any kind of ID system, from basic principles of statistical thermodynamics, statistical fluctuations destroy any kind of long-range order in one dimension [108]. Thus, for instance, in the lamellar structure there cannot be a strict periodicity of local composition along the z-axis, rather there are fluctuations in the size of the A-rich and B-rich domains as one proceeds along the z-axis, these fluctuations are expected to add up in a random fashion. However, in the molecular dynamics simulations of Erukhimovich et al. [99] no attempt could be made to study such effects quantitatively because the backbone contour length L was not very large in comparison with the domain size of an A-rich (or B-rich, respectively) domain. 

So far, we have assumed that the crosslinks pin the smectic layers at a number of points but do not disturb the smectic density wave. However, a sufficient large density of crosslinks might lead to layer distortions that could destroy the quasi-long-range order of ID lamellar lattices [130, 131]. The crosslinks are randomly functionalized into the polymer backbone, and local density variations lead to quenched random disorder. This manifests itself as a mechanical random field that disturbs local layer positions and orientations. The effect of crosslinks on the smectic layer structure can be introduced via a corrugated potential that penalizes deviations of crosslinks from the local layer positions [4,132] ... 

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