Paracrystalline lattice

On the other hand, lattice distortions of the second kind are considered. Assuming [127] that ID paracrystalline lattice distortions are described by a Gaussian normal distribution go (standard deviation ay, its Fourier transform Gd (.S ) = exp (—2n2ols2) describes the line broadening in reciprocal space. Utilizing the analytical mathematical relation for the scattering intensity of a ID paracrys-tal (cf. Sect. 8.7.3 and [127,128]), a relation for the integral breadth as a function of the peak position s can be derived [127,129]... 

Figure 8.42. ID structural models with inherent loss of long-range order, (a) Paracrystalline lattice after HOSEMANN. The lattice constants (white rods) are decorated by centered placement of crystalline domains (black rods), (b) Lattice model with left-justified decoration, (c) Stacking model with formal equivalence of both phases (no decoration principle)... 

Analytical Expressions for Lattice Models. Concerning the aforementioned paracrystalline lattice, an analytical equation has first been deduced by Hermans [128], His equation is valid for infinite extension. Ruland [84] has generalized the result for several cases of finite structural entities. He shows that a master equation... 

The Cluster on Network Model. Previous small angle x-ray (]L-J>) and neutron (4) scattering experiments clearly indicate that ionic clustering is present in "Nation". However, details of the arrangement of matter in these clusters cannot be obtained from these results alone. For hydrocarbon ionomers several different interpretations have been advanced as the cause of the SAXS maximum. These include a model of spherical clusters on a paracrystalline lattice proposed by Cooper et al. (16), the shell core model of Macknight et al. (17) and more recently a lamellar model (18). At present there is no consensus about which of these models best describes clustering in hydrocarbon ionomers. 

This consideration led to hypothesize that the microcrystals of the alternating EN copolymers could be characterized by a similar face-centered mode of packing with ball-like norbornene units arranged in a face-centered paracrystalline lattice, at average distance 6.57 A. The polymer chains in the paracrystalline aggregates could be organized in fringed micelle bundles [121]. 

Interference function scattering was modelled by using a paracrystalline lattice model as a basis, with the adoption of the equations for X-ray scattering from low molecular weight crystalline materials. Thus the Bragg peaks essentially result from a powder diffraction pattern from the block copolymers. Reasonable agreement... 

From our viewpoint, there are three most interesting aspect of those materials to study. First, the particular features of crystallization of these compounds, generally random copolymers, are of quite general interest. Two recent and essentially different models for crystallization of such irregular polymeric materials, that is, the model of nonperiodic layers (1) and the model of the paracrystalline lattice (2), are still under discussion. 

Fig. 3-2. Paracrystalline lattice factor for a fee lattice. Curve f g = 0.05, curve 2 0.07, curve 3 0.09, curve 4 0.11, curve 5 0.13, curve 6 0.15. Taken from [45] with the permission of the American Physical Society...

The first term of Eq.(1), a diffuse back-ground scattering is generally much smaller and less angularly dependent than the second term, and S (s ), the shape amplitude related to the size and shape of the assembly of the particles varies with scattering angle s much more rapidly than Z( ), the paracrystalline lattice factor, so that... 

Figure 2.20. Layout chart of molecules in basis planes of a ciystalline cell (a) and the model paracrystalline lattice (b) of a liquid OTMG.

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