Paracrystalline stack

Figure 8.21. Features of a ID correlation function, yi (x/L) for perfect and disordered topologies. L is the number-average distance of the domains from each other (i.e., long period). Dotted Perfect lattice. Dashed and solid lines Paracrystalline stacks with increasing disorder. a = — v/j / (1 — v/j) with 0 < v/j < 0.5 is a measure of the linear volume crystallinity in the material, which is either or 1 —...

If the statistical model of a paracrystalline stack is assumed, it turns out that the renormalization attenuates the influence of polydispersity on the position of the first zero. In general, the first-zero method is more reliable than the valley-depth method, although it is not perfect. Even the first-zero method is overestimating the value of V . The deviation is smaller than 0.05, if the found crystallinity is smaller than 0.35. If bigger crystallinities are found, the significance of the determination is... 

Figure 8.22. Testing the first-zero method for the determination of the linear crystallinity, V[, from the linear correlation function, yi (x/Lapp) with Lapp being the position of the first maximum in yi (x) (not shown here - but cf. Fig. 8.21). Model tested Paracrystalline stacking statistics with Gaussian thickness distributions. The interval of forbidden zeroes is shown. An additional horizontal non-linear axis permits to determine the linear crystallinity directly. A corresponding vertical axis shows the variation of the classical valley-depth method ... 

If other statistical models of polydispersity should prove more appropriate than the paracrystalline stack, validations of the first-zero method may be carried out in analogy to the one presented here. 

Modeling the Interface Distribution Function for a ID Lamellar Stack As demonstrated in the last section, the nonideality of a real semicrystalline polymer can lead to a broadening and overlapping of the peaks in K (z), which makes it difficult to extract the correct structure parameters simply from the peak positions. The one-dimensional paracrystalline stack has been suggested as an analytical model for the semicrystalline structure [2,13,16], We here present a procedure that allows simulating and modeling the measured IDF based on this model. A simulated IDF... 

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)... 

Figure 34 Average values of the crystal (circles) and amorphous layer (squares) thickness, (fj and (i,), as a function of dwelling time at 205 C (left) and temperature (right). Filled and empty squares correspond to the paracrystalline model fits with a low (A/=3) and high N=20) number of crystals per stack, respectively. With permission from Ivanov, D. A. ...

One of the authors of this article has introduced the paracrystalline concept of above " and also the Q-function method into the evaluation of scattering diagrams of biomembrane stacks. The paracrystalline and Q nction approaches were continuously pursued and completed in the following years " ... 

Mainly two models for biomembrane stacks were defined in the following by paracrystalline concepts, characterized by specific assumptions of the model the vesicle concept and the membrane concept. 

In all paracrystalline studies, with the exception of the present one, the electron denaties of the intermembranous spaces are assumed to be equal to the mean electron density of the biomembrane stacks. 

Q-functions of many biomembranes converge to zero for z > 2 ( = Bragg period). In contrast, electron microscopy of the same stacks usually fliows a much larger number of vesicles. As was mentioned in Chap. 3, the paracrystalline plane-layer lattice model cannot account for this fast convergence of the Qflmction to zero alone. The more general formulas (4.16) should be able to overcome this difficulty by their new concepts ... 

The sample is contaminated by other paracrystalline or crystalline stacks (L d-Crystals for instance, which arise during preparation or measurement). 

The stacking is subject to large paracrystalline distance fluctuations, therefore the stacking structure factor deviates significantly from 1 only for very small values of iQ < 0.025 A ). A detailed account on the structural properties is given in ref. [12]. 

In the Hosemann model, which is based on a paracrystalline model, each crystalline region consists of N alternating stacks of crystalline and amorphous layers. The scattered intensity I(s) calculated from this model as a function of s is given by... 

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