Hosemann model

The details of the lamellar morphology, such as crystalline layer thickness l and amorphous layer thickness /, are quantitatively evaluated using SAXS [18]. For example, they are conveniently derived from the one-dimensional correlation function, that is, Fourier transform of SAXS curves, assuming an ideal lamellar morphology without any distribution for and l [19]. More detailed information on the lamellar morphology can be obtained by fitting theoretical scattering curves (or theoretical one-dimensional correlation functions) calculated from some appropriate model to SAXS curves (or Fourier transform of SAXS curves) experimentally obtained. The Hosemann model in reciprocal space [20] and the Vonk model in real space [7,21] are often employed for such purposes. 

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... 

Because relatively few experimental SANS data are available for IPNs, presently it is difficult to draw any definite conclusions about the structure of IPNs. As is seen from the data considered, the mechanism of phase separation is not mentioned in any work cited above. Meanwhile, this mechanism should determine if the application of any theory is possible for a given system. One may suggest that the Porod and Hosemann models may be used only for the nucleation and growth mechanism of phase separation, most typical for sequential IPNs. For simultaneous IPNs, where spinodal decomposition, as a rule, is more probable, it seems to be more reliable to determine only the heterogeneity parameters, not the radii of particles, if any. It is also necessary to keep in mind the possible changes of the mechanism of phase separation in the course of reaction. 

According to Hosemann-Bonart s model8), an oriented polymeric material consists of plate-like more or less curved folded lamellae extended mostly in the direction normal to that of the sample orientation so that the chain orientation in these crystalline formations coincides with the stretching direction. These lamellae are connected with each other by some amount of tie chains, but most chains emerge from the crystal bend and return to the same crystal-forming folds. If this model adequately describes the structure of oriented systems, the mechanical properties in the longitudinal direction are expected to be mainly determined by the number and properties of tie chains in the amorphous regions that are the weak spots of the oriented system (as compared to the crystallite)9). 

Scattering and Disorder. For structure close to random disorder the SAXS frequently exhibits a broad shoulder that is alternatively called liquid scattering ([206] [86], p. 50) or long-period peak . Let us consider disordered, concentrated systems. A poor theory like the one of Porod [18] is not consistent with respect to disorder, as it divides the volume into equal lots before starting to model the process. He concludes that statistical population (of the lots) does not lead to correlation. Better is the theory of Hosemann [158,211], His distorted structure does not pre-define any lots, and consequently it is able to describe (discrete) liquid scattering. The problems of liquid scattering have been studied since the early days of statistical physics. To-date several approximations and some analytical solutions are known. Most frequently applied [201,212-216] is the Percus-Yevick [217] approximation of the Ornstein-Zernike integral equation. The approximation offers a simple descrip-... 

History. Starting from the ID point statistics of Zernike and Prins [116] J. J. Hermans [128] designs various ID statistics of black and white rods. He applies these models to the SAXS curves of cellulose. Polydispersity of rod lengths is introduced by distribution functions, / , (,r)108. Hermans describes the loss of correlation along the series of rods by a convolution polynomial . One of Hermans lattice statistics is namedparacrystalby Hosemann [5,117]. Hosemann shows that the field of distorted structure is concisely treated by the methods of complex analysis. A controversial subject is Hosemann s extension of ID statistics to 3D [63,131,227,228],... 

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)... 

In an other model, the paracrystalline, (Hosemann, 1936), the amorphous regions are embedded within the crystals, so the other way round (Figure 4.15). Both models are realistic, and occur in combination with each other. 

A second important event was the development by Hosemann (1950) of a theory by which the X-ray patterns are explained in a completely different way, namely, in terms of statistical disorder. In this concept, the paracrystallinity model (Fig. 2.11), the so-called amorphous regions appear to be the same as small defect sites. A randomised amorphous phase is not required to explain polymer behaviour. Several phenomena, such as creep, recrystallisation and fracture, are better explained by motions of dislocations (as in solid state physics) than by the traditional fringed micelle model. 

FIG. 2.11 Diagrammatic representation of the paracrystallinity model (after Hosemann, 1962). 

This discussion has shown that the diffraction pattern can reveal three types of disorder, as discussed fully by several authors (Wiener and White, 1991a Blaurock, 1982 Hosemann and Bagchi, 1962). Thermal disorder is generally referred to as disorder of the first kind and lattice disorder as disorder of the second kind. The disorder due to the mosaic nature of the sample is referred to as orientational disorder. Thermal disorder and small amounts of orientational disorder are not particularly troublesome in the diffraction experiment. Lattice disorder, on the other hand, can be extremely problematic because one can never achieve a fully resolved image of the stmcture since there are too few stmcture factors available to obtain a faithful model. Thermal disorder simply means that the position of the atoms are .smeared in some fashion, determined by the equation of state of the molecules. If the lattice is excellent so that all of the stmcture factors observable within the limits of the... 

Blackwell and coworkers [19] have extensively studied the case of liquid crystalline main-chain copolymers produced by the random polymerization of two or three different monomers. The X-ray scattering patterns of these materials usually display a set of non-periodic diffuse streaks. These diffuse streaks were analyzed by extending the model of disorder of Hosemann to take into account chemical disorder. The description of the data by this model gave the correlation length of the ordering, information on the conformations of the repeat units and proved the random sequence of the copolymers. Subsequent molecular modelling studies revealed the detailed conformation of the repeat units. 

Figure 1.7. The Hosemann paracrystalline model, including the concept of chain folding.

It is worth considering that in [97] an explanation of the origin of the diffraction maxima along the meridian at 0.40 and 0.80 A is provided, consistent with the paracrystalline model proposed by Lyndenmeyer and Hosemann [48] for PAN. The halo at f 0.80 is not necessarily the second-order diffraction of the maximum at f 0.40 A since it is apparent from calculations of Fig. 12 that these two meridional maxima may originate from different contributions. The maximum at 0.40 A arises from the average periodicity of lateral - CN groups alone (Fig. 12C), whereas the maximum at C = 0.8oA- arises from the contribution of only the backbone carbon atoms (Fig. 12D). 

Several structural models have been proposed for semicrystalline polymers, such as the Hosemann-Bonnart model [691], in a polystructure subjected to mechanical stresses. 

R. Hosemann Paracrystalline model with disorder within the lamellae (see Figure 6.38) (1, m, n)... 

Figure 6.37 The paracrystalline model of Hosemann (134). Amorphous structures are illustrated in terms of defects. A radius of gyration approaching amorphous materials might be expected.

It is of interest to compare the results of this modem research with Hosemann s paracrystalline model, first published in 1962. As illustrated in Figure 6.37 (134), this model emphasizes lattice imperfections and disorder, as... 

The mathematical representation of the elastic behavior of oriented heterogeneous solids can be somewhat improved through a more appropriate choice of the boundary conditions such as proposed by Hashin and Shtrikman [66] and Stern-stein and Lederle [86]. In the case of lamellar polymers the formalisms developed for reinforced materials are quite useful [87—88]. An extensive review on the experimental characterization of the anisotropic and non-linear viscoelastic behavior of solid polymers and of their model interpretation had been given by Hadley and Ward [89]. New descriptions of polymer structure and deformation derive from the concept of paracrystalline domains particularly proposed by Hosemann [9,90] and Bonart [90], from a thermodynamic treatment of defect concentrations in bundles of chains according to the kink and meander model of Pechhold [10—11], and from the continuum mechanical analysis developed by Anthony and Kroner [14g, 99]. 

In 1950, as an extension of his work on the X-ray analysis of colloid structures, real crystals and polymers, Hosemann developed his theory of paraciystals, which he intended as a new, comprehensive model for the structure of matter that... 

As is seen from Table 8, there is no correlation between Porod radii and the cross-Unk density. Over most of the composition range studied, the equivalent sphere radii derived from the Hosemaim analysis were significantly larger than the Porod radii. For high PDMS contents, the average domain sizes derived from Hosemann analysis are substantially lower than the Porod radii. The discrepancy between the two methods was ascribed to the invaUd-ity of the models for certain composition ranges. 

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