Correlation in disordered chains

The treatment of correlation, even in periodic polymers, presents a formidable and as yet only partially solved problem if the unit cell is large. This problem is examined by initially discussing localization techniques in larger molecules, and then presenting the first results of the application of localized orbitals for correlation calculations using many-body perturbation theory and the coupled-cluster approach. It is also shown how localization of the orbitals within a large unit cell can be applied for correlation calculations if this unit cell is repeated in a periodic way in the polymer. This chapter also deals with the correlation in polymers with smaller unit cells (like polyacetylene, polydiacetylene, and polyethylene) and includes a detailed discussion of the results obtained. Finally, some ideas about possibilities of treating correlation in disordered chains are noted. 

To solve the replica OZ equations, they must be completed by closure relations. Several closures have been tested against computer simulations for various models of fluids adsorbed in disordered porous media. In particular, common Percus-Yevick (PY) and hypernetted chain approximations have been applied [20]. Eq. (21) for the matrix correlations can be solved using any approximation. However, it has been shown by Given and Stell [17-19] that the PY closure for the fluid-fluid correlations simplifies the ROZ equation, the blocking effects of the matrix structure are neglected in this... 

Mesomorphic forms characterized by conformationally ordered polymer chains packed in lattices with different kinds of lateral disorder have been described for various isotactic and syndiotactic polymers. For instance, for iPP,706 sPP,201 sPS,202 syndiotactic poly(p-methylstyrene) (sPPMS),203 and syndiotactic poly(m -methylstyrene),204 mesomorphic forms have been found. In all of these cases the X-ray fiber diffraction patterns show diffraction confined in well-defined layer lines, indicating order in the conformation of the chains, but broad reflections and diffuse haloes on the equator and on the other layer lines, indicating the presence of disorder in the arrangement of the chain axes as well as the absence of long-range lateral correlations between the chains. 

In conclusion, with no disorder present, the interchain coupling leads to a CDW state, where electron correlations on neighbouring chains are correlated and the material is a highly anisotropic small bandgap semiconductor with electron correlation effects playing a decisive role. 

The Inclusion of Correlation in the Calculation of Density of States of Disordered Chains. - We can easily substitute instead of the matrix blocks Fft-and F,y into the determinant (5) such matrix blocks of dimers which take into account also the major part of the valence correlation. For this we can write for a dimer the more complicated expression given by Liegener21 and by Liegener et al.2X Modifying their expression we can write... 

A consideration of the paracrystalline distortions, primarily the limited lateral correlations of axial chain coordinates ( chain shift disorder ), is found in the paper by Kawaguchi and Petermann [174]. According to the analysis of their electron diffraction patterns of flow-oriented PPP films, the diffraction on the third and higher layer lines is essentially single-chain diffraction. Although indications for an off-meridional position of the (002) reflection are found in some patterns, the authors favour the orthorhombic unit cell. 

Of course, the presence of non-random substitutional, translational or rotational displacement disorder, and hence the presence of short-range lateral correlations between neighboring chains, noticeably complicates the evaluation of the X-ray diffraction intensity distribution. Diffuse scattering in this case appears more localized for instance, in the case of non-random substitutional disorder, for systems where two different structural motifs tend to alternate in neighboring sites, diffuse scattering appears rather peaked near superstructure positions in the diffraction patterns [112]. In these cases, Eq. 3 should be used instead of Eq. 4 for the calculation of diffuse scattering. Equation 3 is usually presented in the form [61,112] ... 

In the analysis of [260], it was argued that the diffuse equatorial diffraction peak, centered at = 0.25 A" (Fig. 31 A), gives information about the mean distance between chain axes, preferentially oriented parallel to the drawing direction, like in a nematic hquid crystal the absence of diffraction peaks off the meridian was ascribed to a substantial absence of correlation in the position and orientation of the chains, besides their (imperfect) parallelism, whereas the substantial absence of layer lines (off the equator) was interpreted as due to the presence of conformational disorder. 

A comparison between calculated Fourier transform of various disordered models and experimental X-ray diffraction pattern have been reported in [262] and [263]. Structural models characterized by small bundles of parallel chains in 3/1 helical conformation packed with lateral disorder and keeping short-range correlations between neighboring chains similar to those of a form and the hexagonal P form of iPP, have been analyzed [262,263]. Since the crystal structure of the form had not been solved at that time, the hexagonal models built up for Fourier transform calculations in [262] and [263] were different from the now accepted structure of P form [264-266] even though the considered models had hexagonal correlations between chains. 

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