Limit ordered model structures

Fig. 2 (A) Chain conformation of isotactic polypropylene in the crystalline state. Symbols R and L identify right- and left-handed helices, respectively, in 3/1 conformations. Subscripts up and dw ( dw standing for down ) identify chains with opposite orientation of C - C bonds connecting tertiary carbon atoms to the methyl groups along the z-axis (B) Limit-ordered model structure (a2 modification, space group P2 /c) [113] (C) Limit-disordered model structure (otl modification, space group Cl/c) [114]. In the a2 modification up and down chains follow each other according to a well-defined pattern. The al modification presents a complete disorder correspon ng to a statistical substitution of up and down isomorphic helices...

Let us consider a structural limiting model, in which the polymer molecules, presenting a periodic conformation, are packed in a crystal lattice with a perfect three-dimensional order. Besides this limiting ordered model, it is possible to consider models of disordered structures having a substantially identical lattice geometry. 

Figure 2.29 (a) Limit-ordered model (space group Ibca or P2i/a) and (b) limit-disordered model (space group Bmcm) for crystal structure of form I of sPP172 (R = right-handed helix, L = left-handed helix). 

The msgority of commercial instrumentation consists of a flood x-ray or uv photon source and an energy analyzer equipped with an energy retardation transfer lens. The analysis spot size in this instrumentation is limited by the image the entrance slit of the analyzer makes on the sample surface in the analysis position. Materials studies are thus limited to model structures or large area surfaces (at least a few millimeters in size). Because semiconductor device structures generally consist of features which are on the order of micrometer dimensions, the trend in modem instmmentation is towards smaller analysis spot size Small analysis spot size permits investigators to employ photoelectron spectroscopy on real devices rather than model device structures. 

As an example, the structure of the a form of isotactic polypropylene may be described with reference to a limit-ordered model (defined a2-form) [113] and a limit-disordered model (defined ai-form) [114], shown in Fig. 2B and C, respectively. 

Three limit-ordered models, shown in Fig. 15, were considered as possible ideal arrangements of EP chains in the mesomorphic bundles. In Fig. 15A and Fig. 15B,B the chains are arranged as in the orthorhombic [194] and monoclinic [196] polymorphs of PE, respectively. In Fig. 15C,C the chains are arranged as in triclinic form of long chains paraffins [197]. These models were chosen as reference, ideal structures, where different kinds of disorder were introduced, in order to better understand their influence on the cal-... 

The polymorphism of SPS is further complicated by the presence of structural disorder in both a and p forms, so that the trans-planar forms are described in terms of disordered modifications intermediate between limit disordered models (a and PO and limit ordered models (a" and P"). 

Crystallographic refinement is a procedure which iteratively improves the agreement between structure factors derived from X-ray intensities and those derived from a model structure. For macro molecular refinement, the limited diffraction data have to be complemented by additional information in order to improve the parameter-to-observation ratio. This additional information consists of restraints on bond lengths, bond angles, aromatic planes, chiralities, and temperature factors. 

Disordered structures belonging to the class (i) are interesting because, in some cases, they may be characterized by disorder which does not induce changes of the lattice dimensions and of the crystallinity, and a unit cell may still be defined. These particular disordered forms are generally not considered as mesomorphic modifications. A general concept is that in these cases the order-disorder phenomena can be described with reference to two ideal structures, limit-ordered and limit-disordered models, that is, ideal fully ordered or fully disordered models. 

In general, doping tends to lead to a loss of x-ray order in polyacetylene and polyphenylene, suggesting that dopant ions may be distributed more or less at random. The structural models shown in Fig. 16 are clearly idealised as only limited order is seen even in cation-doped polymer. The anion dopants are much larger and apparently disrupt the structure too much for any sign of regularity to be seen, except in the case of iodine. 

Equation 1.3 represents a system of usually several thousand coupled differential equations of second order. It can be solved only numerically in small time steps At via finite-difference methods [16]. There always the situation at t + At is calculated from the situation at t. Considering the very fast oscillations of covalent bonds, At must not be longer than about 1 fs to avoid numerical breakdown connected with problems with energy conservation. This condition imposes a limit of the typical maximum simulation time that for the above-mentioned system sizes is of the order of several ns. The limited possible size of atomistic polymer packing models (cf. above) together with this simulation time limitation also set certain limits for the structures and processes that can be reasonably simulated. Furthermore, the limited model size demands the application of periodic boundary conditions to avoid extreme surface effects. 

Because they are so computationally intensive, ab initio and semiempirical studies are limited to models that are about 10 rings or less. In order to study more reahstic carbon structures, approximations in the form of the Hamiltonian (i.e., Schrodinger equation) are necessary. The tight-binding method, in which the many-body wave function is expressed as a product of individual atomic orbitals, localized on the atomic centers, is one such approximation that has been successfully applied to amorphous and porous carbon systems [47]. 

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