Limit-disordered models

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. 

The real crystalline forms are generally intermediate between the limit-ordered and limit-disordered models, the amount of disorder being dependent on the condition of crystallization and thermal and mechanical treatments of the samples. A condition to have more or less disordered modifications, corresponding to the same unit cell, is the substantial equality of steric hindrances in the space regions where a statistical substitution is achieved (Figure 2.29b). 

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). 

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. 

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...

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"). 

In the limit ordered model proposed for the a form of iPP (Fig. 2.25c) up and down chains follow each other according to a well-defined pattern [101,102]. The limit disordered model (Fig. 2.25d) corresponds to a statistical substitution of up and down isomorphic helices in each site of the lattice [29]. The real crystalline modifications of the a form of iPP are intermediate between the limit ordered and limit disordered models of Figure 2.25, the degree of disorder in the positioning of up and down chains being dependent on the thermal and mechanical history of the sample [150]. 

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. 

Crystalline forms corresponding to limiting ordered or disordered models, with equal lattice geometry, can be obtained with different procedures and can present dif-... 

Figure 2.47 Models of packing in limit-ordered forms (a) II and (c) IV of sPP and (b) model of conformationally disordered modification, presenting kink bands, intermediate between limit-ordered models of form II and form IV. In defective region of model (b), delimited by dashed lines, chains are packed as in form IV, whereas in ordered regions chains are packed as in the form II.

Fig. 8 Temperature dependence of the zero field hole mobility in the low carrier density limit in a polyfluorene copolymer. The data are inferred from space-charge-limited current experiments and analyzed in terms of the extended Gaussian disorder model (see Sect. 4.1). From [90] with permission. Copyright (2008) by the American Institute of Physics...

B. Derrida, Random-energy model limit of a family of disordered models. Phys. Rev. Lett. 45, 79-82 (1980). 

The severe computational burden associated with assembling and carrying out adsorption calculations on disordered model microstructures for porous solids, such as those discussed in Sections ILA and II.B, has until recently limited the development of pore volume characterization methods in this direction. While the reahsm of these models is highly appealing, their application to experimental isotherm or scattering data for interpretation of adsorbent pore structure remains cumbersome due to the structural complexity of the models and the computational resources that must be brought to bear in their utilization. Consequently, approximate pore structure models, based upon simple pore shapes such as shts or cylinders, have been retained in popular use for pore volume characterization. 

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-... 

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