Defect region model

Features of the ionic transport in the fluorite-like solid solutions are well described by means of the defect regions model [64,65]. 

It is shown in [67] for Cdo.9Ro.iF2.i (R = La-Lu, Y) that ionic conductivity (ctsook) of fluorites of various RE elements correlates with concentration of interstitial fluoride ions (Fi(48g) + Fi(32Qi + Fi(32f)2) located on the periphery of clusters. These interstitial fluoride ion concentrations were measured by means of single-crystal X-ray diffraction (XRD). 

The percolation limit differs for various RE and depends on both the size factor and the cluster type existing in the structure [63]. 

Conductivity maxima were also observed for anion-deficit solid solutions. For example, such a maximum was found out in Nao.ioNdo.9o(0,F)2 6 solid solution for the composition Nao,ioNdo.9oOo.96Fo,g8 = 0.16) [73]. This fact is in full agreement with data obtained for some oxide fluorite-like solid solutions, for example Zr02 - Yb203 [4]. 

The presence of defects clusters in disordered solid solutions was confirmed by direct experimental methods (neutron diffraction, F NMR, EXAFS, etc.) as well as indirectly, using results of X-ray and electron diffraction studies of ordered fluorite-like phases having close compositions [7]. 

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The ionic conductivity is a characteristic of materials rather sensitive to changes of the defect stmcture and to micro-ordering. The model of micro-ordering developed for description of fluorite-like solid solutions and the defect region model used for description of many objects explain equally discrepancy between the observed charge carriers concentration and the quantity of anion defects [43]. Besides of this, defect area volumes (3000-4000 A ) estimated from the B 2-x x 2+x conductivity data are very close to the associated supercluster size obtained by the micro-ordering areas model [43]. 

The influence of different doping types on the defect structure and conductivity of fluorine-containing phases with fluorite and tysonite structures has been discussed in the current review. The defect-region model including clustering and percolation phenomena has been used for describing the ionic transfer features in soUd solutions. 

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.

The nature of relaxation in glassy materials is investigated via a defect transport model. Relaxation is assumed to occur when a mobile defect reaches a frozen-in region of the material. A many defect calculation leads to the ubiquitous stretched exponential law as a probability limit distribution for the relaxation. The time scale for this relaxation varies inversely with the concentration of uncorrelated defects. The fast disappearance of these defects with decreasing temperature leads to a Vogel-like Taw for the relaxation time scale. 

Figure 2.29 Limit-ordered models of packing of forms II [31] (a) and IV [67] (c) of sPP and model of a conformationally disordered modification, presenting kink-bands (b), intermediate between forms II and IV. A kink-band in the form II of sPP, with (T2G2)n helices, is characterized by a defective region with (T2G2T6G2)n conformational sequence, as in form IV [154, 155]. Reprinted from Reference [143] with permission from American Chemical Society, Copyright 2006.

Brunauer (see Refs. 136-138) defended these defects as deliberate approximations needed to obtain a practical two-constant equation. The assumption of a constant heat of adsorption in the first layer represents a balance between the effects of surface heterogeneity and of lateral interaction, and the assumption of a constant instead of a decreasing heat of adsorption for the succeeding layers balances the overestimate of the entropy of adsorption. These comments do help to explain why the model works as well as it does. However, since these approximations are inherent in the treatment, one can see why the BET model does not lend itself readily to any detailed insight into the real physical nature of multilayers. In summary, the BET equation will undoubtedly maintain its usefulness in surface area determinations, and it does provide some physical information about the nature of the adsorbed film, but only at the level of approximation inherent in the model. Mainly, the c value provides an estimate of the first layer heat of adsorption, averaged over the region of fit. 

Sometimes, the system of interest is not the inhnite crystal, but an anomaly in the crystal, such as an extra atom adsorbed in the crystal. In this case, the inhnite symmetry of the crystal is not rigorously correct. The most widely used means for modeling defects is the Mott-Littleton defect method. It is a means for performing an energy minimization in a localized region of the lattice. The method incorporates a continuum description of the polarization for the remainder of the crystal. 

The existence of kinks was recently explicitly taken into account by Larson (Fig. 14) as a possible model for chain unravelling in the flow [69]. At the same time, Kausch developed a similar model to explain degradation results measured in transient elongational flow (Fig. 15) [70]. With this difference from the Larson model, kinks in the latter model can support compressive stress chain elastic modulii range from 16 to 110 GPa, depending on the number of defects within the kinked region. 

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