Crystalline Models

In the next section we describe a very simple model, which we shall term the crystalline model , which is taken to represent the real, complicated crystal. Some additional, more physical, properties are included in the later calculations of the well-established theories (see Sect. 3.6 and 3.7.2), however, they are treated as perturbations about this basic model, and depend upon its being a good first approximation. Then, Sect. 2.1 deals with the information which one would hope to obtain from equilibrium crystals — this includes bulk and surface properties and their relationship to a crystal s melting temperature. Even here, using only thermodynamic arguments, there is no common line of approach to the interpretation of the data, yet this fundamental problem does not appear to have received the attention it warrants. The concluding section of this chapter summarizes and contrasts some further assumptions made about the model, which then lead to the various growth theories. The details of the way in which these assumptions are applied will be dealt with in Sects. 3 and 4. 

Fig. 2.1. The crystalline model a single crystal in which molecules traverse the lamella perpendicular to the fold surface. Cilia are formed at the end of the molecules outside the crystalline core. The folds are predominantly adjacent and the loop sizes may vary...

Many workers have offered the opinion that the isokinetic relationship is confined to reactions in condensed phase (6, 122) or, more specially, may be attributed to solvation effects (13, 21, 37, 43, 56, 112, 116, 124, 126-130) which affect both enthalpy and entropy in the same direction. The most developed theories are based on a model of the half-specific quasi-crystalline solvation (129, 130), or of the nonideal conformal solutions (126). Other explanations have been given in terms of vibrational frequencies involving solute and solvent (13, 124), temperature dependence of solvent fluidity in the quasi-crystalline model (40), or changes of enthalpy and entropy to produce a hole in the solvent (87). 

Table 19.2 Thermal properties of liquid crystalline model compounds derived from BB and TA [25]...

Fluorapatite (FA) corresponds to the chemical formula Caio(P04)eF2 and crystallises in the hexagonal space group PGs/m, with Z = 1 and unit-cell parameters a = b = 9.367 A and c = 6.884 A [1] (Fig. 2). From a structural viewpoint, fluorapatite is often considered as a crystalline model for other apatites and is seen as a reference apatitic array [2]. It is one of the very first apatite structures to have been solved. It has been thoroughly studied since the 1930s [3] and is well documented in the literature. In particular, Sudarsanan et al. [1] reported the single crystal refinement of X-ray diffraction (XRD) data, and the detailed description of atomic positions and local symmetry is fully available [4,5],... 

In crystalline oxides and hydroxides of iron (III) octahedral coordination is much more common than tetrahedral 43). Only in y-FegOs is a substantial fraction of the iron (1/3) in tetrahedral sites. The polymer isolated from nitrate solution is the first example of a ferric oxyhydroxide in which apparently all of the irons are tetrahedrally coordinated. From the oxyhydroxide core of ferritin, Harrison et al. 44) have interpreted X-ray and electron diffraction results in terms of a crystalline model involving close packed oxygen layers with iron randomly distributed among the eight tetrahedral and four octahedral sites in the unit cell. In view of the close similarity in Mdssbauer parameters between ferritin and the synthetic poljmier it would appear unlikely that the local environment of the iron could be very different in the two materials, whatever the degree of crystallinity. Further study of this question is needed. 

Plansible explanations for the current experimental observations can provide the quasi-crystalline model proposed by Zhdanov et al. [28], Shimakawa et al. [29],... 

Flexible, linear macromolecules frequently show on cooling only partial crystallization. Such materials can often be described by the crystallinity model (typical crystallinities of 30-90%) 18-19). Two of the questions in the description of macro-molecular mesophases must thus be what is the crystallinity of the sample and which phases are present ... 

In the following, three crucial experiments are described which will show the validity of the crystalline model. 

Table 2.6 Constants for Carreau-WLF (Amorphous) and Carreau-Arrhenius (Semi-Crystalline) Models for Various Common Thermoplastic...

Lutskii and Mikhailenko99 attempted a theoretical calculation of the correlation factor based on a quasi-crystalline model for liquids. To obtain a reasonable... 

Some experimentally derived values of yy for foam and emulsion bilayers are listed in Table 3.16. Values of yy for BLMs are also given for comparison. These data are obtained on the basis of an experiment in which the rupture of BLM is caused by an external electric field of intensity U [456,463]. Using the i(U) dependence the value of yy for bilayers from lyso PC and lyso PE is found to be 0.5 to 1.510"11 J m 1 (Table 3.16). For egg lecithin BLM in n-decane yy is also evaluated [459,464], Depending on the adopted model, packing model [465] or liquid-crystalline model [464] two values of yy are obtained yy = 0.75-10" J m 1 and % = 2.M011 J m1. The latter value is also determined in [466] by studying microscopic holes in tube liposomes in electric field (Table 3.16). 

Fig. 1.11 Diagrams of polymer crystallinity models (a) fringed micelles, (b) a chain-folded lamella.

Analytical approach. An alternative to this Monte-Carlo like procedure leading to individual crystalline models fitting only relatively well the observed data, is an analytical description of diffraction patterns based on the stacking fault probabilities and thus on the resulting layer-layer correlation probability distribution, which has been tried for the description of ice in mesopores. Analytical approaches have been widely used for the de-... 

C. Diffraction-Crystal simulates powder, fiber, and single-crystal diffraction from crystalline models, which helps interpret the experimental data from molecular, inorganic, and polymeric crystalline materials. 

Few studies have been made on transport processes involving concentrated solutions. In the concentrated solutions, in the range of dehydrated melt formation, incompletely hydrated melts and anhydrous salt melts, various structural models are described to define their properties, i.e. the free-volume model, the lattice-model and the quasi-crystalline model. Measured and calculated transport phenomena do not always represent simple ion migration of individual particles, but instead we sometimes find them to be complicated cooperative effects (27). 

These three observations lead to the conclusion that a good crystalline model consists of a sheet-like arrangement of chains parallel to the unit-cell axis as first proposed by Palmer and Ballantyne.(19)... 

VI. Melts for which Quasi-Crystalline Models are Inadequate. . . 467... 

VI. MELTS FOR WHICH QUASI-CRYSTALLINE MODELS ARE INADEQUATE... 

A second mechanism for increasing disorder on melting which cannot be conveniently represented by a quasi-crystalline model for the melt involves the formation of association complexes. Quite generally, these can be defined as clusters of the units of structure (e.g., molecules or ions) in the crystal which have approximately the same distance between nearest neighbours as in the crystal lattice, but which need not have the full regularity of crystal packing. As already stated, only one particular form of cluster, the crystal nucleus can normally be extended indefinitely... 

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