Three-dimensional crystal defects

Pores and macroscopic inclusions are three-dimensional crystal defects. From the standpoint of the reactivity of solids, pores can be very important. Consider, for instance, the formation of porous scales during oxidation (tarnishing) [11]. (For example, the decarburization of iron cannot occur if a non-porous oxide scale without grain boundaries is formed on its surface.) Or consider the direct reduction of ore [10] in which the reduction rate is greatly dependent upon the formation of porous metal surface layers. In many so-called solid state reactions, gaseous products are formed as well as solid reaction products as, for example, during the reaction of TiOa with BaCOs to produce BaTiOs with the formation of In such cases, just as in the case of ore reduction, the formation of a porous product surface layer is of decided importance for the progress of the reaction. 

Closed isolated pores which, during transport processes through the crystal, can act essentially either as infinitely large resistances or as short circuits, depending upon the vapour pressure and rate of vaporization of the component being transported. 

Open pores, which are connected with each other and also with the surface of the crystal. The form and distribution of the pores are most important for transport phenomena such as diffusion, Knudsen flow, and surface diffusion. Because of the technological importance of this field, it possesses an extensive literature, especially on procedures for determining porosity. We can only make mention of this literature here [12]. 

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Every one-, two- or three-dimensional crystal defect gives rise to a potential field in which the various lattice constituents (building elements) distribute themselves so that their thermodynamic potential is constant in space. From this equilibrium condition, it is possible to determine the concentration profiles, provided that the partial enthalpy and entropy quantities and jj(f) of the building units i are known. Let us consider a simple limiting case and assume that the potential field around an (planar) interface is symmetric as shown in Figure 10-15, and that the constituent i dissolves ideally in the adjacent lattices, that is, it obeys Boltzmann statistics. In this case we have... 

To a good approximation, only atoms within the dotted circles in Figs. 20.30a and b are displaced from their equilibrium position in a real, three-dimensional crystal the diameter d of these circles would be very much less than the length / of the dislocation, i.e. the length, perpendicular to the page, of the extra half plane of atoms ab in Fig. 20.30a, or of the line cd in Fig. 20.306. Dislocations strictly, therefore, are cylindrical defects of diameter d and length / however, since I d they are referred to as line defects. 

Some important aspects of topochemical polymerizations can be understood by inspection of Eq. (1), All reactivity comes about by very specific rotations of the monomers and by 1,4-addition of adjacent units and an extended, fully conjugated polymer chain is formed. The unique feature of the topochemical polymerization of diacetylenes is the fact that in many cases the reaction can be carried out as a single phase process. This leads to macroscopic, defect-free polymer single crystals which cannot be obtained, in principle, by crystallization of ready-made polymers by conventional methods. Thus, polydiacetylenes are ideal models for the investigation of the behaviour of macromolecules in their perfect three dimensional crystal lattice. 

The optical treatment in Chapter 1 is concerned with the nature of the image of a simple two-dimensional grid formed by a perfect lens, in focus, using monochromatic light. However, to understand the nature of the lattice image formed in the electron microscope, we must take into account the thickness and the orientation of the three-dimensional crystal, the defect of focus and the aberrations of the objective lens (see Chapter 2), and the beam convergence, because all these factors influence the relative phases of the diffracted beams that are permitted to pass through the objective aperture. 

Disclinations are rotation defects, and are rare or absent in three-dimensional crystals, owing to their prohibitive energies, but are in general compatible with liquid crystalline structures. They were initially called disinclinations [2, 3, 54], but the term was later simplified to disclination . 

A crystal represents a complex quantum mechanical system with an enormous amount of particles with a strong interaction between them. If all the particles are located in space strictly ordered with the formation of an ideal three-dimensional crystal structure, then such a system possesses minimal free energy. In a real crystal, however, the ideal periodicity is often broken due to inevitable thermal fluctuations some atoms break periodic array, abandon their ideal position and produce a defect. The frequency and amount of fluctuations are defined by the Boltzmann factor, i.e., they depend on temperature and binding strength or, in other words, on the depth of the potential well corresponding to the regular position of atoms. 

Secondly, the ultimate properties of polymers are of continuous interest. Ultimate properties are the properties of ideal, defect free, structures. So far, for polymer crystals the ultimate elastic modulus and the ultimate tensile strength have not been calculated at an appropriate level. In particular, convergence as a function of basis set size has not been demonstrated, and most calculations have been applied to a single isolated chain rather than a three-dimensional polymer crystal. Using the Car-Parrinello method, we have been able to achieve basis set convergence for the elastic modulus of a three-dimensional infinite polyethylene crystal. These results will also be fliscussed. 

Obviously this model is very simplified, compared with the real crystal which contains many defects, dislocations and entanglements. In particular, it neglects many aspects of the true three-dimensional nature of the lamella which one may have thought to be important the influence of the stacking of folds, which is... 

A particular kind of disorder, characterized by maintaining three-dimensional long-range periodicity only for some points of the structure, has been found in samples of syndiotactic polypropylene having a relatively low degree of stereoregularity.189 190 In these samples the chains present conformational disorder, which produces defects frozen in the crystals. 

The various types of point defect found in pure or almost pure stoichiometric solids are summarized in Figure 1.17. It is not easy to imagine the three-dimensional consequences of the presence of any of these defects from two-dimensional diagrams, but it is important to remember that the real structure of the crystal surrounding a defect can be important. If it is at all possible, try to consult or build crystal models. This will reveal that it is easier to create vacancies at some atom sites than others, and that it is easier to introduce interstitials into the more open parts of the structure. 

Considering the crystal imperfections that are typically found in all crystals, the crystal quality of organic pigments is a major concern. The external surface of any crystal exhibits a number of defects, which expose portions of the crystal surface to the surrounding molecules. Impurities and voids permeate the entire interior structure of the crystal. Stress, brought about by factors such as applied shear, may change the cell constants (distances between atoms, crystalline angles). It is also possible for the three dimensional order to be incomplete or limited to one or two dimensions only (dislocations, inclusions). 

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