Crystal, defect, point deformation

An extension of the kinetic theory on cases when a mechanical pressure interacts with kinetic processes inside solid volume and on interfaces has wide application interests. The elastic deformations in solid are presented from influence of external forces and from presence of internal defects of crystal structure point defects (vacancy, intersite atoms, complexes of atoms, etc.), extended defects (dislocations and inner interfaces in polycrystals), and three-dimensional defects (heterophases crystals, polycrystals). 

The potential at a point r in a perfect crystal is given by K(r). If the crystal is now deformed by the presence of a defect, then the potential at point r will be different from V(t) because the potential at a point depends on the positions of the atoms in the neighborhood. If we assume that the deformation is not too severe (i.e., the deformation is a slowly varying function of position), then the potential at point r in the deformed crystal will be equivalent to the potential at point (r—R) in the undeformed crystal. R, in general, is a function of position and is called the displacement function. Thus, in a deformed crystal, the potential at a point r is given by... 

The singularities in the liquid crystals cause the deformation of the director field of liquid crystals and thus affect the symmetry of liquid crystals. This idea provides an approach to analyze the characteristics of the defects. The order vectors (or scalars, or tensors) of various liquid crystals are not the same. The director n is the order vector of the nematic liquid crystals, but the order for the cholesteric liquid crystals is a symmetric matrix, i.e., a tensor. Because the order vector space is thus a topological one, any configuration of the director field of liquid crystals is thus represented by a point in the order vector space. The order vector space (designated by M) is associated with the symmetry of liquid crystals. The topologically equivalent defects in liquid crystals constitutes the homotopy class. The complete set of homotopy classes constitutes a homotopy group, denoted Hr(M). r is the dimension of the sub-space surrounding a defect, which is related to the dimension of the defect (point, line or wall) d, and the dimension of the liquid crystal sample d by... 

Point defects and dislocations are shown in Sect. 5.3 to appear as nonequilibrium defects, usually introduced during crystal growth or deformation, and as equilibrium defects, generated thermally. The dislocation density can be determined by the moir6 method (see Appendix 17 and Fig. 5.93). On annealing of crystals above their crystallization temperature, the nonequilibrium dislocation density increases, as is demonstrated in the left graph of Fig. 6.75. Since the number of such nonequilibrium... 

It is well known [73] that plastic deformation in crystals can occur when the applied shear stress can cause one plane of atoms to slip over another plane because there is an imperfect match between these adjacent planes at a particular point in the crystal lattice. These points of imperfection are called dislocations [74] and were identified by electron diffraction techniques to relate to specific crystal defects. Dislocations are observed in polyethylene single crystals by Peterman and Gleiter [75] and give credence to the idea that yield in crystalline polymers can be understood in similar terms to those used by metallurgists for crystalline solids. 

Dislocation theory as a portion of the subject of solid-state physics is somewhat beyond the scope of this book, but it is desirable to examine the subject briefly in terms of its implications in surface chemistry. Perhaps the most elementary type of defect is that of an extra or interstitial atom—Frenkel defect [110]—or a missing atom or vacancy—Schottky defect [111]. Such point defects play an important role in the treatment of diffusion and electrical conductivities in solids and the solubility of a salt in the host lattice of another or different valence type [112]. Point defects have a thermodynamic basis for their existence in terms of the energy and entropy of their formation, the situation is similar to the formation of isolated holes and erratic atoms on a surface. Dislocations, on the other hand, may be viewed as an organized concentration of point defects they are lattice defects and play an important role in the mechanism of the plastic deformation of solids. Lattice defects or dislocations are not thermodynamic in the sense of the point defects their formation is intimately connected with the mechanism of nucleation and crystal growth (see Section IX-4), and they constitute an important source of surface imperfection. 

Dislocations are line defects. They bound slipped areas in a crystal and their motion produces plastic deformation. They are characterized by two geometrical parameters 1) the elementary slip displacement vector b (Burgers vector) and 2) the unit vector that defines the direction of the dislocation line at some point in the crystal, s. Figures 3-1 and 3-2 show the two limiting cases of a dislocation. If b is perpendicular to s, the dislocation is named an edge dislocation. The screw dislocation has b parallel to v. Often one Finds mixed dislocations. Dislocation lines close upon themselves or they end at inner or outer surfaces of a solid. 

This chapter is concerned with the influence of mechanical stress upon the chemical processes in solids. The most important properties to consider are elasticity and plasticity. We wish, for example, to understand how reaction kinetics and transport in crystalline systems respond to homogeneous or inhomogeneous elastic and plastic deformations [A.P. Chupakhin, et al. (1987)]. An example of such a process influenced by stress is the photoisomerization of a [Co(NH3)5N02]C12 crystal set under a (uniaxial) chemical load [E.V. Boldyreva, A. A. Sidelnikov (1987)]. The kinetics of the isomerization of the N02 group is noticeably different when the crystal is not stressed. An example of the influence of an inhomogeneous stress field on transport is the redistribution of solute atoms or point defects around dislocations created by plastic deformation. 

The influence of plastic deformation on the reaction kinetics is twofold. 1) Plastic deformation occurs mainly through the formation and motion of dislocations. Since dislocations provide one dimensional paths (pipes) of enhanced mobility, they may alter the transport coefficients of the structure elements, with respect to both magnitude and direction. 2) They may thereby decisively affect the nucleation rate of supersaturated components and thus determine the sites of precipitation. However, there is a further influence which plastic deformations have on the kinetics of reactions. If moving dislocations intersect each other, they release point defects into the bulk crystal. The resulting increase in point defect concentration changes the atomic mobility of the components. Let us remember that supersaturated point defects may be annihilated by the climb of edge dislocations (see Section 3.4). By and large, one expects that plasticity will noticeably affect the reactivity of solids. 

This interaction arises from the overlap of the deformation fields around both defects. For weakly anisotropic cubic crystals and isotropic point defects, the long-range (dipole-dipole) contribution obeys equation (3.1.4) with a(, ip) oc [04] (i.e., the cubic harmonic with l = 4). In other words, the elastic interaction is anisotropic. If defects are also anisotropic, which is the case for an H centre (XJ molecule), in alkali halides or crowdions in metals, there is little hope of getting an analytical expression for a [35]. The calculation of U (r) for F, H pairs in a KBr crystal has demonstrated [36] that their attraction energy has a maximum along an (001) axis with (110) orientation of the H centre reaching for 1 nn the value -0.043 eV. However, in other directions their elastic interaction was found to be repulsive. 

The resolution of the atomic force microscope depends on the radius of curvature of the tip and its chemical condition. Solid crystal surfaces can often be imaged with atomic resolution. At this point, however, we need to specify what Atomic resolution is. Periodicities of atomic spacing are, in fact, reproduced. To resolve atomic defects is much more difficult and usually it is not achieved with the atomic force microscope. When it comes to steps and defects the scanning tunneling microscope has a higher resolution. On soft, deformable samples, e.g. on many biological materials, the resolution is reduced due to mechanical deformation. Practically, a real resolution of a few nm is achieved. 

Although point defects certainly occur in nanoparticles (and unusual coordination sites are probably common in some very small nanoparticles), it is generally agreed that nanocrystals do not contain dislocations or other extended defects because the energetics of these features are significant and diffusion distances are small. So (in the absence of deformation), given that all big crystals start out small, where do dislocations and planar defects in macroscopic materials come from ... 

By measuring the shape of the quadrupolar echo in single crystals of RbBr and Rbl, Mehring and Kanert showed that a quadrupolar distribution function could be determined.Once this fimction was known, the authors could quantitate the dislocation density as a fimction of shear stress and presented a model to determine the density of point defects and dislocations in the lattice. It was concluded that the EFG in imdeformed RbBr single crystals was due to point defects, while plastic deformation induced dislocations. Discussion pertaining to sample doping is delayed to Section 4.3.2. 

It is worth pointing out here that if material that is subject to deformation is soluble in the liquid into which it has been immersed, one may observe the so-called Ioffe effect. This effect is, for instance, revealed when brittle crystals of sodium chloride undergo plastic deformation in a pool of water that is not saturated with salt and dissolves the surface. In this case plasticity occurs not due to a decrease in resistance to plastic flow, as in the case of adsorption plasticizing, but rather due to an increase in the strength of crystals because of the dissolution of surface layer containing structural defects. 

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