Three-dimensional lattice diffusion

The random walk approach is based on the random-walk concept, which was originally apphed to the problem of diffusion and later adopted by Flory [3] to deduce the conformations of macromolecules in solution. The earliest analysis was by Simha et al. [4], who neglected volume effects and treated the polymer as a random walk. Basically, the solution was represented by a three-dimensional lattice. 

The versatility of lattice models to describe encounter-controlled reactions in systems of more complicated geometries can be illustrated in two different applications. In this subsection layered diffusion spaces as a model for studying reaction efficiency in clay materials are considered and in the following subsection finite, three-dimensional lattices of different symmetries as a model for processes in zeolites are studied. Now that the separate influences of system size N, dimensionality d (integral and fractal), and valency v have been established, and the relative importance of d = 3 versus surface diffusion (and reduction of dimensionality ) has been quantified, the insights drawn from these studies will be used to unravel effects found in these more structured systems. 

Although the above diagram is a simple model, this is not how it occurs in Nature. We have presented the above concept because it is easier to understand than the actual conditions which occur. It should be clear that the overall solid state reaction is dependent upon the rate of diffusion of the two (2) species, and the two rates may, or may not, be the same. The reason that A and/or B do not react in the middle, i.e.- the phase AB, is that AB has a certain ordered structure which probably differs from either A or B. But there is a more important reason which is not easily illustrated in any two-dimensional diagram, particularly since each compound has a three-dimensional lattice structure. 

Unlike PAN-based carbon fibers, mesophase pitch-based fibers experience significant graphitization during which dislocations in the turbostratic carbon stacks are gradually annealed, resulting in the formation of a three-dimensional lattice. Fischbach (75) presented a detailed study of graphitization which characterizes the process as a combination of atomic diffusion and crystallite growth. Table 4 provides a comparison of properties for some commercial pitch- and PAN-based carbon fibers. 

Like crystals, glasses must be made of an extended three-dimensional lattice, but the diffuse character of x-rays scattering shows that this lattice is neither symmetric nor periodic, unlike in crystals in other words, there is no long-distance order. 

Ionic transport in solid electrolytes and electrodes may also be treated by the statistical process of successive jumps between the various accessible sites of the lattice. For random motion in a three-dimensional isotropic crystal, the diffusivity is related to the jump distance r and the jump frequency v by [3] ... 

Volume diffusion refers to the transport of atoms through the body of a solid. It is also called lattice or bulk diffusion. In amorphous or glassy solids and in cubic crystals, the speed of diffusion in all directions is the same and is said to be isotropic. In all other crystals, the rate of volume diffusion depends upon the direction taken and is anisotropic. Volume diffusion is usually much slower than short-circuit diffusion, which refers to diffusion along two- and three-dimensional imperfections in the material. 

When the charge-transfer step in an electrodeposition reaction is fast, the rate of growth of nuclei (crystallites) is determined by either of two steps (I) the lattice incorporation step or (2) the diffusion of electrodepositing ions into the nucleus (diffusion in the solution). We start with the first case. Four simple models of nuclei are usually considered (a) a two-dimensional (2D) cylinder, (b) a three-dimensional (3D) hemisphere, (c) a right-circular cone, and (d) a truncated four-sided pyramid (Fig. 7.2). 

A characteristic feature of these partially oxidized bis(oxalato)platinate salts of divalent cations is the coexistence of two modulations of the lattice over a wide temperature range (a) a one-dimensional modulation, as detected by the appearance of diffuse lines on X-ray films, perpendicular to the [Pt(C204)2] anion stacking direction and surrounding the even Bragg reflection layer lines of non-zero order (b) a three-dimensional modulation which gives rise to a complicated pattern of fine satellite spots in the neighbourhood of every reciprocal layer line. 

Let us now consider a random walker in a three-dimensional cubic lattice. The atom will jump between sites of the normal lattice for a substitutional diffuser, and from interstitial to interstitial site for an interstitial diffuser. In the present case, the Einstein-Smoluchovskii equation for the diffusion coefficient in three dimensions which is a generalization of Equation 5.36, that is,... 

Consider, for example, the case of the quasi-one-dimensional organic metal TTT2I3+5. This material often exhibits a set of disordered iodine sublattices either commensurate or incommensurate with the main lattice. Lowe-Ma et al. [85] have analyzed the intensity of the corresponding set of diffuse reciprocal layers and postulated that, in their samples, part of the It ions are substituted by I2 and I- moieties. In fact, the intensity of the zero layer is not negligible and no three-dimensional ordering of iodine chains is observed at low temperature. However, TTT2I3+8 crystals are often characterized by a varying amount of positional disorder [136] of iodine columns rather than by a chemical disorder. 

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