Diffusion in Crystalline Solids

The proportionality D is a constant, which is known as the diffusion coefficient or diffusivity, with a unit of m s (SI) or cm s. The diffusion coefficient is a property of materials, which is the most useful parameter to characterize the rate of diffusive mass transport, showing a strong dependence on temperature. Although it is also a function of composition, if the diffusing species are significantly diluted, it can be assumed to be independent on the composition. 

In practical experiments, it is difficult to maintain a concentration to be independent of time. Therefore, it is more often to characterize the change in concentration as a function of time t, thus leading to Pick s second law. For one dimension, the Pick s second law is given by  

The Pick s second law can be derived from the first law, together with the principle of matter conservation. For one-dimension, a region between the two 

Equation (5.37) can be solved with certain boundary conditions that are determined by experiments. 

For instance, a common technique to measure diffusion coefficient D is to deposit a very thin layer of a radioactive isotope (or mass isotope) on a flat surface of a thick sample, which is then annealed at a given temperature for a given time duration. By measuring the concentration of the diffusing species as a function of distance, the diffusion coefficient can be determined. In this case, the experimental system is a semi-inflnite solid. If the initial thickness of the radioisotope layer is sufficiently small as compared with the diffusing distance of the radioisotope, the solution of Eq. (5.37) is given by  

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Manning, R.J. (1968) Diffusion Kinetics for Atoms in Crystals, van Nostrand, Princeton Murch, G. E. (1984) Diffusion in Crystalline Solids (Eds. Murch, G. E., Nowick, A. S.), Academic, Orlando... 

W. Prank, U. Gosele, H. Mehrer, and A. Seeger. Diffusion in silicon and germanium. In G.E. Murch and A.S. Nowick, editors, Diffusion in Crystalline Solids, pages 63-142, Orlando, Florida, 1984. Academic Press. 

G.E. Murch and A.S. Nowick (eds), Diffusion in Crystalline Solids (Associated Press, New York, 1984). 

Diffusion through nonporous crystalline solids depends markedly on the crystal lattice structure and the diffusing entity. The mechanisms of diffusion in crystalline solids include (Seader and Henley, 2006) ... 

Nowick AS (1984) Atom transport in oxides of the fluorite structure, hi Diffusion in crystalline solids. Academic, Orlando, pp 143-188... 

M. Nastar, V. Y. Dobretsov, and G. Martin, Self-consistent formulation of configurational kinetics close to equilibrium the phenomenological coefficients for diffusion in crystalline solids, Phil. Mag. A, vol. 80, p. 155, 2000. 

The chemical and physical stability of a solid drug decreases with decreasing crystallinity and increasing amorphous character, corresponding to an increase in molecular mobility (i.e., diffusivity) in the solid state. This phenomenon is of particular significance to proteins, peptides, and other biological materials. Certain additives other than water may stabilize proteins in the solid state, perhaps by locking in the defects. 

As with thermal conductivity, we see in this section that disorder can greatly affect the mechanism of diffusion and the magnitude of diffusivities, so that crystalline ceramics and oxide glasses will be treated separately. Finally, we will briefly describe an important topic relevant to all material classes, but especially appropriate for ceramics such as catalyst supports—namely, diffusion in porous solids. 

A number of diffusion mechanisms in crystalline solids are possible. Atoms vibrate in their equilibrium sites after that, periodically, these oscillations turn out to be large enough to give rise to a jump from one site to the other. The order of magnitude of the frequency of these oscillations is about 1012-1013 Hz. In this regard, it has been shown that the jump rate at which an atom jumps into an empty neighboring site is given by [30]... 

Diffusion of atoms or ions in crystalline solids can occur by at least three possible mechanisms, as shown schematically in Figure 2.7. In some solids, transport proceeds primarily by the vacancy mechanism, in which an atom jumps into an adjacent, energetically equivalent vacant lattice site. The vacancy mechanism is generally much slower than the interstitial mechanism (discussed below). Nonetheless, it is thought to be responsible for self-diffusion in all pure metals and for most substitutional alloys (Shewmon, 1989). 

Figure 2.7 Diffusion of atoms or ions in crystalline solids can occur by at least three possible mechanisms illustrated here. In the vacancy mechanism (bottom arrow), an atom in a lattice site jumps to an adjacent vacant lattice site. In the interstitial mechanism (middle arrow), an interstitial atom jumps to an adjacent vacant interstitial site. In the intersitialcy mechanism (top two arrows), an interstitial atom pushes an atom residing in a lattice site into an adjacent vacant interstitial site and occupies the displaced atom s former site. (After Lalena and Cleary, 2005. Copyright John Wiley Sons, Inc. Reproduced with permission.)...

Due to efficient spin diffusion, homogeneous crystalline solids present a unique relaxation time, regardless of chemical differences. Then different polymorphs can be differentiated by using pulse sequences able to discriminate substances on the basis of their different relaxation properties. In particular cases it has been possible to extract the individual subspectra from a mixture of polymorphs, by modifying the inversion recovery pulse sequence to decompose the spectra [43]. 

In theory, heat in crystalline solids is transferred by three mechanisms (i) electrons (ii) lattice vibrations and (iii) radiation [44], Since zirconia is an electronic insulator (electrical conductivity occurring at high temperatures by oxygen ion diffusion), electrons play no part in the total thermal conductivity of the system. Hence, thermal conduction in zirconia-based ceramics is mainly by lattice vibrations (phonons) or by radiation (photons). 

Broadly viewed, there are two main mechanisms of solid-state diffusion in crystalline materials ... 

In crystalline solids, diffusion occurs by the hopping of atoms from lattice site to lattice site. This hopping process can occur by the hopping of either atoms into unoccupied (i.e., vacant ) sites in the lattice or between unoccupied interstitial spaces in the lattice. Interstitial diffusion is more common for small impurity atoms that can fit into the interstitial spaces in the lattice while vacancy diffusion is more common for larger atoms that can only occupy the regular lattice sites. 

The driving forces provide a motivation for sintering but the actual occurrence of sintering requires transport of matter, which in crystalline solids occurs by a process of diffusion involving atoms, ions, or molecules. Crystalline solids are not ideal in structure. At any temperature they contain various imperfections... 

Numerous chemical reactions or micro-structural changes in solids take place through solid state diffusion, i.e. the movement and transport of atoms in solid phases. In crystalline solids, the diffusion takes place because of the presence of defects. Point defects, e.g. vacancies and interstitial ions, are responsible for lattice diffusion. Diffusion also takes place along line and surface defects which include grain boundaries, dislocations, inner and outer surfaces, etc. As diffusion along linear, planar and surface defects is generally faster than in the lattice, they are also termed high diffiisivity or easy diffusion paths. Another frequently used term is short circuit diffusion. 

The mass transport of charge species owing to concentration gradient in a solid electrolyte is governed by solid-state diffusion. In a solid-state diffusion process, atoms and ions transport through lattice of crystalline structures like in ceramic and other solid nonmetals like polymers owing to the presence of a nonuniform concentration distribution of the migrating elements. 

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