Diffusion in spinel

Bhattacharya, J. and A. Van der Ven. 2011. First principles study of competing mechanisms of non-dilute Li diffusion in spinel Li -TiSj. Physical Reviezvs B, 83,.l-9. 

Armijo [1198] has discussed a number of features of the kinetics and mechanisms of spinel formation. Under suitable conditions, the rates of some high temperature spinel syntheses can be studied gravimetrically [1199]. Holt [1180] has been concerned with the role of oxygen diffusion in CoA1204 formation. 

Let us now turn to diffusion in the general case, without worrying about the exact mechanism or the rates of diffusion of the various species. As an example to illustrate how we would analyze a diffusion-limited solid state reaction, we use the general equation describing formation of a compound with spinel (cubic) structure and stoichiometry ... 

We have now presented all of the possible diffusion reactions in spinel synthesis. These are summarized as follows ... 

If the diffusion medium is isotropic in terms of diffusion, meaning that diffusion coefficient does not depend on direction in the medium, it is called diffusion in an isotropic medium. Otherwise, it is referred to as diffusion in an anisotropic medium. Isotropic diffusion medium includes gas, liquid (such as aqueous solution and silicate melts), glass, and crystalline phases with isometric symmetry (such as spinel and garnet). Anisotropic diffusion medium includes crystalline phases with lower than isometric symmetry. That is, most minerals are diffu-sionally anisotropic. An isotropic medium in terms of diffusion may not be an isotropic medium in terms of other properties. For example, cubic crystals are not isotropic in terms of elastic properties. The diffusion equations that have been presented so far (Equations 3-7 to 3-10) are all for isotropic diffusion medium. 

The diffusion couple discussed above consists of two halves of the same phase. If the two halves are two minerals, such as Mn-Mg exchange between spinel and garnet (Figure 3-5), there would be both partitioning and diffusion. Define the diffusivity in one half (x < 0) to be D, and in the other half (x > 0) to be D. Both and are constant. Let w be the concentration (mass fraction) of a minor element (such as Mn). The initial condition is... 

Often it is necessary to treat diffusion between different layers as three dimensional diffusion. For isotropic minerals such as garnet and spinel (including magnetite), diffusion across different layers may be considered as between spherical shells, here referred to as "spherical diffusion couple." Oxygen diffusion in zircon may also be treated as isotropic because diffusivity c and that Tc are roughly the same (Watson and Cherniak, 1997). If each shell can be treated as a semi-infinite diffusion medium, the problem can be solved (Zhang and Chen, 2007) as follows ... 

Ryerson F.J. and McKeegan K.D. (1994) Determination of oxygen self-diffusion in aker-manite, anorthite, diopside, and spinel implications for oxygen isotopic anomalies and the thermal histories of Ca-Al-rich inclusions. Geochim. Cosmochim. Acta 58, 3713-3734. 

Let us systematize the possible boundary conditions for cation diffusion in a spinel. Since in the ternary system (at a given P and T) the chemical potentials of two components are independent, we may distinguish between three different transport situations. If A denotes a change across the product layer and O and AO are chosen as the independent components, the possibilities are... 

Intercalation of cations into a framework of titanium dioxide is a process of wide interest. This is due to the electrochromic properties associated with the process (a clear blue coloration results from the intercalation) and to the system s charge storage capabilities (facilitated by the reversibility of the process) and thus the potential application in rocking-chair batteries. We have studied alkali-metal intercalation and ion diffusion in the Ti02 anatase and spinel crystals by theoretical methods ranging from condensed-phase ab initio to semiempirical computations [65, 66]. Structure relaxation, electron-density distribution, electron transfer, diffusion paths and activation energies of the ion intercalation process were modeled. 

Just as the edges of cation-centred polyhedra represent anion diffusion paths in a crystal, the edges of anion centred polyhedra represent cation diffusion paths. The polyhedra shown in Figures 7.18 reveal that cation diffusion in cubic close-packed structures will take place via alternative octahedral and tetrahedral sites. Direct pathways, across the faces of the polyhedron, are unlikely, as these mean that a cation would have to squeeze directly between two anions. There is no preferred direction of diffusion. For ions that avoid either octahedral or tetrahedral sites, for bonding or size reasons, diffusion will be slow compared to ions which are able to occupy either site. In solids in which only a fraction of the available metal atom sites are filled, such as the spinel structure, clear and obstructed diffusion pathways can easily be delineated. 

There are two (2) different cases that we can distinguish, both of which represent possible diffusion mechanisms in spinel. These are Ion diffusion or Gaseous. Let us consider the ion-diffusion mechanism first. This is shown in the following diagram. 

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