Diffusion in garnets

Careful readers might notice that diffusion in garnet is three-dimensional with spherical geometry, and should not be treated as one-dimensional diffusion. Section 5.3.2.1 addresses this concern. 

Chakraborty S. and Ganguly G. (1991) Compositional zoning and cation diffusion in garnets. In Diffusion, Atomic Ordering, and Mass Transport Selected Topics in Geochemistry (ed. J. Ganguly). Springer, New York. 

Loomis T. P. (1978b) Multicomponent diffusion in garnet II. Comparison of models with natural data. Am. J. Sci. 278, 1119-1137. 

Chakraborty S. and Ganguly J. (1992). Cation diffusion in aluminosilicate garnets Experimental determination in spessartine-almandine diffusion couples, evaluation of effective binary diffusion coefficients, and applications. Contrib. Mineral Petrol, 111 74-86. 

Diffusion in a system with three or more components is called multicomponent diffusion. One example is diffusion of Ca, Fe, Mn, and Mg in a zoned garnet (Ganguly et al., 1998a). Another example is diffusion between an andesitic melt... 

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... 

Figure 3-5 MnO partition between and diffusion in two minerals, olivine and garnet. Diffusional anisotropy of olivine is ignored. Initially, MnO in both phases were 0.2 wt%. As the two minerals come into contact, there will be diffusion to try to reach the equilibrium state. The partition coefficient (Mn)oiiv/(Mn)gt is assumed to be 0.59. The diffusivity in olivine is assumed to be 10 times that in garnet, resulting in a wider diffusion profile with a smaller slope in olivine.

Example 4.1. Suppose olivine and garnet are in contact and olivine is on the left-hand side (x<0). Ignore the anisotropic diffusion effect in olivine. Suppose Fe-Mg interdiffusion between the two minerals may be treated as one dimensional. Assume olivine is a binary solid solution between fayalite and forsterite, and garnet is a binary solid solution between almandine and pyrope. Hence, Cpe + CMg= 1 for both phases, where C is mole fraction. Let initial Fe/(Fe- -Mg) = 0.12 in olivine and 0.2 in garnet. Let Xq = (Fe/Mg)gt/ (Fe/Mg)oi = 3, >Fe-Mg,oi = 10 ° mm+s, and Dpe-Mg,gt =... 

Some minerals display miscibility gaps. If the boundary between two compositional zones corresponds to a miscibility gap, then there will be both partitioning and diffusion across the boundary. Because the miscibility gap widens as temperature decreases, the concentrations on the two sides are further separated rather than smoothed out. Figure 5-25b shows a hypothetical example. Wang et al. (2000) used this property to make the first direct observation of immisci-bility in garnet. 

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 ... 

The following table gives measured Fe concentrations in garnet as a function of distance from the center. Treat the diffusion profile as a spherical diffusion couple. Fit the data to find jDdt... 

Elphick S.C., Ganguly J., and Loomis T.P. (1985) Experimental determination of cation diffusivities in aluminosilicate garnets, I experimental methods and interdiffusion data. Contrib. Mineral. Petrol. 90, 36-44. 

Ganguly J., Cheng W., and Chakraborty S. (1998a) Cation diffusion in aluminosilicate garnets experimental determination in pyrope-almandine diffusion couples. Contrib. Mineral. Petrol. 131, 171-180. 

Ganguly J., Tirone M., and Hervig R.L. (1998b) Diffusion kinetics of samarium and neodymium in garnet, and a method for determining cooling rates of rocks. Sdence 281, 805-807. 

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