Effective binary diffusivity

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. 

Interdiffusion, effective binary diffusion, and multicomponent diffusion may be referred to as chemical diffusion, meaning there are major chemical concentration gradients. Chemical diffusion is defined relative to self diffusion and tracer diffusion, for which there are no major chemical concentration gradients. 

When one refers to the diffusion equation, it is usually the binary diffusion equation. Although theories for multicomponent diffusion have been extensively developed, experimental studies of multicomponent diffusion are limited because of instrumental analytical error and theoretical complexity, and there are yet no reliable diffusivity matrix data for practical applications in geology. Multicomponent diffusion is hence often treated as effective binary diffusion by treating the component under consideration as one component and combining all the other components as the second component. 

Despite the various drawbacks, the effective binary approach is still widely used and will be widely applied to natural systems in the near future because of the difficulties of better approaches. For major components in a silicate melt, it is possible that multicomponent diffusivity matrices will be obtained as a function of temperature and melt composition in the not too distant future. For trace components, the effective binary approach (or the modified effective binary approach in the next section) will likely continue for a long time. The effective binary diffusion approach may be used under the following conditions (but is not limited to these conditions) with consistent and reliable results (Cooper, 1968) ... 

If initial concentration gradients are in two complementing components, such as FeO and MgO (with the sum of FeO and MgO molar concentrations being a constant), and all other components have uniform concentration, the diffusion between the two components may be treated as interdiffusion, or effective binary diffusion. 

In all of the three cases above, components other than the discussed major components often cannot be treated as effective binary diffusion, especially for a component whose concentration gradient is small or zero compared to other components. Such a component often shows uphill diffusion. 

As long as care is taken so that effective binary diffusivity obtained from experiments under the same set of conditions is applied to a given problem, the approach works well. Although the limitations mean additional work, because of its simplicity and because of the unavailability of the diffusion matrices, the effective binary diffusion approach is the most often used in geological systems. Nonetheless, it is hoped that effort will be made in the future so that multi-component diffusion can be handled more accurately. 

If the difference in concentration is in one component only, e.g., one side contains a dry rhyolite, and the other side is prepared by adding H2O to the rhyolite, then the main concentration gradient is in H2O, and all other components have smaller concentration gradients. The diffusion of H2O may be treated fairly accurately by effective binary diffusion. In other words, the diffusion of the component with the largest concentration gradient may be treated as effective binary, especially if the component also has high diffusivity. The diffusion of other components in the system may or may not be treated as effective binary diffusion. 

If the difference in concentration is in only two exchangeable components, such as FeO and MgO, the interdiffusion in a multicomponent system may be treated as effective binary. The diffusion of other components in the system may or may not be treated as effective binary diffusion. 

In principle, the diffusion matrix approach can be extended to trace elements. My assessment, however, is that in the near future diffusion matrix involving 50 diffusing components will not be possible. Hence, simple treatment will still have to be used to roughly understand the diffusion behavior of trace elements the effective binary diffusion model to handle monotonic profiles, the modified effective binary diffusion model to handle uphill diffusion, or some combination of the diffusion matrix and effective binary diffusion model. 

Even in the absence of uphill diffusion, a trace element concentration profile often does not match that for a constant diffusivity by using the effective binary diffusion treatment. Hence, the effective binary diffusivity depends on the chemical composition, which is expected. 

If diffusivity is extracted from the profile of the isotopic fraction of an element, it may differ significantly from, and often greater than, the effective binary diffusivity obtained from the concentration profile of the trace or minor element. 

To quantify the diffusion profiles is a difficult multicomponent problem. The activity-based effective binary diffusion approach (i.e. modified effective binary approach) has been adopted to roughly treat the problem. In this approach. 

Tracer diffusivities are often determined using the thin-source method. Self-diffusivities are often obtained from the diffusion couple and the sorption methods. Chemical diffusivities (including interdiffusivity, effective binary diffusivity, and multicomponent diffusivity matrix) may be obtained from the diffusion-couple, sorption, desorption, or crystal dissolution method. 

For inter diffusion between same-valence ions (ionic exchange) in an aqueous solution, or a melt, or a solid solution such as olivine (Fe +, Mg +)2Si04, an equation similar to Equation 3-135c has been derived from the Nemst-Planck equations first by Helfferich and Plesset (1958) and then with refinement by Barter et al. (1963) with the assumption that (i) the matrix (or solvent) concentration does not vary and (ii) cross-coefficient Lab (phenomenological coefficient in Equation 3-96a) is negligible, which is similar to the activity-based effective binary diffusion treatment. The equation takes the following form ... 

For the calculation of convective dissolution rate of a falling crystal in a silicate melt, the diffusion is multicomponent but is treated as effective binary diffusion of the major component. The diffusivity of the major component obtained from diffusive dissolution experiments of the same mineral in the same silicate melt is preferred. Diffusivities obtained from diffusion-couple experiments or other types of experiments may not be applicable because of both compositional effect... 

Table 4-4 Effective binary diffusivities ((im /s) in "dry" silicate meits at 1573 K...

Crystai growth distance and behavior of major component This problem is similar to diffusive crystal dissolution. Hence, only a summary is shown here. Consider the principal equilibrium-determining component, which can be treated as effective binary diffusion. The density of the melt is often assumed to be constant. The density difference between the crystal and melt is accounted for. 

Behavior of trace element that can be treated as effective binary diffusion The above discussion is for the behavior of the principal equilibrium-determining component. For minor and trace elements, there are at least two complexities. One is the multicomponent effect, which often results in uphill diffusion. This is because the cross-terms may dominate the diffusion behavior of such components. The second complexity is that the interface-melt concentration is not fixed by thermodynamic equilibrium. For example, for zircon growth, Zr concentration in the interface-melt is roughly the equilibrium concentration (or zircon saturation concentration). However, for Pb, the concentration would not be fixed. 

If the diffusion of a minor or trace element can be treated as effective binary (not uphill diffusion profiles) with a constant effective binary diffusivity, the concentration profile may be solved as follows. The growth rate u is determined by the major component to be n D ff, and is given, not to be solved. Use i to denote the trace element. Hence, w, and Dt are the concentration and diffusivity of the trace element. Note that Di for trace element i is not necessarily the same as D for the major component. The interface-melt concentration is not fixed by an equilibrium phase diagram, but is to be determined by partitioning and diffusion. Hence, the boundary condition is the mass balance condition. If the boundary condition is written as w x=o = Wifl, the value of Wi must be found using the mass balance condition. In the interface-fixed reference frame, the diffusion problem can be written as... 

Cooper A.R. and Varshneya A.K. (1968) Diffusion in the system K20-Sr0-Si02,1 effective binary diffusion coefficients. /. Am. Ceram. Soc. 51, 103-106. 

A modified effective binary diffusion model. /. Geophys. Res. 98, 11901-... 

For ideal gases the effective binary diffusion coefficient can be calculated from molecular properties (see Appendix A). The film thickness, 5, is determined by hydrodynamics. Correlations are given in the literature which allow the calculation of the transfer coefficient in the case of equimolar counterdiffusion, kf, rather than the film thickness, 5 ... 

Hence the so-called effective binary diffusion coefficient, D m is introduced. The latter is defined as ... 

For ideal gases an expression for the effective binary diffusion coefficient, D m, as a function of the binary diffusion coefficient, DA], can be derived as in this case the Stefan-Maxwell equation holds ... 

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