Isotopic Constraints on Earth System Processes. Группа авторов

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coefficient (i.e., n = 2 in equation 1.3) and thus given a diffusion profile of a component i one can in most cases fit the data using equation 1.2 to determine the effective binary diffusion coefficient for component i. Cooper (1968) showed how the effective binary diffusion coefficient of a given component is related to the full diffusion matrix of the system, and he also pointed out that effective binary diffusion coefficients depend not only on the local composition and thermodynamic state (e.g., temperature and pressure) but also on the direction of the diffusive flux in composition space. It follows from this that in order to determine the appropriate effective binary diffusion coefficient for modeling a natural diffusion profile, one needs to run an experiment with a diffusion couple juxtaposing compositions that are as close as possible to those of the far‐field values of the natural system.

      1.2.3. Self‐Diffusion Coefficients

A very basic type of diffusion coefficient is the self‐diffusion coefficient that measures of the intrinsic mobility of a species. A common experimental approach for measuring self‐diffusion coefficients is to locally alter the isotopic composition of the element of interest in an otherwise homogeneous medium and observe the rate at which the isotopic difference spreads. In effect, this eliminates all diffusive coupling to the other species because their gradients are all zero. The self‐diffusion coefficient

of elements in silicate liquids can be very different from each other. For example, Liang et al. (1996a) reported a large set of experimentally determined self‐diffusion coefficients as function of composition in molten CaO‐Al₂O₃‐SiO₂ showing that ~ 10×, ~2×, and ~1 to 2× . These differences in self‐diffusion are in marked contrast to the very similar magnitude of the effective binary diffusion coefficients in a molten rhyolite‐basalt diffusion couple. Fig. 1.1 shows this by overlaying diffusion profiles measured by Richter et al. (2003) in a diffusion couple in which natural rhyolite liquid and a natural basaltic liquid were juxtaposed and annealed in a piston cylinder experiment. The remarkably similar diffusive behavior the major oxides shown in Fig. 1.1 is the result of the fluxes of MgO, CaO, FeO, and K₂O all being strongly coupled to the concentration gradient of SiO₂ via the off‐diagonal terms of the diffusion matrix. There are two reasons for this strong coupling with SiO₂. One reason is that the sum of volume fluxes of all the components must add up to zero and thus the fluxes are rate limited by the large and sluggish volume flux of SiO₂ that has to balance the fluxes of the other components. The second reason for the coupling is that the chemical potential gradients of the major oxides, which drive their diffusion, depend on the local SiO₂ content of the melt (see Liang et al., 1967 for a detailed discussion of the causes of diffusive coupling in silicate liquids).

      1.2.4. Thermal (Soret) Diffusion Coefficients

      The flux equation of a system that is inhomogeneous in both composition and temperature will have terms proportional to both the chemical gradient and the temperature gradient. Expanding the binary diffusion representation of the flux to include the effect of a temperature gradient results in a binary flux equation of the form (Tyrell, 1961)

Figure 1.1 Normalized concentration of CaO, MgO, K2O, FeO, and SiO2 measured across glass recovered from a molten rhyolite‐basalt diffusion experiment are superimposed to show the similarity of the effective diffusivity of all the major oxide components. For the purpose of this comparison, the concentration differences were normalized by (C‐CL)/(CH‐CL), where C is the concentration of a component along the profile, CL is its concentration in the low end‐member, and CH is its concentration in high end‐member.

      Figure taken from Richter et al. (2003).

      (1.6)

      where ρ is the bulk density,

is the effective binary diffusion coefficient of i in a mixture of components i and j, X_i and X_j are the mass fractions of i and j, and σ _i is the Soret coefficient. For present purposes, the term “Soret diffusion” will be used when the mass transport in a silicate liquid is due to fluxes driven by both chemical and temperature differences.

1.3. KINETIC ISOTOPE FRACTIONATION DURING DIFFUSION BETWEEN NATURAL MELTS

The earliest report of measurable differences in the mobility of isotopes of a major element in a silicate liquid was reported by Richter et al. (1999) based on the results of a diffusion experiment involving an isothermal and isochemical CaO‐Al₂O₃‐SiO₂ melt with an initial step with different concentration of ⁴⁸Ca and ⁴⁰Ca but with the same isotopic ratio of ⁴⁸Ca/⁴⁰Ca on both sides of the step. When annealed, this setup would cause ⁴⁸Ca and ⁴⁰Ca to race each other as they diffused into the low concentration side of the couple. A well‐resolved isotopic fractionation in the glass recovered from this experiment showed that ⁴⁰Ca had diffused measurably faster than ⁴⁸Ca. The relative mobility of the calcium isotopes was reported as D₄₈ /D₄₀ = (40/48) ^β with β = 0.075 ± 0.025. Subsequent laboratory experiments by Richter et al. (2003; 2008; 2009b; 2014a) addressed the question of whether the relative mobility of isotopes seen in isochemical experiments would also arise in molten systems where chemical potential gradients are present and where there can be significant coupling between the various diffusing components of the melt.

      1.3.1. Laboratory Experiments Documenting Ca Isotope

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