Isotopic Constraints on Earth System Processes. Группа авторов

), the modified EBD model is capable of reproducing the sign and magnitude of uphill diffusion in the K₂O and CaO profiles (Fig. 2.7a–f). The positive C_f for CaO implies that CaO partitions into mafic liquids whereas the negative C_f for K₂O implies that K₂O partitions into felsic liquids, which is consistent with the known partitioning behavior of these two oxide components (Ryerson & Hess, 1978; Watson, 1976). The misfit on the ends of the CaO profiles suggest that the activity of CaO depends on melt components in addition to SiO₂. The next level of complexity would be to include the alkali components in the activity‐composition model for CaO.

      The goodness‐of‐fit for the K₂O profile is even less satisfying due to the unusual shape of the measured K₂O profile, which is nearly flat on the mafic side and linear to nearly convex on the felsic side. This shape cannot be reproduced using the simplified EBD models discussed herein. Nevertheless, the inferred is a reasonable average value, being somewhat overestimated on the rhyolite side and somewhat underestimated on the phonolite side. A better fit could be obtained by casting as a function of SiO₂ concentration (Richter et al., 2003, 2009; Watkins et al., 2009) or by introducing some other additional ad hoc complexity into the model. We did not undertake such efforts because the calculated concentration profiles provide an adequate description of the K₂O fluxes to be used as a baseline for modeling the isotope ratio profiles.

Figure 2.7 The modified EBD model applied to the data. Top panels: SiO₂ diffusion is well‐approximated as effective binary in an infinite medium. Middle panels: Both CaO and K₂O exhibit uphill diffusion, which is attributed in the model to transient partitioning between high‐silica and low‐silica liquids. Bottom panels: Despite the simplifying assumptions, the model captures the sign and magnitude of diffusive isotope effects.

       Model for the Isotope Ratio Profiles

Model isotope ratio profiles are generated by invoking a difference in diffusivity between the isotopes of an element. The dependence of diffusivity on mass can be written as (Richter et al., 1999):

      (2.9)

      where β is a dimensionless empirical parameter. Here, m refers to the isotopic mass and not the mass of an isotopically substituted molecule such as, for example, CaAl₂O₄. This functional form is based on principles from the kinetic theory of gases, but it’s also convenient because it normalizes out the fractional mass difference between isotopes, thereby allowing for direct comparison between different isotopic systems.

      To model the isotope ratio profiles, we assume the rhyolite and phonolite have the same isotopic composition since they are within error of each other with respect to δ ⁴⁴Ca. Although we did not directly measure the δ ⁴¹K of either starting material, the assumption that they are the same can be justified on the basis that the global range in δ ⁴¹K of igneous rocks and minerals is < 1‰ (Morgan et al., 2018). We assume further that the equilibrium isotope ratio profile is uniform at all times; i.e., C_{e, 44} is defined to be proportional to C_{e, 40} such that the ratio C_{e, 44}/C_{e, 40} is constant, which implies that there is no equilibrium isotopic partitioning between the liquids. Hence, any isotopic fractionations in the model arise solely from diffusive fluxes and the difference in isotopic diffusion coefficients.

      Model isotopic profiles are compared to the data in the bottom panels of Fig. 2.7. For Ca isotopes, no single set of parameters ( , D_b, C_f, and β ) can explain the full profile or its time evolution reflected in the 2.5‐ and 6‐hour runs. Nevertheless, the fits are qualitatively no worse than those used to estimate β factors and associated uncertainties in other natural silicate liquid diffusion couple experiments (Richter et al., 2003; Watkins et al., 2009). By plotting model curves corresponding to different values of β against the data, we estimate β = 0.10 ± 0.02. This is a somewhat crude estimate, however, and appears to be on the low side for the rhyolite while fitting the phonolite in the 2.5‐hour run, and on the low side for the phonolite while fitting the rhyolite in the 6‐hour run. For K isotopes, we estimate β = 0.25 ± 0.03, which is comparable to the value for Li ( β = 0.21–0.23) in basalt‐rhyolite and rhyolite‐rhyolite diffusion couples (Holycross et al., 2018; Richter et al., 2003).

      2.5.3. Comparison to Previous Studies

It was shown previously that β factors correlate with the diffusivity of the cation normalized by that of Si (or SiO₂) (Fig. 2.8). Since Si is strongly bound in multi‐atom complexes with O (as well as Al and other Si atoms), it diffuses more slowly than other elements, and hence the ratio D_i/D_Si is generally greater than unity. For elements present in major quantities, the ratio D_i/D_Si is typically close to unity because of the cooperative motion that is required to yield a net flux of counter‐diffusing major elements. The regime where D_i/D_Si is high tends to occur in more silica‐rich liquids and with decreasing temperature (Dingwell, 1990).

Figure 2.8 β factors from this study (points with an asterisk) compared to literature values. The quantity /D_Si is used as a proxy for extent to which a diffusing cation is coupled to the melt matrix. The Ca β factor is offset from the overall trend, but this may be because is overestimated. The β factor for K is the largest value yet reported for diffusion in silicate liquids.

      The ratio /D_Si is inferred to be about 20, and the β factor for Ca is offset from the overall trend. The ratio /D_Si is inferred to be about 35 and the β factor for K is the highest value yet determined for diffusion in silicate liquids. Despite K₂O being present in major quantities, it behaves like a trace element insofar as its diffusion is kinetically decoupled from the motion of the slower‐moving aluminosilicate structures (cf. Dingwell, 1990). The high β factor suggests that large