where the units of concentration may be in weight percent if density can be assumed to be constant. The reason for the approximation symbol in equation 2.5 is that Zhang (1993) uses equation 2.8 to calculate concentration profiles and then calculates activity profiles from the relation a = kC/Ce, where k is the proportionality constant between Ce and 1/ γ . A point that was nuanced by Zhang (1993) is that equations 2.5 and 2.7 are not equivalent, but rather are scaled by a factor k, which can be determined from the constraint that a = C at x = 0 (interface).
Figure 2.5 The Zhang model and its parameters are based on the concept of elemental partitioning in an SiO2 gradient. The initial concentration distribution is C+ for x > 0 and C− for x < 0. At equilibrium, the concentration contrast may be reversed (depending on the sign of Cf) such that C+, eq for x < 0 and C−, eq for x > 0.
Model Validation and Behavior
The Zhang (1993) model deviates significantly from the EBD model when the component of interest diffuses significantly faster than SiO2. In this situation, which is common in silicate liquids, the component of interest will not reach homogeneity quickly, but rather, will partition along the compositional continuum between the high‐ and low‐silica liquids and reach compositional homogeneity only as fast as SiO2 diffuses.
The behavior of the model is shown in Fig. 2.6 for an infinite diffusion couple where the initial concentration gradient is small and = 8. If Cf is negative (Fig. 2.6a), the element partitions into the high‐silica liquid and the hypothetical Ce gradient opposes the concentration gradient. In this case, mass diffuses from the high concentration to the low concentration side at a higher rate than if Cf=0, resulting in a pair of bumps in the concentration profile. If Cf is positive (Fig. 2.6b), the activity gradient is opposite the concentration gradient and uphill diffusion occurs. Note that in both cases, the Ce profile evolves at a rate given by Db (equation 2.6) and is thus less evolved than the concentration profiles exhibiting uphill diffusion.
Figure 2.6 Model behavior showing concentration, activity, and transient equilibrium concentration profiles for C− = 6 wt%, C+ = 8 wt%, /Db = 8, and variable Cf. The top panel is identical to Fig. 2c in Zhang (1993) to validate the Matlab code used in calculations herein. For an element with small initial concentration contrast, the sign and magnitude of uphill diffusion depends on the sign and magnitude of Cf, which describes how the element partitions between high silica versus low silica liquids.
Model Applied to the Rhyolite‐Phonolite Couples
The rhyolite‐phonolite diffusion couple spans a broad range of melt compositions. Although the diffusion coefficients are expected to be sensitive to melt composition across this range, we assume and Db are constant. We also treat the diffusion couple as infinite with respect to SiO2 even though we see some uphill diffusion of SiO2 at the ends of the couple in the 2.5 hour run. This, however, may just be due to mass balance and dilution effects caused by the faster diffusing components. For the faster diffusing K2O and CaO components, the diffusion couple is not treated as infinite; i.e., ∂C/∂x = 0 at the boundaries.
Despite the many simplifying assumptions that are needed to distill a general multicomponent diffusion problem down to a three‐parameter model (Cf, Db, and ), 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 Cf for CaO implies that CaO partitions into mafic liquids whereas the negative Cf 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.
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