Isotopic Constraints on Earth System Processes. Группа авторов

(see Supplemental Information 2.A). Regardless, for a given transient profile of SiO₂, there exists an associated hypothetical “quasi‐equilibrium” profile for the equilibrium concentration of the component of interest, C_e, that scales with 1/ γ as well as the concentration of SiO₂. Assuming that diffusion of SiO₂ can be characterized by a concentration‐independent EBD, D_b, and that diffusion of SiO₂ does not reach either end of the diffusion couple, the “equilibrium” concentration profile for C_e is an error function:

(2.6)

where C₋ and C₊ are the initial concentrations at x < 0 and x > 0 for the diffusion couple and C_f is the difference in the equilibrium concentration between the two liquids, as depicted in Fig. 2.5. The parameter C_f can be positive or negative, depending on whether the component preferentially partitions into the high‐silica or low‐silica liquid. A large C_f implies strong preference for one liquid versus the other. If C_f = 0, the model reverts back to the simplified EBD model. Replacing γ with 1/C_e in equation 2.5 yields

(2.7)

      which leads to the one‐dimensional diffusion equation (Eq. 9a in Zhang, 1993):

(2.8)

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/C_e, where k is the proportionality constant between C_e 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 C_f) 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 C_f is negative (Fig. 2.6a), the element partitions into the high‐silica liquid and the hypothetical C_e 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 C_f=0, resulting in a pair of bumps in the concentration profile. If C_f is positive (Fig. 2.6b), the activity gradient is opposite the concentration gradient and uphill diffusion occurs. Note that in both cases, the C_e profile evolves at a rate given by D_b (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%, /D_b = 8, and variable C_f. 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 C_f, 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 D_b 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 (C_f, D_b,