Partitioning solid-melt partition coefficients

A series of studies has been made by Yalkowsky and co-workers. The so-called general solubility equation was used for estimating the solubility of solid nonelectrolytes [17, 18]. The solubility log S (logarithm of solubility expressed as mol/L) was formulated with log P logarithm of octanol/water partition coefficient), and the melting point (MP) as shown in Eq. (11). This equation generally... 

One possibility for increasing the minimum porosity needed to generate disequilibria involves control of element extraction by solid-state diffusion (diffusion control models). If solid diffusion slows the rate that an incompatible element is transported to the melt-mineral interface, then the element will behave as if it has a higher partition coefficient than its equilibrium partition coefficient. This in turn would allow higher melt porosities to achieve the same amount of disequilibria as in pure equilibrium models. Iwamori (1992, 1993) presented a model of this process applicable to all elements that suggested that diffusion control would be important for all elements having diffusivities less than... 

If chemical equilibrium between the melt and the solid is assumed throughout the melting column, the definition of the partition coefficient (D) ... 

Where /yand ps are the melt and solid densities, respectively, melt velocity. W is the solid upwelling velocity and A is the bulk partition coefficient. Note that if the partition coefficient A is 1, the effective velocity approaches the melt velocity, and also that the difference in effective velocity between elements with different Di decreases at larger porosities. In the following text, the subscripts 0, 1, and 2 are taken to refer to °Th, and Ra, respectively. 

Sediment may be added by bulk mixing via imbricate thnisting (Bebout and Barton 2002), dehydration (Class et al. 2000), or melting (Johnson and Plank 1999). The latter two may differ in their P-T conditions and, therefore, residual mineralogy as well as relevant partition coefficients. In general, fluids are less effective transport agents than melts (i.e., trace elements are more soluble in melt than in pure water or even brine), but fluid/solid partitioning can fractionate some elements, notably Ba-Th and U-Th, more than melt/solid. However, as pressure increases, the distinction between fluid and melt decreases as their mutual solubility increases and they approach a critical end-point. 

Since we do not know the solid-liquid oxygen partition coefficient, we must resort to an approximation. Most oxygen atoms are ieO and crystallization rarely changes the total oxygen concentration of the residual silicate melt very significantly. We can assume... 

The approach just used for fractional crystallization can be transposed immediately to fractional melting, a process by which each packet of melt is withdrawn from the source thereby prevented from equilibration with the solid. Again, these equations will be developed in Chapter 9, but the present section emphasizes a representation which does not require constant Berthelot-Nernst partition coefficients, and therefore is more useful for major elements. 

Figure 4.7 Assumed probability density function for the degree of melting F (top). Resulting probability density functions for the reduced solid concentration of element i upon fractional melting (middle) and batch melting (bottom) for different solid-liquid partition coefficients D,.

The Uj and w, deviates are normal deviates, the deviates vf are log-normal. See text for the description of the computed random variables K represents mineral-liquid partition coefficients, F the fraction of residual melt, x the fraction of a mineral in the cumulate, D bulk solid-liquid partition coefficients. 

The best way to convince ourselves that this rather convoluted technique works well is to build a synthetic example that we invert in a second stage. We use four elements (m = 4, elj to el4), five lavas (s = 5) for which we assume the melt fraction and residual mineral abundances listed in Table 9.2, and two non-sterile residual minerals (Mini and Min2) whose partition coefficients are listed in Table 9.3. The assumed source composition is listed in Table 9.3 which also shows the assumed bulk solid-liquid partition coefficients for each lava. 

Figure 9.6 Comparison of the equilibrium [equation (9.2.2)] and fractional melting [equation (9.3.15)] models for a bulk solid-liquid partition coefficient Dt of 0.1 (top) and 2 (bottom). Although the concentrations predicted by the two models diverge rapidly for incompatible elements in instantaneous melts, they remain virtually identical for compatible elements.

These equations converge towards those of the fractional melting model for tp Dh and, contrary to McKenzie (1985) equation (29), CUq tends to C0 when porosity partition coefficient are of the same order of magnitude, large variability is achieved in both the solid and the residue, a point which will be returned below. Considerable attention has been recently focussed on this model which may explain the fractionation of some strongly incompatible nuclides in the U decay series (McKenzie, 1985 Williams and Gill, 1989 Beattie, 1993). 

In this case, trace element and carrier occupy the same structural position both in the solid phase and in the melt and are subject to the same compositional effects in both phases (i.e., extension of the cation matrix in the melt and amount of anorthite component in the solid). Figure 10.9A shows the effect of normalization the conventional partition coefficient of Sr between plagioclase and liquid varies by about one order of magnitude under equal P-T conditions, with increasing anorthite component in solid solution, whereas normalized distribution coefficient D is virtually unaffected. Figure 10.9B shows the same effect for the Ba-Ca couple. 

In the case of interface equilibrium (open system conditions), the partition coefficient is valid only at the interface between solid and liquid (or at zero distance from the interface) and at time of crystallization (or melting) t ... 

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