Fractional melting

During batch melting, melt always remains in equilibrium with solid. In contrast, during fractional melting, melt is extracted as soon as it is generated and only the last drop of extracted melt is in equilibrium with the solid. 

There are several ways to derive the fundamental equation for fractional melting. Here we choose an approach that can be easily understood. 

Cj is the concentration in the solid and C is the concentration in the accumulated extracted melt and is related to the instantaneous melt C, by 

The solid is in equilibrium with the instantaneous melt, or, the last drop of the extracted (aggregated) melt by the following relationship 

Note that the residual solid is in equilibrium with instantaneous melt, rather thtui extracted melt. The relationship between C and C, can be rewritten as 

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crystallization can be transposed immediately to fractional melting, a process by which each packet of melt is withdrawn from the source thereby prevented from equilibration with the solid.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. 

As a parallel to equation (1.5.2), we define Dt as the ratio of the concentration mJ/Msol of species i in the residual solid to the concentration dm iq /dMliq in the last increment of liquid 

If Dni2 and Dn 3 are constant, the two ratios define a straight-line in a logarithmic plot and equation (1.5.32) can be integrated as 

Although less compatible than Mg, Al apparently is still well retained by the residue during melting, which supports the presence of an aluminous phase in the residue as suggested by Johnson et al. (1990). If we knew the Al203/Mg0 ratio of the whole rock prior to melting, it would be possible to calculate that ratio in the last liquid extracted and compare its value with mid-ocean ridge basalt (MORB) values. = 

The general expressions for the more probable case where minerals do not enter the melt in their modal proportions are given by 

Three types of fracdonal crystallizadon are considered here — equilibrium crystalHzadon, Rayleigh firacdonadon and in situ crystaUi2adon. 

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Several models relating the isotopic effects of U-series disequilibria to the timescales of the melting process have now been proposed (e.g., McKenzie 1985 Williams and Gill 1989 Spiegelman and Elliott 1993 Qin 1992 Iwamori 1994 Richardson and McKenzie 1994). While these models differ mainly in their treatment of the melt extraction process (i.e., reactive porous flow vs near fractional melting), because they incorporate the effect... 

When plotted in a In (Al203/Mg0) vs ln(FeO/MgO) (Figure 1.12), the clinopyroxene data define a unique trend which is reasonably linear, thereby supporting the hypothesis of fractional melting made by Johnson et al. (1990). In addition, the linear array supports a homogeneous source and rather constant 0MgoFeO = 0-3 and >MgoAl2° fractionation coefficients. 

Figure 1.12 Al203/Mg0 vs FeO/MgO plot of peridotite clinopyroxenes from near the Antarctica-Africa-America triple junction (Tablel.13 data from Johnson et al, 1990). The linear array [equation (1.5.32)] supports the assumption that a homogeneous peridotitic source has experienced fractional melting with DMgOA,2° 0.18.

The concentration of an element i in the residual solid after extraction of a liquid fraction F by fractional melting is given by equation (9.3.14)... 

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,.

Batch partial melting will hereafter be understood as equilibrium melting, which is in contrast to fractional melting discussed in Section 9.3.3. The foundation of this model is remarkably simple and was first laid down by Schilling and Winchester (1967). A number of more or less complex modifications enabling useful information to be extracted from the data were later introduced by Gast (1968), Shaw (1970) and Albarede (1983). Bulk equilibrium crystallization of a liquid batch can be handled with equations identical to those for batch-melting. 

Fractional melting processes are even more efficient than equilibrium melting in fractionating incompatible elements for small fractions of melt since... 

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.

We can again introduce Shaw s Pt variables, which we assume to be constant (eutectic melting), and change variables according to equation (9.2.14). Thereupon, the differential form of the fractional melting equation can be rewritten... 

The difficulty of applying the fractional melting model is the discontinuous character of the melting process (e.g., Presnail, 1969). Whenever a mineral phase is exhausted, the progress of fractional melting requires temperature jumps of expectedly large amplitude and discontinuous variations in melt chemistry which are not in general well-documented in natural examples. 

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). 

The value of C,iq at z=0 can be the concentration of a liquid generated by batch- or fractional melting from the same source or that of an exotic liquid introduced at the... 

As for fractional crystallization and fractional melting, element-element plots with a logarithmic scale should show straight lines for the solid as well as for the liquid, since both differ by a constant coefficient. Contrary to fractional crystallization but similar to fractional melting, discussed above, and to percolation, to be presented below, zone-melting is a very powerful process to separate incompatible elements. 

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