Elements incompatible

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... 

Figure 8.19 Evolution of the liquid concentration at the interface with a solid growing at the constant rate v from a solution initially at C0. K is the solid-liquid partition coefficient. Steady-state takes longer to establish for incompatible elements.

Incompatible-element ratios (e.g., Th/La, Nb/Zr, Ce/Yb in basalts) are therefore expected to be very insensitive to mineral separation from the melt and, for differentiated lavas, can be used as a parameter characteristic of their parent liquid see below). 

Changes in concentrations induced by fractional crystallization are much more visible for compatible than for incompatible elements, o... 

For very small partition coefficients (incompatible elements), equation (9.3.13) becomes... 

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.

Clearly, the averaging process decreases the efficiency of fractionation between incompatible elements. 

The residence time of a trace element is xJaLi. compatible elements can be thought of as reactive and have shorter residence times than inert incompatible elements. As shown in Chapter 7, equation (9.4.4) can be integrated from 0 to t into equation (7.2.12)... 

The corresponding expression for isotopic (or incompatible-element) ratios is given by DePaolo (1981). Let us label i 1 and /2 the two isotopes of the same element. We further assume that their partition coefficient is identical, as are their r and z, values. Dividing equation (9.4.17) for isotope i2 by the corresponding equation for isotope il, we get... 

Incompatible elements can achieve very large enrichment in the liquid. Steady-state is achieved over a characteristic length in proportion with (ktL/ktR)L. For small porosities, this length is in the order of (4>/)L for incompatible elements, in the order of L for compatible elements (Figure 9.12). A very small fraction

limit concentration ( C0 / >) and the characteristic length of incompatible elements. 

Figure 9.12 The zone-refining model described by simplified equation (9.4.27) for a completely molten zone. Concentration in the solid left behind the zone for different values of the bulk solid-liquid D,. Steady-state is achieved over distances much shorter for compatible than for incompatible elements.

Figure 9.13 The zone-refining model with an infinite number of passes determination through equation (9.4.32) of the length in the exponential distribution of solid concentrations described by equation (9.4.31). Incompatible elements are such that the kf/kf ratio is nearly equal to the ratio of residual porosity to the degree of melting and therefore are efficiently skimmed downstream ( , L).

This relationship has been displayed in Figure 9.13. For small values of d> and compatible elements are such that /ciRssfclL. This means that , L and compatible elements such as Ni, Cr, or Mg are virtually unaffected by zone-refining. Incompatible elements are such that ktR/ktLat

efficient scavenging by ascending molten zones. Again, residual porosity is a critical factor for incompatible-element distributions. 

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. 

Incompatible elements keep pace with the liquid, compatible elements lag significantly behind. 

The more compatible the elements, the more they lag behind. Note the quite efficient separation of incompatible elements. An interesting property of trace-element ratios is their change around the initial value since Nd is more incompatible than Sm, the Sm/Nd ratio is expected first to decrease and then increase below the initial Sm/Nd value as the liquid progresses in the rock column, o... 

For small extents of crystallization, the maximum change, and thereby the most valuable information on F, will be obtained from elements with high Dt (compatible elements) such as Ni in basaltic olivine. Elements with ), 1 (incompatible elements), such as Th, Ba or rare-earth elements in basaltic systems, will provide basically no clue to F variations. In addition, information carried by incompatible elements, which do not fractionate with respect to each other, is entirely redundant. This is better shown by taking the relative change in the ratio of two elements il and i2 per increment of crystallization... 

In Figure 9.15, the relationship between the fractional change in the elemental ratio and the extent of crystallization F is plotted for different values of AD=Di2—Dn for partition coefficients less than 0.1, several tens of percent fractionation are needed before a change of a few percent in the ratio becomes visible. Crystal fractionation does not change incompatible-element ratios such as La/Yb, Zr/Nb,. .. except in extremely residual melts. 

Figure 9.15 Fractionation of two trace elements il and i2 during fractional crystallization according to equation (9.5.2). AD is the difference Dn — Di2. Incompatible elements are not fractionated efficiently even for large extents of solid removal.

Similarly, when using incompatible elements to address fractionation processes... 

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