Hi Zack,
Short answer: negative molar abundances for solid solution end members are fine. Negative mole fractions in the liquid would be a big problem!
Longer answer: when formulating a solid solution model there are two choices. One is to take an independent set of end members that can be combined to make all required compositions. The other is to use end members that represent the limit of each cation substitution, or combination thereof, and solve for the proportions of those end members. Both approaches are valid, though not exactly equivalent (especially if there is non-ideal mixing). MELTS uses the first convention; some schemes, such as THERMOCALC, tend to use the second.
For example, for spinels in the system FeO-MgO-Fe2O3-Cr2O3-Al2O3-TiO2 there are eight possible compositional end members:
Quote
| spinel (sensu stricto) | MgAl2O4 |
| hercynite | FeAl2O4 |
| magnetite | Fe3O4 |
| chromite | FeCr2O4 |
| ulvospinel | Fe2TiO4 |
| magnesiochromite | MgCr2O4 |
| magnesioferrite | MgFe2O4 |
| qandilite | Mg2TiO4 |
In addition, each end member has two possible ordering states (though in practice Ti and Cr end members are almost perfectly normally ordered). As magnesiochromite can be formed by the reaction MgCr2O4 = FeCr2O4 + MgAl2O4 - FeAl2O4 etc. only five end members are actually needed to describe the compositional space. MELTS uses the first five end members above (leading to four independent compositional parameters, as
Xhc is calculated by difference) plus three ordering parameters. The end member properties of the three dependent end members place constraints on the values of the interaction parameters, usually denoted by 'W'. Note that the configurational entropy in MELTS spinel is calculated using the site fractions, which are always between 0 and 1, but the non-ideal interactions are expressed in terms of mole fractions of independent end members and the ordering parameters. For pyroxenes it is the same idea, just complicated slightly in that the independent compositional parameters are not simply
Xen,
Xdi...
MELTS calculates the equilibrium state at given
P-T conditions using a global free energy minimisation whereas programs like THERMOCALC solve ÃŽâ€G = 0 for a set of reactions between phase end members. So to formulate a MELTS-like spinel model for THERMOCALC you would use more than eight end members, including normally ordered and inversely ordered ones, but fewer interaction parameters (i.e. the 'W' values). When calculating the equilibrium state, THERMOCALC would include internal reactions between spinel end members, like the one given above, in its set of reactions; then solving ÃŽâ€G = 0 would give the composition and ordering state of both spinel and coexisting phases. For a THERMOCALC-style formulation the end-member mole fractions should be positive.
You can get more information about the way the spinel solid solution is implemented in MELTS at the xMELTS Supplemental Calculator (http://magmasource.caltech.edu/calculator/) site. Although the calculator is mainly a way to try out the new spinel volume model, it also allows you to convert between various compositional and ordering parameters. This includes independent and / or dependent end-member proportions (including negative mole fractions), cation site fractions and wt% oxides (plus the MELTS ordering model). Aside from minor differences in notation (see Hamecher
et al., 2012 (http://dx.doi.org/10.1007/s00410-012-0790-0)) the compositional and ordering variables are identical to those used in MELTS and pMELTS. Note: to get the spinel ordering state as calculated in (p)MELTS, i.e. without the pressure dependence, just input
Ps = 1.
Hope that helps,
Paula