Negative molar abundances of phases

[zackg@berkeley.edu]

Hi,

I notice I am sometimes getting negative molar abundances of some phases in MELTS.  Is this expected?  What is the meaning of it?  For example, I'm attaching an output for a computation where the essenite and hercynite components of the cpx and spinel are negative.

If this is a computational problem, then perhaps something about how I am inputting data is excacerbating it?

I'm using MELTS V5.0.0 - Revision: 1.13 on OS X.

Cheers,

Zack Gainsforth
Space Sciences Laboratory
UC Berkeley


Title: iris_iw

T = 700.00 (C)  P = 0.001 (kbars)  log(10) f O2 = -21.10  delta HM = -10.08  NNO = -4.85  QFM = -4.28  COH = -2.78  IW = 0.00

Constraint Flags: fO2 path = IW 

Liquid           mass = 0.69 (gm)  density = 2.35 (gm/cc)  viscosity = 6.28 (log 10 poise)     (analysis in wt %)
      G = -10758.08 (J)  H = -9188.35 (J)  S = 1.61 (J/K)  V = 0.30 (cc)  Cp = 0.92 (J/K) 
        SiO2   TiO2  Al2O3  Fe2O3  Cr2O3    FeO    MnO    MgO    NiO    CoO    CaO   Na2O    K2O   P2O5    H2O    CO2    SO3 Cl2O-1  F2O-1
       72.28   0.01   2.62   0.15   0.00   0.77   0.00   0.22   0.00   0.00   0.20  23.68   0.07   0.00   0.00   0.00   0.00   0.00   0.00

olivine          mass = 62.88 (gm)  density = 3.60 (gm/cc)     (analysis in mole %)
                 (Ca0.00Mg0.62Fe''0.37Mn0.01Co0.00Ni0.00)2SiO4
                 G = -803169.86 (J)  H = -687689.87 (J)  S = 118.67 (J/K)  V = 17.48 (cc)  Cp = 68.44 (J/K) 
          tephroite      fayalite    co-olivine    ni-olivine  monticellite    forsterite
               0.59         37.25          0.00          0.00          0.29         61.88

clinopyroxene    mass = 6.28 (gm)  density = 3.27 (gm/cc)     (analysis in mole %)
                 cpx Na0.08Ca0.89Fe''0.14Mg0.80Fe'''0.01Ti0.02Al0.09Si1.97O6
                 G = -96407.07 (J)  H = -85092.38 (J)  S = 11.63 (J/K)  V = 1.92 (cc)  Cp = 7.08 (J/K) 
           diopside clinoenstatit  hedenbergite alumino-buffo     buffonite      essenite       jadeite
              72.06          3.08         14.33          2.16          1.93         -1.29          7.72

feldspar         mass = 11.86 (gm)  density = 2.56 (gm/cc)     (analysis in mole %)
                 K0.00Na0.96Ca0.03Al1.03Si2.97O8
                 G = -192752.16 (J)  H = -169174.36 (J)  S = 24.23 (J/K)  V = 4.64 (cc)  Cp = 15.49 (J/K) 
             albite     anorthite      sanidine
              96.29          3.25          0.46

nepheline        mass = 13.38 (gm)  density = 2.49 (gm/cc)     (analysis in mole %)
                 neph Na3.35K0.03Ca0.00[]0.62Al3.38Si4.62O16
                 G = -215628.41 (J)  H = -188041.06 (J)  S = 28.35 (J/K)  V = 5.37 (cc)  Cp = 16.04 (J/K) 
       na-nepheline   k-nepheline  vc-nepheline  ca-nepheline
              37.24          0.66         61.62          0.49

spinel           mass = 4.90 (gm)  density = 4.94 (gm/cc)     (analysis in mole %)
                 Fe''0.88Mg0.15Fe'''0.09Al0.09Cr1.75Ti0.04O4
                 G = -38871.18 (J)  H = -31338.92 (J)  S = 7.74 (J/K)  V = 0.99 (cc)  Cp = 4.13 (J/K) 
           chromite     hercynite     magnetite        spinel    ulvospinel
              87.46        -10.60          4.54         15.09          3.51

Total solids     mass = 99.30 (gm)  density = 3.27 (gm/cc)
                 G = -1346828.68 (J)  H = -1161336.59 (J)  S = 190.61 (J/K)  V = 30.40 (cc)  Cp = 111.19 (J/K) 

[Paula]

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:

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 X_hc 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 X_en, X_di...

MELTS calculates the equilibrium state at given P-T conditions using a global free energy minimisation whereas programs like THERMOCALC solve ΔG = 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 ΔG = 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 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) 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 P_s = 1.

Hope that helps,
Paula

[zackg@berkeley.edu]

Thank you Paula for the terrific answer!

Using the spinel calculator you linked to, then I can clearly see how the ordering parameters play out with the partitioning of Fe2+ and Fe3+ between the tetrahedral and octahedral sites in the chromite.   This is very useful indeed, and clears up the question for me.

Thank you,

Zack