Quantifying garnet-melt trace element partitioning using lattice-strain theory: assessment of statistically significant controls and a new predictive model
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DOI: 10.1007/s00410-007-0235-3
- Cite this article as:
- Draper, D.S. & van Westrenen, W. Contrib Mineral Petrol (2007) 154: 731. doi:10.1007/s00410-007-0235-3
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Abstract
As a complement to our efforts to update and revise the thermodynamic basis for predicting garnet-melt trace element partitioning using lattice-strain theory (van Westrenen and Draper in Contrib Mineral Petrol, this issue), we have performed detailed statistical evaluations of possible correlations between intensive and extensive variables and experimentally determined garnet-melt partitioning values for trivalent cations (rare earth elements, Y, and Sc) entering the dodecahedral garnet X-site. We applied these evaluations to a database containing over 300 partition coefficient determinations, compiled both from literature values and from our own work designed in part to expand that database. Available data include partitioning measurements in ultramafic to basaltic to intermediate bulk compositions, and recent studies in Fe-rich systems relevant to extraterrestrial petrogenesis, at pressures sufficiently high such that a significant component of majorite, the high-pressure form of garnet, is present. Through the application of lattice-strain theory, we obtained best-fit values for the ideal ionic radius of the dodecahedral garnet X-site, r_{0}(3+), its apparent Young’s modulus E(3+), and the strain-free partition coefficient D_{0}(3+) for a fictive REE element J of ionic radius r_{0}(3+). Resulting values of E, D_{0}, and r_{0} were used in multiple linear regressions involving sixteen variables that reflect the possible influence of garnet composition and stoichiometry, melt composition and structure, major-element partitioning, pressure, and temperature. We find no statistically significant correlations between fitted r_{0} and E values and any combination of variables. However, a highly robust correlation between fitted D_{0} and garnet-melt Fe–Mg exchange and D_{Mg} is identified. The identification of more explicit melt-compositional influence is a first for this type of predictive modeling. We combine this statistically-derived expression for predicting D_{0} with the new expressions for predicting E and r_{0} outlined in the first of our pair of companion papers into an updated set of formulae that use easy-to-measure quantities (e.g. garnet composition, pressure, temperature) to predict variations in E, r_{0}, and D_{0}. These values are used in turn to calculate D values for those garnets. The updated model substantially improves upon a previous model (van Westrenen et al. in Contrib Mineral Petrol 142:219–234, 2001), and accounts well for trivalent cation partitioning in nominally anhydrous systems up to at least 15 GPa, including for eclogitic bulk compositions and for Fe-rich systems appropriate to magmagenesis on the Moon and Mars. The new model is slightly less successful in predicting partitioning with strongly majoritic garnets, although the mismatch is much less than with the original 2001 model. Although it also improves upon the 2001 model in predicting partitioning in hydrous systems, the mismatch between model and observation is still unacceptably large. The same statistical tools were applied in an attempt to predict tetravalent partitioning as well, because lattice-strain based techniques are not applicable to such partitioning. However, no statistically significant predictive relationships emerged from that effort. Our analyses show that future efforts should focus on filling the gap in partitioning data between ∼10 and 25 GPa to evaluate more closely the gradual transition of garnet to majorite, and on systematically expanding the hydrous partitioning database to allow extension of our model to water-bearing systems.
Introduction
As was the case for the original WvW01 model, the goal of this work is to provide a means by which trivalent element partitioning between coexisting garnet and melt can be predicted from easy-to-evaluate parameters. In addition, we hope to elucidate constraints on the fundamental control(s) on D values: is it melt composition and structure, crystal composition, or both, and which has the largest effect? Some recent studies, e.g. Mysen and Dubinsky (2004) and Gaetani (2004), have raised questions about the lack of incorporation of an explicit term reflecting melt composition (and hence melt structure) in predictive models that use the lattice-strain approach. However, melt composition does play an implicit role in those models via the incorporation of terms such as D_{Mg} in WvW01, and D_{Fe} in the van Westrenen and Draper (this issue) formulation for predicting D_{0} for trivalent cation partitioning between garnet and melt; similar but more complicated involvement appears in the predictive expressions for clinopyroxene presented by Wood and Blundy (1997). The success of these models shows that incorporating such an explicit account does not appear to be required in lattice-strain expressions to obtain predictions of D values that are accurate enough to be used in geochemical modeling. One purpose of the addition of statistical assessments as presented here is to provide an opportunity for the explicit evaluation of the effect of melt composition and/or melt structure on partitioning, and then compare the accuracy of predictions resulting from both approaches.
We investigated a wide range of combinations of parameters to seek statistically significant correlations, and combined those results with the revised theoretical formulations presented in the first of this pair of companion papers to yield an updated model that successfully predicts anhydrous partitioning of trivalent cations to within a few tens of percent relative. As we show below, this approach provides a statistical expression for D_{0} that performs as well as the D_{0} expression formulated from thermodynamic considerations in van Westrenen and Draper (this issue), while requiring only two controlling parameters, D_{Mg} and garnet-melt Fe–Mg exchange. Both terms involve melt composition, a first for this type of modeling. As pointed out by Gaetani (2004), it is in predictive expressions for D_{0} that the influence of melt composition and structure is most likely to be manifest. The revised model, consisting of this statistical D_{0} expression combined with expressions for r_{0} and E based on crystal-chemical considerations, accounts for trivalent partitioning as well as did the original WvW01 version when applied to the same datasets as was that original model. In addition, it successfully accounts for partitioning in more Fe-rich compositional systems, which the WvW01 model could not, and much more closely predicts partitioning in majorite-rich garnets. No terms expressly linked to majorite content were found to be statistically significant and additional data coverage is needed to make better predictions for majoritic garnets, although in the crystal-chemical E model (van Westrenen and Draper, this issue), there is explicit incorporation of a majorite component in the form of (Al + Cr)_{apfu} [the number of atoms per formula unit (12 oxygen) of Al and Cr in garnet].
The new model fails badly, however, at predicting trivalent partitioning in hydrous systems, both for lower pressure, generally pyropic garnets like those produced by Nicholls and Harris (1980) and Barth et al. (2002), and majoritic garnet grown from a hydrous pyrolite composition by Inoue et al. (2000). In the following sections, we detail the choice of parameters used in the statistical evaluations, present the results of multiple linear regression analysis, and document their statistical significance. We then present the preferred model expressions at this stage of our work, show the degree to which the new preferred model accounts for a range of garnet-melt trivalent partitioning data, and outline gaps that remain to be filled. Finally, we show that partitioning of tetravalent cations (e.g. Zr, Hf, Ti) is not predictable at present using either lattice-strain or statistical formulations.
Approach and computations
In this section, we summarize the data sources used in our analysis, and outline the compositional variables chosen for statistical evaluation. We then present the calculations themselves and their results, showing which of the chosen variables can produce statistically defensible expressions for the lattice-strain parameters used in the predictive model. All statistical calculations presented here were performed using the statistical software package SPSS© version 15.0 for Windows.
Expanded database
List of garnet-melt partitioning experimental studies used in this work
Reference | Conditions |
---|---|
Sc-bearing experiments incorporated in statistically-evaluated database | |
Barth et al. (2002) | 1.8 GPa, ∼1,000°C, hydrous, eclogitic/tonalitic compositions |
Bennett et al. (2004) | 3 GPa, 1,300–1,400°C, anhydrous, CMAS with Na, Ti |
Corgne and Wood (2004) | 25 GPa, 2,300°C, anhydrous, majoritic garnet |
Draper et al. (2003) | 5–9 GPa, 1,750–1,950°C, anhydrous, majoritic garnet, Fe-rich |
Draper et al. (2006) | 2–7 GPa, 1,600–1,800°C, anhydrous, majoritic garnet, Fe-rich |
Dwarzski et al. (2006) | 5–7 GPa, 1,600–1,700°C, anhydrous, Fe-, Ti-rich |
Hauri et al. (1994) | 2.5 GPa, 1,430°C, anhydrous |
Jenner et al. (1993)^{a} | 2.5 GPa, 1,000–1,100°C, hydrous |
Kato et al. (1988)^{a} | 15–24 GPa, 2,100–2,260°C, anhydrous, majoritic garnet |
Pertermann et al. (2004) | ∼3 GPa, 1,325–1,400°C, anhydrous, eclogitic composition |
Sisson and Bacon (1992) | ∼1 GPa, 980°C, hydrous, rhyolitic composition |
van Westrenen et al. (1999) | 3 GPa, ∼1,550°C, anhydrous, CMAS |
van Westrenen et al. (2000b) | 3 GPa, ∼1,550°C, anhydrous, CMAS-Fe |
Yurimoto and Ohtani (1992) | 16–20 GPa, 1,900–2,000°C, anhydrous, majoritic garnet |
Sc-absent experiments, not included in statistically-evaluated database | |
Johnson (1994) | 3 GPa, 1,430°C, anhydrous |
Klein et al. (2000) | 1–3 GPa, 900–1,150°C, hydrous, andesitic to tonalitic melts |
Klemme et al. (2002) | 3 GPa, 1,400°C, anhydrous |
Nicholls and Harris (1980) | 2–3 GPa, 940–1,420°C, dacitic melts, hydrous |
Salters and Longhi (1999) | 2.8 GPa, 1,500–1,550°C, anhydrous |
Salters et al. (2002) | 2.8–3.4 GPa, 1,465–1,660°C, anhydrous |
Shimizu and Kushiro (1975) | 3 GPa, 1,275°C, anhydrous |
Walter et al. (2004) | ∼23 GPa, ∼2,300°C, anhydrous, majoritic garnet |
The data set listed in Table 1 is not identical to the one used in van Westrenen and Draper (this issue) for several reasons. First, Table 1 lists several garnet-melt partitioning experiments in hydrous systems, which were used to gauge the success of the statistical model in predicting hydrous D values. Our companion paper focuses exclusively on anhydrous experiments. Second, only data sets that include D values for scandium can produce best-fit values for D_{0}, r_{0}, and E simultaneously without making additional assumptions, such as those described in the following paragraph. Scandium’s ionic radius of 0.87 Å for coordination number 8 (Shannon 1976) is lower than the r_{0} values that result from garnet-melt partitioning data (van Westrenen et al. 1999). For natural garnets, r_{0} values are typically 0.91–0.93 Å ; all the other trivalent cations that partition into garnet’s X-site (Y, REE) have radii larger than this value. As a result, in the absence of Sc data the left-hand limbs of parabolae formed by plotting D values versus ionic radius are unconstrained. For example, consider three experiments from the studies listed in Table 1: run MP240 of Pertermann et al. (2004), run 16 of van Westrenen et al. (2000b), and runs 180 + 252 from Draper et al. (2003). Sc was included in these experiments and hence Eq. 1 yields values for D_{0}, r_{0}, and E. If D values for Sc are omitted in those fits, i.e. using only the limbs of the parabolae at higher ionic radius values than expected r_{0}, drastically different and unrealistic results are obtained. For run MP240, the Sc-included fits yield values for D_{0}, E, and r_{0} of 11.95 ± 2.41, 563 ± 54 GPa, and 0.914 ± 0.012 Å, respectively; the Sc-excluded values are 408 ± 726, 250 ± 76 GPa, and 0.711 ± 0.109 Å. For run 16, we have 6.20 ± 1.66, 406 ± 77 GPa, and 0.912 Å with Sc versus 491 ± 3,010, 127 ± 139 GPa, and 0.546 ± 0.577 Å without; and for 180 + 252, we have 2.51 ± 0.30, 480 ± 53 GPa, and 0.911 ± 0.010 Å versus 131 ± 265, 138 ± 57 GPa, and 0.578 ± 0.193 Å. Similar relationships were also described by van Westrenen et al. (2000b) (their Fig. 5 and p. 196).
Many garnet-melt partitioning studies did not include Sc (see Table 1). Using additional constraints, meaningful fits to Sc-absent data sets can still be obtained. For example, a predictive model for r_{0} based on Sc-bearing data can be combined with Eq. 1 to obtain best-fit values for E and D_{0} in Sc-absent experiments, e.g. van Westrenen et al. (2000b). These values can then be used to construct predictive models for E and D_{0}. Although we consider this a perfectly valid approach for the crystal-chemical and thermodynamic models of van Westrenen and Draper (this issue), this could introduce artificial bias into our statistical evaluations. Therefore, only Sc-bearing experiments for which physically-realistic results for D_{0}, r_{0}, and E can be precisely determined simultaneously were included in our statistical analyses.
Finally, although van Westrenen and Draper (this issue) opted to omit data from some earlier studies on partitioning in majoritic garnet, e.g. Kato et al. (1988) and Yurimoto and Ohtani (1992), we chose to include data from Sc-bearing experiments from those two studies in our statistical analysis largely because no other data were available at pressures from ∼16 to 20 GPa for majoritic garnets. As we show below, our revised models nevertheless significantly improve predicted D values for such studies.
For these experiments, fitting the data to Eq. 1 yields values for D_{0}, r_{0}, and E for each experiment; we refer to these three values as the lattice-strain fit parameters. We then selected 16 variables intended to reflect garnet or liquid composition, garnet crystal chemistry, melt structure, temperature, and pressure; we refer to these simply as variables. Possible correlations between these variables were first assessed to find combinations in which all variables were independent of one another. Then, each individual lattice-strain fit parameter (D_{0}r_{0}, and E) was multiply regressed against combinations of these independent variables using both standard and “stepwise” approaches. Multiple linear regression results for these permutations show which are robust predictors and which are not.
Choosing variables for statistical evaluations
Variables chosen for possible use in statistical regressions
Quantity | Description |
---|---|
D_{Ca}, D_{Mg}, D_{Fe} | Garnet-liquid D values for major constituents of X-site |
K_{D} for Fe–Mg, Al–Si | \({\rm K}^{{{\rm Fe} - {\rm Mg}}}_{D} = \frac{{{\rm FeO}^{{{\rm gt}}} {\rm MgO}^{{{\rm liq}}}}}{{{\rm FeO}^{{{\rm liq}}} {\rm MgO}^{{{\rm gt}}}}};{\rm K}^{{{\rm Al} - {\rm Si}}}_{D} = \frac{{{\rm Al}_{2} {\rm O}^{{{\rm gt}}}_{3} {\rm SiO}^{{{\rm liq}}}_{2}}}{{{\rm Al}_{2} O^{{{\rm liq}}}_{3} {\rm SiO}^{{{\rm gt}}}_{2}}}\) |
Gr, Py, Alm | Mol% grossular, pyrope, almandine in garnet |
Si, Al per formula unit (pfu) | Number Si and Al cations per 12-oxygen formula unit |
Melt Mg# | Molar Mg/Mg + Fe in melt |
Melt MgO/SiO_{2} | Ratio of wt% oxide MgO to SiO_{2} in melt |
Melt FeO/SiO_{2} | Ratio of wt% oxide FeO to SiO_{2} in melt |
Melt NBO/t | Ratio of non-bridging oxygens to tetrahedral cations in melt (Mysen 1983) |
Melt ionic porosity | Measure of melt “empty space”; Dowty (1980), Carroll and Draper (1994) |
Temperature | |
Pressure |
Assessing bivariate correlations among variables
Before using any of the variables chosen for statistical analysis, it must be demonstrated that they are not correlated with one another; only independent variables should be used in predicting garnet-melt partitioning. We accomplished this by computing correlation matrices consisting of Pearson’s correlation coefficients for all possible variable pairs with their associated p-values. The p-values (two-tailed) denote the statistical significance of the correlation coefficients; at the 95% confidence limit, those correlations with p-values less than or equal to 0.05 are considered significant, whereas those with p-values greater than 0.05 are not. We computed correlation matrices for two versions of the dataset. The first includes every experiment from the Sc-bearing studies listed in Table 1. The second excludes the data from the seven Sc-bearing experiments that were performed under hydrous conditions (Jenner et al. 1993; Barth et al. 2002); recall that the hydrous experiments of Nicholls and Harris (1980) and Klein et al. (2000) lacked Sc and so cannot be fitted to Eq. 1. We proceeded in this fashion because the hydrous experiments were run at hundreds of degrees cooler temperatures than all the other experiments, which may result in discontinuities in some variables, and because the role of water in affecting partitioning in general is still poorly understood.
Matrix showing correlations between variables chosen for statistical analysis; data from hydrous experiments included
| 10,000/T | P (GPa) | Fe–Mg K_{D} | Al–Si K_{D} | FeO/SiO_{2} | MgO/SiO_{2} | Mg# | Ionic Porosity | NBO/t | Mol% Gr | Mol% Py | Mol% Alm | D_{Mg} | D_{Ca} | D_{Fe} | Si per 12 Oxy | Al per 12 Oxy |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
10,000/T | −0.759 | −0.021 | −0.278 | −0.272 | −0.844 | −0.501 | −0.783 | −0.827 | 0.585 | −0.728 | 0.521 | 0.737 | 0.808 | 0.824 | −0.758 | 0.731 | |
0.000 | 0.889 | 0.061 | 0.067 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | ||
P (GPa) | −0.759 | 0.007 | 0.085 | 0.048 | 0.760 | 0.442 | 0.694 | 0.628 | −0.532 | 0.669 | −0.470 | −0.365 | −0.363 | −0.460 | 0.921 | −0.899 | |
0.000 | 0.965 | 0.573 | 0.749 | 0.000 | 0.002 | 0.000 | 0.000 | 0.000 | 0.000 | 0.001 | 0.013 | 0.013 | 0.003 | 0.000 | 0.000 | ||
Fe–Mg K_{D} | −0.021 | 0.007 | −0.286 | −0.353 | 0.256 | 0.282 | −0.135 | 0.098 | 0.095 | 0.015 | −0.110 | −0.224 | 0.092 | 0.131 | 0.113 | −0.085 | |
0.889 | 0.965 | 0.054 | 0.016 | 0.086 | 0.057 | 0.372 | 0.515 | 0.530 | 0.922 | 0.468 | 0.135 | 0.542 | 0.421 | 0.456 | 0.572 | ||
Al–Si K_{D} | −0.278 | 0.085 | −0.286 | 0.817 | 0.251 | −0.269 | 0.643 | 0.516 | −0.597 | 0.082 | 0.258 | −0.204 | −0.281 | −0.344 | 0.084 | −0.309 | |
0.061 | 0.573 | 0.054 | 0.000 | 0.092 | 0.070 | 0.000 | 0.000 | 0.000 | 0.587 | 0.083 | 0.174 | 0.059 | 0.030 | 0.578 | 0.037 | ||
FeO/SiO_{2} | −0.272 | 0.048 | −0.353 | 0.817 | 0.202 | −0.474 | 0.571 | 0.483 | −0.609 | −0.081 | 0.450 | −0.328 | −0.290 | −0.486 | 0.064 | −0.314 | |
0.067 | 0.749 | 0.016 | 0.000 | 0.178 | 0.001 | 0.000 | 0.001 | 0.000 | 0.592 | 0.002 | 0.026 | 0.051 | 0.001 | 0.670 | 0.034 | ||
MgO/SiO_{2} | −0.844 | 0.760 | 0.256 | 0.251 | 0.202 | 0.558 | 0.801 | 0.921 | −0.694 | 0.822 | −0.551 | −0.584 | −0.620 | −0.637 | 0.867 | −0.857 | |
0.000 | 0.000 | 0.086 | 0.092 | 0.178 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | ||
Mg# | −0.501 | 0.442 | 0.282 | −0.269 | −0.474 | 0.558 | 0.234 | 0.344 | −0.001 | 0.795 | −0.968 | −0.341 | −0.508 | −0.352 | 0.500 | −0.306 | |
0.000 | 0.002 | 0.057 | 0.070 | 0.001 | 0.000 | 0.117 | 0.019 | 0.995 | 0.000 | 0.000 | 0.020 | 0.000 | 0.026 | 0.000 | 0.038 | ||
Ionic porosity | −0.783 | 0.694 | −0.135 | 0.643 | 0.571 | 0.801 | 0.234 | 0.891 | −0.880 | 0.670 | −0.256 | −0.481 | −0.624 | −0.674 | 0.698 | −0.794 | |
0.000 | 0.000 | 0.372 | 0.000 | 0.000 | 0.000 | 0.117 | 0.000 | 0.000 | 0.000 | 0.086 | 0.001 | 0.000 | 0.000 | 0.000 | 0.000 | ||
NBO/t | −0.827 | 0.628 | 0.098 | 0.516 | 0.483 | 0.921 | 0.344 | 0.891 | −0.788 | 0.694 | −0.336 | −0.648 | −0.704 | −0.752 | 0.716 | −0.776 | |
0.000 | 0.000 | 0.515 | 0.000 | 0.001 | 0.000 | 0.019 | 0.000 | 0.000 | 0.000 | 0.022 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | ||
Mol% Gr | 0.585 | −0.532 | 0.095 | −0.597 | −0.609 | −0.694 | −0.001 | −0.880 | −0.788 | −0.524 | 0.017 | 0.307 | 0.434 | 0.557 | −0.540 | 0.707 | |
0.000 | 0.000 | 0.530 | 0.000 | 0.000 | 0.000 | 0.995 | 0.000 | 0.000 | 0.000 | 0.913 | 0.038 | 0.003 | 0.000 | 0.000 | 0.000 | ||
Mol% Py | −0.728 | 0.669 | 0.015 | 0.082 | −0.081 | 0.822 | 0.795 | 0.670 | 0.694 | −0.524 | −0.843 | −0.342 | −0.652 | −0.482 | 0.743 | −0.621 | |
0.000 | 0.000 | 0.922 | 0.587 | 0.592 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.020 | 0.000 | 0.002 | 0.000 | 0.000 | ||
Mol% Alm | 0.521 | −0.470 | −0.110 | 0.258 | 0.450 | −0.551 | −0.968 | −0.256 | −0.336 | 0.017 | −0.843 | 0.249 | 0.514 | 0.353 | −0.529 | 0.318 | |
0.000 | 0.001 | 0.468 | 0.083 | 0.002 | 0.000 | 0.000 | 0.086 | 0.022 | 0.913 | 0.000 | 0.095 | 0.000 | 0.025 | 0.000 | 0.031 | ||
D_{Mg} | 0.737 | −0.365 | −0.224 | −0.204 | −0.328 | −0.584 | −0.341 | −0.481 | −0.648 | 0.307 | −0.342 | 0.249 | 0.769 | 0.855 | −0.397 | 0.396 | |
0.000 | 0.013 | 0.135 | 0.174 | 0.026 | 0.000 | 0.020 | 0.001 | 0.000 | 0.038 | 0.020 | 0.095 | 0.000 | 0.000 | 0.006 | 0.006 | ||
D_{Ca} | 0.808 | −0.363 | 0.092 | −0.281 | −0.290 | −0.620 | −0.508 | −0.624 | −0.704 | 0.434 | −0.652 | 0.514 | 0.769 | 0.909 | −0.391 | 0.334 | |
0.000 | 0.013 | 0.542 | 0.059 | 0.051 | 0.000 | 0.000 | 0.000 | 0.000 | 0.003 | 0.000 | 0.000 | 0.000 | 0.000 | 0.007 | 0.023 | ||
D_{Fe} | 0.824 | −0.460 | 0.131 | −0.344 | −0.486 | −0.637 | −0.352 | −0.674 | −0.752 | 0.557 | −0.482 | 0.353 | 0.855 | 0.909 | −0.446 | 0.488 | |
0.000 | 0.003 | 0.421 | 0.030 | 0.001 | 0.000 | 0.026 | 0.000 | 0.000 | 0.000 | 0.002 | 0.025 | 0.000 | 0.000 | 0.004 | 0.001 | ||
Si per 12 O | −0.758 | 0.921 | 0.113 | 0.084 | 0.064 | 0.867 | 0.500 | 0.698 | 0.716 | −0.540 | 0.743 | −0.529 | −0.397 | −0.391 | −0.446 | −0.922 | |
0.000 | 0.000 | 0.456 | 0.578 | 0.670 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.006 | 0.007 | 0.004 | 0.000 | ||
Al per 12 O | 0.731 | −0.899 | −0.085 | −0.309 | −0.314 | −0.857 | −0.306 | −0.794 | −0.776 | 0.707 | −0.621 | 0.318 | 0.396 | 0.334 | 0.488 | −0.922 | |
0.000 | 0.000 | 0.572 | 0.037 | 0.034 | 0.000 | 0.038 | 0.000 | 0.000 | 0.000 | 0.000 | 0.031 | 0.006 | 0.023 | 0.001 | 0.000 |
Matrix showing correlations between variables chosen for statistical analysis; data from hydrous experiments omitted
| 10,000/T | P (GPa) | Fe–Mg K_{D} | Al–Si K_{D} | FeO/SiO_{2} | MgO/SiO_{2} | Mg# | Ionic Porosity | NBO/t | Mol% Gr | Mol% Py | Mol% Alm | D_{Mg} | D_{Ca} | D_{Fe} | Si per 12 Oxy | Al per 12 Oxy |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
10,000/T | −0.870 | −0.082 | −0.255 | −0.207 | −0.900 | −0.434 | −0.859 | −0.847 | 0.699 | −0.799 | 0.455 | 0.756 | 0.576 | 0.807 | −0.876 | 0.873 | |
0.000 | 0.621 | 0.117 | 0.206 | 0.000 | 0.006 | 0.000 | 0.000 | 0.000 | 0.000 | 0.004 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | ||
P (GPa) | −0.870 | 0.017 | 0.038 | 0.005 | 0.735 | 0.387 | 0.662 | 0.588 | −0.511 | 0.663 | −0.423 | −0.444 | −0.264 | −0.501 | 0.917 | −0.893 | |
0.000 | 0.917 | 0.820 | 0.976 | 0.000 | 0.015 | 0.000 | 0.000 | 0.001 | 0.000 | 0.007 | 0.005 | 0.104 | 0.003 | 0.000 | 0.000 | ||
Fe–Mg K_{D} | −0.082 | 0.017 | −0.289 | −0.382 | 0.309 | 0.332 | −0.124 | 0.126 | 0.065 | 0.098 | −0.192 | −0.260 | 0.077 | 0.181 | 0.130 | −0.093 | |
0.621 | 0.917 | 0.074 | 0.016 | 0.055 | 0.039 | 0.453 | 0.444 | 0.695 | 0.553 | 0.242 | 0.110 | 0.640 | 0.313 | 0.431 | 0.572 | ||
Al–Si K_{D} | −0.255 | 0.038 | −0.289 | 0.840 | 0.205 | −0.371 | 0.632 | 0.510 | −0.582 | 0.001 | 0.382 | −0.282 | −0.282 | −0.452 | 0.033 | −0.278 | |
0.117 | 0.820 | 0.074 | 0.000 | 0.211 | 0.020 | 0.000 | 0.001 | 0.000 | 0.997 | 0.016 | 0.078 | 0.082 | 0.008 | 0.841 | 0.086 | ||
FeO/SiO_{2} | −0.207 | 0.005 | −0.382 | 0.840 | 0.151 | −0.561 | 0.585 | 0.456 | −0.627 | −0.122 | 0.544 | −0.314 | −0.212 | −0.467 | 0.032 | −0.297 | |
0.206 | 0.976 | 0.016 | 0.000 | 0.359 | 0.000 | 0.000 | 0.004 | 0.000 | 0.459 | 0.000 | 0.052 | 0.195 | 0.006 | 0.849 | 0.066 | ||
MgO/SiO_{2} | −0.900 | 0.735 | 0.309 | 0.205 | 0.151 | 0.500 | 0.773 | 0.908 | −0.694 | 0.834 | −0.495 | −0.775 | −0.556 | −0.728 | 0.861 | −0.858 | |
0.000 | 0.000 | 0.055 | 0.211 | 0.359 | 0.001 | 0.000 | 0.000 | 0.000 | 0.000 | 0.001 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | ||
Mg# | −0.434 | 0.387 | 0.332 | −0.371 | −0.561 | 0.500 | 0.116 | 0.256 | 0.088 | 0.773 | −0.975 | −0.468 | −0.511 | −0.518 | 0.439 | −0.247 | |
0.006 | 0.015 | 0.039 | 0.020 | 0.000 | 0.001 | 0.480 | 0.116 | 0.596 | 0.000 | 0.000 | 0.003 | 0.001 | 0.002 | 0.005 | 0.130 | ||
Ionic porosity | −0.859 | 0.662 | −0.124 | 0.632 | 0.585 | 0.773 | 0.116 | 0.894 | −0.889 | 0.629 | −0.125 | −0.731 | −0.641 | −0.898 | 0.667 | −0.782 | |
0.000 | 0.000 | 0.453 | 0.000 | 0.000 | 0.000 | 0.480 | 0.000 | 0.000 | 0.000 | 0.447 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | ||
NBO/t | −0.847 | 0.588 | 0.126 | 0.510 | 0.456 | 0.908 | 0.256 | 0.894 | −0.816 | 0.699 | −0.249 | −0.810 | −0.651 | −0.844 | 0.698 | −0.775 | |
0.000 | 0.000 | 0.444 | 0.001 | 0.004 | 0.000 | 0.116 | 0.000 | 0.000 | 0.000 | 0.126 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | ||
Mol% Gr | 0.699 | −0.511 | 0.065 | −0.582 | −0.627 | −0.694 | 0.088 | −0.889 | −0.816 | −0.487 | −0.096 | 0.583 | 0.496 | 0.805 | −0.517 | 0.703 | |
0.000 | 0.001 | 0.695 | 0.000 | 0.000 | 0.000 | 0.596 | 0.000 | 0.000 | 0.002 | 0.561 | 0.000 | 0.001 | 0.000 | 0.001 | 0.000 | ||
Mol% Py | −0.799 | 0.663 | 0.098 | 0.001 | −0.122 | 0.834 | 0.773 | 0.629 | 0.699 | −0.487 | −0.803 | −0.681 | −0.740 | −0.765 | 0.739 | −0.624 | |
0.000 | 0.000 | 0.553 | 0.997 | 0.459 | 0.000 | 0.000 | 0.000 | 0.000 | 0.002 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | ||
Mol% Alm | 0.455 | −0.423 | −0.192 | 0.382 | 0.544 | −0.495 | −0.975 | −0.125 | −0.249 | −0.096 | −0.803 | 0.417 | 0.507 | 0.546 | −0.479 | 0.265 | |
0.004 | 0.007 | 0.242 | 0.016 | 0.000 | 0.001 | 0.000 | 0.447 | 0.126 | 0.561 | 0.000 | 0.008 | 0.001 | 0.001 | 0.002 | 0.104 | ||
D_{Mg} | 0.756 | −0.444 | −0.260 | −0.282 | −0.314 | −0.775 | −0.468 | −0.731 | −0.810 | 0.583 | −0.681 | 0.417 | 0.788 | 0.904 | −0.552 | 0.562 | |
0.000 | 0.005 | 0.110 | 0.078 | 0.052 | 0.000 | 0.003 | 0.000 | 0.000 | 0.000 | 0.000 | 0.008 | 0.000 | 0.000 | 0.000 | 0.000 | ||
D_{Ca} | 0.576 | −0.264 | 0.077 | −0.282 | −0.212 | −0.556 | −0.511 | −0.641 | −0.651 | 0.496 | −0.740 | 0.507 | 0.788 | 0.885 | −0.329 | 0.273 | |
0.000 | 0.104 | 0.640 | 0.082 | 0.195 | 0.000 | 0.001 | 0.000 | 0.000 | 0.001 | 0.000 | 0.001 | 0.000 | 0.000 | 0.041 | 0.092 | ||
D_{Fe} | 0.807 | −0.501 | 0.181 | −0.452 | −0.467 | −0.728 | −0.518 | −0.898 | −0.844 | 0.805 | −0.765 | 0.546 | 0.904 | 0.885 | −0.556 | 0.598 | |
0.000 | 0.003 | 0.313 | 0.008 | 0.006 | 0.000 | 0.002 | 0.000 | 0.000 | 0.000 | 0.000 | 0.001 | 0.000 | 0.000 | 0.001 | 0.000 | ||
Si per 12 O | −0.876 | 0.917 | 0.130 | 0.033 | 0.032 | 0.861 | 0.439 | 0.667 | 0.698 | −0.517 | 0.739 | −0.479 | −0.552 | −0.329 | −0.556 | −0.923 | |
0.000 | 0.000 | 0.431 | 0.841 | 0.849 | 0.000 | 0.005 | 0.000 | 0.000 | 0.001 | 0.000 | 0.002 | 0.000 | 0.041 | 0.001 | 0.000 | ||
Al per 12 O | 0.873 | −0.893 | −0.093 | −0.278 | −0.297 | −0.858 | −0.247 | −0.782 | −0.775 | 0.703 | −0.624 | 0.265 | 0.562 | 0.273 | 0.598 | −0.923 | |
0.000 | 0.000 | 0.572 | 0.086 | 0.066 | 0.000 | 0.130 | 0.000 | 0.000 | 0.000 | 0.000 | 0.104 | 0.000 | 0.092 | 0.000 | 0.000 |
Multiple linear regressions
Multiple linear regressions were performed on each subset of independent variables in two ways. In the first, all variables in the subset were included in the regression. In the second, three “stepwise” techniques were employed. The use of the stepwise approach to augment the standard technique allows us to determine if all variables in the regression are truly required. All approaches use an F statistic and its associated significance to assess the contributions from each variable in a regression. The significance (p-value) is defined in the same way as for the correlation matrices described above. In the traditional stepwise technique, at each step the independent variable not in the equation that has the smallest significance of F is entered into the regression, if that significance is sufficiently small. Variables already in the regression equation are removed if their significance of F becomes sufficiently large. The method terminates when no more variables are eligible for inclusion or removal. The threshold significance values used for inclusion or removal were 0.05 and 0.10, respectively.
Two additional stepwise techniques were also employed. The first is backward elimination, a variable selection procedure in which all variables are entered into the equation and then sequentially removed. The variable with the smallest partial correlation with the dependent variable is considered first for removal, which occurs if it meets the criterion for elimination. After the first variable is removed, the variable remaining in the equation with the smallest partial correlation is considered next. The procedure stops when there are no variables in the equation that satisfy the removal criteria. The second additional technique, forward selection, is a selection procedure in which variables are sequentially entered into the model. The first variable considered for entry into the equation is the one with the largest positive or negative correlation with the dependent variable. This variable is entered into the equation only if it satisfies the criterion for entry. If the first variable is entered, the independent variable not in the equation that has the largest partial correlation is considered next. The procedure stops when there are no variables that meet the entry criterion. In both these latter techniques, the same criteria for entry and removal were used, namely significance of F of 0.05 and 0.10, respectively. In most cases, the results of the three stepwise techniques yield identical conclusions regarding which variables are required and which can be eliminated.
After performing these four types of multiple linear regression on each of the variable subsets identified using the bivariate correlation procedure, for each of the three lattice-strain fit parameters, we find that for E and r_{0} none of the regressions produced useful predictive expressions. In all cases for those two parameters, multiple correlation coefficients (R) were well below 0.5 with coefficients of determination (R^{2}) below 0.25. Plots of predicted versus measured values of E and r_{0} from these regressions produce “shotgun” patterns instead of anything resembling one-to-one trends. This lack of correlation is likely due to the rather more complicated formulations and manipulations required to produce the crystal-chemical expressions that do produce useful predictors for these two quantities (van Westrenen and Draper, this issue). Clearly there are nonlinear components to these relationships, and these do not manifest themselves in the multiple-linear correlations tested in this analysis. We therefore conclude that at present, available garnet-melt and majorite-melt partitioning data cannot be used to define statistically-significant predictive relationships for these lattice-strain parameters of the type tested for here. That no statistical models could be found for r_{0} shows, in our view, that it is essential to base r_{0} models on systematic experiments in simple systems, as discussed previously in Van Westrenen et al. (1999, 2000a, b). The crystal-chemical r_{0} model of Van Westrenen and Draper (this issue) was obtained in a step-wise approach, building upon these simple system experiments to derive crystal-chemistry effects prior to deriving pressure and temperature terms.
Coefficients of determination (= square of multiple correlation coefficients) for multiple linear regressions performed on subsets of variables
| All variables | Stepwise | Backward | Forward |
---|---|---|---|---|
Regression subsets from datablock analyzed including results from hydrous experiments | ||||
10,000/T, Fe–Mg K_{D}, Al–Si K_{D}, melt FeO/SiO_{2} | 0.706 | 0.691 | 0.691 | 0.691 |
10,000/T, Fe–Mg K_{D}, Al–Si K_{D} | 0.682 | 0.651 | 0.674 | 0.651 |
P (GPa), Fe–Mg K_{D}, Al–Si K_{D}, melt FeO/SiO_{2} | 0.471 | 0.429 | 0.470 | 0.429 |
P (GPa), Fe–Mg K_{D}, Al–Si K_{D} | 0.409 | 0.385 | 0.385 | 0.385 |
Fe–Mg K_{D}, Al–Si K_{D}, melt MgO/SiO_{2} | 0.577 | 0.536 | 0.572 | 0.536 |
Fe–Mg K_{D}, Al–Si K_{D}, garnet mol% Alm, D_{Mg} | 0.731 | 0.731 | 0.731 | 0.731 |
Fe–Mg K_{D}, Al–Si K_{D}, melt Mg# | 0.489 | 0.487 | 0.487 | 0.487 |
Fe–Mg K_{D}, Al–Si K_{D}, garnet mol% Py | 0.520 | 0.498 | 0.498 | 0.498 |
Fe–Mg K_{D}, Al–Si K_{D}, garnet mol% Alm | 0.463 | 0.441 | 0.441 | 0.441 |
Fe–Mg K_{D}, Al–Si K_{D}, D_{Mg} | 0.609 | 0.608 | 0.608 | 0.608 |
Fe–Mg K_{D}, Al–Si K_{D}, D_{Ca} | 0.698 | 0.642 | 0.698 | 0.642 |
Fe–Mg K_{D}, melt ionic porosity, melt Mg# | 0.641 | 0.637 | 0.637 | 0.637 |
Fe–Mg K_{D}, garnet mol% Gr, melt Mg# | 0.555 | 0.555 | 0.555 | 0.555 |
Fe–Mg K_{D}, Al–Si K_{D}, Si per 12 oxygens | 0.409 | 0.400 | 0.400 | 0.400 |
Regression subsets from datablock analyzed excluding results from hydrous experiments | ||||
10,000/T, Fe–Mg K_{D}, Al–Si K_{D} | 0.631 | 0.586 | 0.631 | 0.586 |
P (GPa), Fe–Mg K_{D}, Al–Si K_{D} | 0.416 | 0.371 | 0.416 | 0.371 |
P (GPa), melt FeO/SiO_{2}, D_{Ca} | 0.696 | 0.696 | 0.696 | 0.696 |
Fe–Mg K_{D}, Al–Si K_{D}, melt MgO/SiO_{2} | 0.549 | 0.548 | 0.548 | 0.548 |
Fe–MgK_{D}, Al–Si K_{D}, D_{Mg} | 0.867 | 0.867 | 0.867 | 0.867 |
Fe–Mg K_{D}, melt ionic porosity, garnet mol% Alm | 0.680 | 0.659 | 0.659 | 0.659 |
Fe–Mg K_{D}, melt NBO/t, garnet mol% Alm | 0.608 | 0.608 | 0.608 | 0.608 |
Fe–Mg K_{D}, Al–Si K_{D}, Al per 12 oxygens | 0.416 | 0.353 | 0.353 | 0.353 |
Fe–Mg K_{D}, Al–Si K_{D}, Si per 12 oxygens | 0.422 | 0.399 | 0.399 | 0.399 |
Fe–Mg K_{D}, Al–Si K_{D}, D_{Ca}, Al per 12 oxygens | 0.737 | 0.707 | 0.737 | 0.707 |
Fe–Mg K_{D}, D_{Ca}, Al per 12 oxygens | 0.721 | 0.707 | 0.707 | 0.707 |
Fe–Mg K_{D}, Al–Si K_{D}, D_{Ca} | 0.619 | 0.54 | 0.619 | 0.54 |
Al–Si K_{D}, D_{Ca}, Al per 12 oxygens | 0.714 | 0.707 | 0.707 | 0.707 |
D_{Ca}, Al per 12 oxygens, melt FeO/SiO_{2} | 0.722 | 0.707 | 0.707 | 0.707 |
Successful D_{0} model
The apparently successful model listed in boldface in Table 5 relates D_{0} to the variable subset Fe–Mg K_{D}, Al–Si K_{D}, and D_{Mg}. The R^{2} values for this subset for all four methods of regression are identical at 0.867. Part of the output for these regressions is a t-statistic for each of the coefficients determined by the regression and its significance (p-value). As in the computations above, p-values less than 0.05 connote statistical significance for the regression coefficients, whereas values above that threshold signify non-significance. When all three variables are included in the regression, the significance of the t-statistic associated with the coefficient for Al–Si K_{D} is 0.722, greatly in excess of the threshold value of 0.05. Furthermore, in each of the three stepwise methods, that variable is excluded by the regression procedure as not being truly required in the regression. Finally, the three stepwise regression procedures also give identical results for the regression coefficients and significance values for the two surviving variables, Fe–Mg K_{D} and D_{Mg}, as well as for the F statistic associated with the overall regression result. From these results, it is clear that Al–Si K_{D} is not in fact required in predicting D_{0}.
Details of successful statistical D_{0} model
Variable | Coefficient | t Statistic | P value |
---|---|---|---|
Fe–Mg K_{D} | 3.24 ± 1.46 | 2.22 | 0.033 |
D_{Mg} | 3.01 ± 0.20 | 15.19 | <0.001 |
Intercept | −2.59 ± 1.04 | −2.48 | 0.018 |
Updated model and comparisons
New model equations
Preferred predictive expressions
Equation 3 of van Westrenen et al. (this issue): |
r_{0} = 0.9302 Py + 0.993 Gr + 0.916 Alm + 0.946 Spes + 1.05 (And + Uv) – 0.0044(±3) (P – 3) + 0.000058(±7) (T – 1817) [±least significant unit] |
Equation 6 of van Westrenen and Draper (this issue): |
E = 2,826(±174) (1.38 + r_{0})^{−3} + 12.4(±13) (P) − 0.072(±7) (T) + 237(±8) (Al + Cr)_{apfu} |
Thermodynamic D_{0}: |
Equation 18 of van Westrenen et al. (this issue): |
\(D_{0} ({\rm REE}) = \raise0.7ex\hbox{${\exp{\left({\frac{{400,290{\left({\pm 1,100} \right)} + 4,586{\left({\pm 150} \right)}P - 218{\left({\pm 7} \right)}T}}{{RT}}} \right)}}$} \!\mathord{\left/ {\vphantom {{\exp {\left({\frac{{400290{\left({\pm 1100} \right)} + 4586{\left({\pm 150} \right)}P - 218{\left({\pm 7} \right)}T}}{{RT}}} \right)}} {{\left({\gamma^{{{\rm garnet}}}_{{{\rm Fe}}} D_{{{\rm Fe}}}} \right)}^{2}}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${{\left({\gamma^{{{\rm garnet}}}_{{{\rm Fe}}} D_{{{\rm Fe}}}}\right)}^{2}}$}\) |
Statistical D_{0}: |
D_{0} = (3.24 ± 1.46) Fe–Mg K_{D} + (3.01 ± 0.20) D_{Mg} – (2.59 ± 1.04) |
The crystal-chemical expression for r_{0} has only very small uncertainties associated with the pressure and temperature terms, derived from mineral physics data on variations of the dimensions of the garnet X-site as a function of pressure, and from linear regression of pressure-corrected r_{0} values against temperature, that when propagated result in uncertainties on r_{0} of 0.01–0.001 Å, compared to typical absolute values of 0.88–0.99 Å. Model E values incorporate propagated uncertainties in r_{0}, in addition to errors in best-fit parameters associated with pressure, temperature, and majorite content terms (Table 7). Thus model E values are uncertain to within approximately 80 GPa out of 465–730 GPa typical of the garnet X-site. Propagation of the uncertainties in the statistical D_{0} expression yields estimates that have uncertainties of approximately 20% relative, illustrated by the error bars on Fig. 2. We demonstrate below that the updated model, using the nominal predicted D_{0} values, performs very well for all but hydrous compositional systems.
Users of this predictive model should be aware that they can choose between either the statistical or thermodynamic D_{0} expressions. Advantages of the statistical D_{0} expression include its simplicity (relying on fewer parameters) and its more explicit incorporation of the melt-compositional effect on predicted D values. Accordingly, use of the statistical D_{0} expression may be preferable to some users. One major advantage of the thermodynamic expression is that P and T are explicitly incorporated. This allows direct application to polythermal, polybaric partial melting processes involving garnet or majorite, whereas an additional model for the Fe–Mg K_{D} dependence on P and T is required to use the statistical D_{0} expression in these cases.
At the completion of our statistical evaluations, we were unable to isolate a variable that can tie D_{0} explicitly to majorite content in garnet. The variables we chose that might have yielded such a link were Si and Al per formula unit, Al–Si K_{D}, and perhaps pressure. None of these survived our criteria for successful regression analysis. This is fully consistent with the thermodynamic analysis in our companion paper (van Westrenen and Draper, this issue).
Comparison between model and measurements
Percent relative mismatches between measured partitioning values and model predictions
| Sc | Y | Ce | Nd | Sm | Eu | Gd | Dy | Er | Yb | Lu |
---|---|---|---|---|---|---|---|---|---|---|---|
Thermodynamic E, D_{0} | |||||||||||
Minimum mismatch | 1.2 | 2.8 | 39.0 | 3.9 | 5.9 | 0.3 | 1.7 | 0.1 | 0.5 | 0.6 | 6.6 |
Maximum mismatch | 121.1 | 115.2 | 94.2 | 176.1 | 129.4 | 120.7 | 43.7 | 176.6 | 57.6 | 128.7 | 74.4 |
Average mismatch | 35.8 | 42.5 | 75.0 | 59.8 | 45.9 | 37.3 | 22.2 | 43.6 | 30.3 | 40.3 | 42.8 |
Thermodynamic E, statistical D_{0} | |||||||||||
Minimum mismatch | 0.3 | 0.6 | 17.0 | 0.1 | 1.2 | 3.0 | 5.5 | 0.4 | 1.3 | 0.0 | 4.6 |
Maximum mismatch | 168.8 | 284.3 | 94.2 | 92.6 | 98.6 | 103.5 | 46.7 | 272.1 | 74.5 | 220.9 | 69.5 |
Average mismatch | 54.8 | 45.0 | 58.8 | 40.8 | 41.1 | 41.3 | 21.1 | 42.3 | 28.3 | 46.9 | 26.7 |
Partitioning data for trivalent cations from nine experimental runs selected (essentially at random) from the studies listed in Table 1 are plotted against ionic radius in Fig. 3 and compared with model curves derived both from the original WvW01 formulation and from the current work. In Fig. 3a, data from two experiments performed under nominally anhydrous conditions at ∼3 GPa on basaltic (Salters and Longhi 1999) and eclogitic (Pertermann et al. 2004) starting materials are well matched by the new model curves, with significant improvement over the WvW01 model. Note that both of these experiments lack data for Sc, and hence were not included in the statistical database. A slight mismatch persists between the model predictions and the data from the Salters et al. (2002) experiment, although we note that the mismatch is less than it was for the WvW01 model. This panel illustrates that the new model performs as well or better than did the original WvW01 formulations for generally basaltic systems typical of melting in Earth’s mantle.
In Fig. 3b, we plot data from two nominally anhydrous experiments on Fe-rich bulk compositions relevant to extraterrestrial petrogenesis, performed at 3–5 GPa. Both data sets were included in the statistical database, because the number of garnet partitioning data from such systems is extremely limited (Draper et al. 2003, 2006). Unsurprisingly, the new model does an excellent job of accounting for these measurements, and again constitutes a substantial improvement over the mismatches with the WvW01 formulations (see also Fig. 1). Thus it would appear that garnet partitioning in anhydrous, Fe-rich systems appropriate for lunar or martian petrogenesis can be accounted for by the new formulation, although full confirmation awaits the appearance of new data in Fe-rich systems not included in the present database.
Figure 3c, d illustrate where the new model succeeds less well than in the above cases. In Fig. 3c, we plot data from two hydrous experiments on basaltic (Barth et al. 2002) to intermediate (Nicholls and Harris 1980) compositions at 2–3 GPa. As before, we see a very significant improvement over the WvW01 model. That model drastically overestimated D_{0} (4,000 to 24,000 instead of a more realistic 10–20), causing the dashed curves on this figure representing the original model to exit from the depicted D versus ionic radius space. Still, the new model yields unacceptable mismatches with the measured D values. As mentioned above, Wood and Blundy (2002) attempted to model the effect of H_{2}O on garnet-melt partitioning using a thermodynamic approach. The D_{0} values used in their calibration were largely based on constrained fits to Sc-free experiments, using WvW01 model values for r_{0} and E as boundary conditions. Owing to the relatively low temperatures involved, our new crystal-chemical formulations predict significantly lower r_{0} and E values than did WvW01 for experiments containing hydrous melts. This difference is illustrated in Fig. 3c by the offset of model curves to lower ionic radius values compared to the experimental measurements, and implies that the model of Wood and Blundy (2002) may need to be re-evaluated using values of D_{0} that are constrained by the new rather than old models for r_{0} and E.
In Fig. 3d, we plot data from studies that produced garnets having a significant majorite content in nominally anhydrous experiments performed at 9–25 GPa. These include one experiment on Homestead L5 ordinary chondrite at 9 GPa (Draper et al. 2003); the experiment reported by Corgne and Wood (2004); and an experiment performed by Walter et al. (2004). The latter study lacked Sc data and so was not included in the statistical database, but the former two runs were included. Once again, the new model does a much better job accounting for the measured data than did the WvW01 formulation. Significant mismatches remain, particularly for high-ionic radius elements like the light rare earths, which are expected to have the smallest D values. Such elements are notoriously difficult to work with in partitioning studies, owing to the effect of incorporation of even the smallest amounts of melt in the garnet trace element analyses. Such incorporation of melt, much richer in low-D elements than is coexisting garnet, can artificially raise the apparent D values for those elements (Draper et al. 2006). We attribute the departure of the LREE partitioning data of Walter et al. (2004) and Corgne and Wood (2004) from the parabolic curve to that very phenomenon. Finally, applying the new model to the single study in which majoritic garnets were produced in hydrous experiments (Inoue et al. 2000) (not shown) yields even worse mismatches than for the lower pressure hydrous data shown in Fig. 3c. The mismatches with hydrous partitioning data constitute our updated model’s largest shortcoming.
Tetravalent cation partitioning
Finally, it would be highly desirable to be able to predict garnet-melt partitioning of 4+ cations to a similar level of success as we have now demonstrated for trivalent cations. Ratios of trivalent cations like the heavy rare earths to high field strength 4+ cations (e.g. Lu/Hf), as well as their isotopic systematics, are widely used geochemical fingerprints for a variety of petrologic processes in terrestrial-planet basaltic magmatism (Salters and Hart 1989; Hirschmann and Stolper 1996; Salters 1996; Beard et al. 1998; Blichert-Toft et al. 1999; Green et al. 2000; Chauvel and Blichert-Toft 2001; Klemme et al. 2002). It is much more difficult, if not impossible, to apply the lattice-strain methodology for predicting 4+ partitioning for a variety of reasons. First, there are fewer elements that meet the requirement to apply the lattice-strain model, namely having the same charge and entering the same crystallographic site, in this case likely the garnet Y-site. Hf, Ti, and Zr meet this requirement, but the ionic radii of Hf and Zr are extremely similar and are close to the expected r_{0} value for 4+ partitioning into garnet’s Y-site (van Westrenen et al. 2001a). Thus, data for additional elements with radii significantly lower and higher than these are needed. Elements with larger ionic radii include Th and U, but these are too large to fit into the Y-site, and U in addition can exist in several valence states over the range of oxygen fugacities common to most magmas. On the smaller side of the lattice-strain parabola, there are virtually no 4+ elements to work with; one possibility would be to estimate the D value for octahedral Si entering into the Y-site for majoritic garnets, but doing so is complicated by our lack of knowledge of how much of the Si in the liquid phase is also octahedrally coordinated. In addition, there is some evidence that some 4+ cations, such as Zr and Hf, may enter into both X- and Y-sites in the garnet structure (van Westrenen et al. 2001a). It is therefore not feasible to evaluate lattice-strain fit parameters for 4+ cations in the same manner as we outlined above for 3+ cations.
In the absence of lattice-strain fit parameters, we used the same variable subsets identified as being linearly independent by our bivariate correlation analysis (Tables 3, 4, 5) in multiple linear regressions with the literature Ti, Hf and Zr D values themselves. These regressions sought to determine whether those subsets could be used to predict directly these partition coefficients. Unfortunately, none of these attempts was successful. In all cases, multiple correlation coefficients and their associated coefficients of determination were very low, ranging from approximately 0.1 to 0.2. It is therefore clear that more detailed work will be required if a predictive model for garnet-melt partitioning of 4+ cations is to be achieved.
Conclusions and future directions
We have developed a new predictive model for garnet-melt partitioning of trivalent cations to be used to model anhydrous partial melting processes under a variety of conditions and bulk compositions relevant to Earth, Moon, and Mars. Data from water-bearing experiments are only poorly reproduced, both for lower pressure systems where garnet is largely pyropic, and for higher pressure conditions where garnet incorporates a significant majorite component.
The model has been successfully applied to anhydrous data from experiments up to 9 GPa. There is a lack of accurate partitioning data between this pressure and the majoritic garnet stability limit of ∼25 GPa that remains to be filled by subsequent experimentation. No parameter explicitly connected to the increasing majorite content in garnet with increasing pressure was found to be statistically significant, although our crystal-chemical model for E (van Westrenen and Draper this issue) does incorporate such a term. Nevertheless, mismatches between measurements in majoritic garnet and our updated model are much smaller than the mismatches between measurements and the WvW01 version. We suggest that when the pressure gap is filled, we will have data reflecting the gradual transition to majorite from lower pressure pyropic garnets, and this enhanced coverage may allow us to account better for the onset of the majorite transition.
Accordingly, the next steps in achieving a predictive model that can handle virtually all compositional factors and apply to petrogenesis anywhere on the terrestrial planets should include (1) greater coverage of the pyrope-to-majorite transition; (2) more complete coverage of pressures between ∼9 and 25 GPa; and (3) a systematic effort to understand the effect of water of a scale similar to the one mounted thus far in anhydrous systems, aimed at improving upon the initial garnet-hydrous melt calibration of Wood and Blundy (2002) as described earlier. We speculate that the role of water will also help elucidate the effects of melt composition and structure, given the profound effects the presence of water has on melting and crystallization reactions in magmatic systems.
Acknowledgments
We are grateful to Dr. Charles K. Shearer and Mr. Paul V. Burger for invaluable assistance in performing trace element analyses using the University of New Mexico SIMS facility, and to Mr. Michael Spilde for his help with electron microprobe analyses in UNM’s electron beam laboratory. Journal reviews by an anonymous reviewer and, especially, by Dr. Marc Hirschmann greatly improved this paper. This research was supported by National Science Foundation grant EAR-0337237 to DSD and a European Young Investigator (EURYI) award to WvW.