Contributions to Mineralogy and Petrology

, Volume 154, Issue 6, pp 731–746

Quantifying garnet-melt trace element partitioning using lattice-strain theory: assessment of statistically significant controls and a new predictive model

Authors

    • Institute of Meteoritics1 University of New Mexico
  • Wim van Westrenen
    • Faculty of Earth and Life SciencesVrije Universiteit
Original Paper

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

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, r0(3+), its apparent Young’s modulus E(3+), and the strain-free partition coefficient D0(3+) for a fictive REE element J of ionic radius r0(3+). Resulting values of E, D0, and r0 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 r0 and E values and any combination of variables. However, a highly robust correlation between fitted D0 and garnet-melt Fe–Mg exchange and DMg 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 D0 with the new expressions for predicting E and r0 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, r0, and D0. 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

In the first paper in this companion pair (van Westrenen and Draper, this issue), we reviewed progress made in updating the predictive model of van Westrenen et al. (2001b) for garnet-liquid trivalent trace element partitioning, referred to as WvW01 hereafter, in terms of improved crystal-chemical and thermodynamic formulations made possible by the inclusion of new partitioning data produced since the initial publication of WvW01. The original model did not account for the presence of a majorite component in garnet, particularly in Fe-rich compositions typical of lunar and martian magmas (Draper et al. 2003, 2006; Dwarzski et al. 2006), leading to unacceptably large mismatches between model and observation. Figure 1 shows examples of these mismatches. This shortcoming has prompted our efforts to expand the database for garnet-melt partitioning, combining data we have produced for that purpose with those generated by other groups since WvW01. Success in this effort would allow the predictive model to be used to constrain the geochemical role of garnet in the generation of lunar and martian magmas, as well as those on Earth.
https://static-content.springer.com/image/art%3A10.1007%2Fs00410-007-0235-3/MediaObjects/410_2007_235_Fig1_HTML.gif
Fig. 1

Illustration of mismatch in Fe-rich systems between measured garnet-melt partitioning and that predicted by WvW01 model. Data are from experiments on Apollo 15 green C, a low-Ti lunar picrite glass composition (Draper et al. 2006) and on Homestead L5 ordinary chondrite (Draper et al. 2003). Solid curves are fits of data to Eq. 1, dashed curves are predictions given by WvW01 model

In this contribution, we augment the theoretical formulations presented by WvW01 and in van Westrenen and Draper (this issue) to include statistical evaluations of empirical relationships between garnet-melt partitioning values for trivalent cations, as expressed by fits of experimental data to the lattice-strain equation, and parameters that are relatively easy to measure or estimate, such as garnet and melt major element compositions, pressure, and temperature. We define D values as the weight ratio of an element of interest between coexisting garnet and melt, i.e. D = concentrationgarnet/concentrationmelt, following the terminology of Beattie et al. (1993). We use D values to obtain values for E, the apparent Young’s modulus of the garnet X-site, where trivalent trace elements such as the rare earth elements typically reside (Quartieri et al. 1999a, b; 2002), r0, the ionic radius of the “ideal” trace element cation that dissolves into that site without straining the crystal lattice, and D0, the maximum D value possible for a cation of a given charge (in this case, 3+) occupying that site. D values are related to these three parameters by the lattice-strain equation, first promulgated by Blundy and Wood (1994) and Wood and Blundy (1997) to rationalize plagioclase-melt and clinopyroxene-melt D values, and later applied to garnet in the WvW01 model:
$$ D_{i} = D_{0} \times {\rm exp}{\left[{\frac{{- 4\pi EN{\left[{\frac{{r_{0}}}{2}{\left({r_{i} - r_{0}} \right)}^{2} + \frac{1}{3}{\left({r_{i} - r_{0}} \right)}^{3}} \right]}}}{{RT}}} \right]}, $$
(1)
where Di is the measured partition coefficient for element i, ri is the ionic radius of that element, R is the gas constant, N is Avogadro’s number, and T is in Kelvin. This expression helps attach physical meaning to the long-known near-parabolic trends formed when D values for elements having the same charge, partitioning into the same crystallographic site, are plotted against ionic radius, as was first detailed by Onuma et al. (1968).

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 DMg in WvW01, and DFe in the van Westrenen and Draper (this issue) formulation for predicting D0 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 D0 that performs as well as the D0 expression formulated from thermodynamic considerations in van Westrenen and Draper (this issue), while requiring only two controlling parameters, DMg 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 D0 that the influence of melt composition and structure is most likely to be manifest. The revised model, consisting of this statistical D0 expression combined with expressions for r0 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

Experimental data used in our modeling come from three categories of sources: those that were already available for inclusion in the original WvW01 model; those we have produced in studies intended in part for the specific purpose of expanding the compositional range available for WvW01; and those that have been published in the intervening years by other research groups. Table 1 lists these data sources along with the general range of compositions (basaltic, sensu lato, unless otherwise noted) and experimental conditions.
Table 1

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

aNot all experiments in these studies included data for Sc, and hence not all were included in statistical database

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 D0, r0, 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 r0 values that result from garnet-melt partitioning data (van Westrenen et al. 1999). For natural garnets, r0 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 D0, r0, 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 r0, drastically different and unrealistic results are obtained. For run MP240, the Sc-included fits yield values for D0, E, and r0 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 r0 based on Sc-bearing data can be combined with Eq. 1 to obtain best-fit values for E and D0 in Sc-absent experiments, e.g. van Westrenen et al. (2000b). These values can then be used to construct predictive models for E and D0. 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 D0, r0, 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 D0, r0, 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 (D0r0, 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

The 16 specific variables used in our statistical calculations, listed in Table 2, were chosen to reflect the potential influence of intensive and extensive parameters on garnet-melt partitioning. Variables expressing garnet compositional variation include contents of the endmember components pyrope (Py), grossular (Gro), and almandine (Alm), calculated from electron microprobe determinations of the major-element compositions of garnets grown in the experiments. Variables chosen to express the transition from “normal” garnet to higher-pressure, majoritic garnet are the stoichiometric values of silicon and aluminum per 12-oxygen formula unit; with increasing majorite content, Si increases from the canonical 3.0 atoms per formula unit (apfu) typical of lower pressure, pyropic garnets, while Al decreases from the canonical 2.0 apfu. These changes reflect the enhanced solubility with increasing pressure of a pyroxene component, R22 +Si2O6, in the Y-site of garnet, R 32 +R 23 +[SiO4]3, via the substitution VI[R2+ + Si4+] = 2VIR3+ (Xirouchakis et al. 2002; Draper et al. 2003); this change can be envisioned as a solid solution between pyrope and enstatite. Variables chosen to address garnet-melt major element equilibrium include D values for Ca, Mg, and Fe (the major constituent cations for the garnet X-site) and KD values for both Fe–Mg and Al–Si exchange, i.e. \({\rm K}^{{{\rm Fe} -{\rm Mg}}}_{D} = \frac{{{\rm FeO}^{{{\rm gt}}} {\rm MgO}^{{{\rm liq}}}}}{{{\rm FeO}^{{{\rm liq}}} {\rm MgO}^{{{\rm gt}}}}};K^{{{\rm Al} - {\rm Si}}}_{D} = \frac{{{\rm Al}_{2} {\rm O}^{{{\rm gt}}}_{3} {\rm SiO}^{{{\rm liq}}}_{2}}}{{{\rm Al}_{2} {\rm O}^{{{\rm liq}}}_{3} {\rm SiO}^{{{\rm gt}}}_{2}}}\) where gt and liq refer to contents in coexisting garnet and liquid, respectively. Melt composition is addressed using melt FeO/SiO2, MgO/SiO2, and Mg# [molar Mg/(Mg + Fe)]. Melt structure is represented by the variables NBO/t, the ratio of non-bridging oxygens to tetrahedrally coordinated cations in a liquid composition (Mysen 1983), and ionic porosity, a measure of the “empty space” in a silicate melt (Dowty 1980; Fortier and Giletti 1989; Carroll and Draper 1994). Although the data set (Table 1) contains several garnet-hydrous melt partitioning experiments, H2O was not included as a parameter in our statistical assessments. This is because only seven experimental runs in extant studies measuring garnet partitioning under hydrous conditions included Sc in the suite of analyte elements (Jenner et al. 1993; Barth et al. 2002). As discussed above, inclusion of Sc is essential for assessing all three lattice-strain fit parameters simultaneously. Again, it would be possible to constrain D0 values from r0 and E models based on Sc-bearing experiments, as was done by Wood and Blundy (2002) in their attempt to quantify the effect of H2O on garnet-melt partitioning. However, doing so introduces an additional level of model-dependence, which we were eager to avoid. In short, the currently available data for garnet-melt partitioning in hydrous systems are insufficient to include H2O as a variable for statistical evaluation, and such experiments should clearly be a focus for future work.
Table 2

Variables chosen for possible use in statistical regressions

Quantity

Description

DCa, DMg, DFe

Garnet-liquid D values for major constituents of X-site

KD 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/SiO2

Ratio of wt% oxide MgO to SiO2 in melt

Melt FeO/SiO2

Ratio of wt% oxide FeO to SiO2 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.

Tables 3 and 4 give the correlation matrices for these two cases. Each entry in these tables consists of the Pearson coefficient (upper line) and two-tailed significance (lower line) for each possible variable pair. Entries for which correlations are not significant (p > 0.05) are denoted in boldface type. In both sets of results, the garnet-melt exchange coefficients for Fe–Mg and for Al–Si are the least correlated with the others; some variables (e.g. DFe) are significantly correlated with almost all of the other chosen variables. From these two tables, subsets of variables can be chosen that contain only variables that are linearly independent of one another. For the assessment that included the hydrous-experiment data, fourteen such subsets can be identified; for the hydrous-excluded assessment, 15 subsets can be used. Some of the subsets are common to both groups, whereas others appear in only one group (these will be presented below). Our next step was therefore to perform multiple linear regressions on each of these subsets to determine whether any of them usefully predict any of the lattice-strain fit parameters D0, r0, and E.
Table 3

Matrix showing correlations between variables chosen for statistical analysis; data from hydrous experiments included

 

10,000/T

P (GPa)

Fe–Mg KD

Al–Si KD

FeO/SiO2

MgO/SiO2

Mg#

Ionic Porosity

NBO/t

Mol% Gr

Mol% Py

Mol% Alm

DMg

DCa

DFe

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 KD

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 KD

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/SiO2

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/SiO2

−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

DMg

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

DCa

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

DFe

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

 

Upper line for each entry is the Pearson correlation coefficient; lower line is p-value for significance of correlation (2-tailed). Variable pairs having p < 0.05 are correlated at 95% confidence. Tabulated p values of 0.000 signify < 0.0001. Entries in boldface denote variable pairs that do not show significant correlations with one another and are thus suitable for further statistical analyses

Table 4

Matrix showing correlations between variables chosen for statistical analysis; data from hydrous experiments omitted

 

10,000/T

P (GPa)

Fe–Mg KD

Al–Si KD

FeO/SiO2

MgO/SiO2

Mg#

Ionic Porosity

NBO/t

Mol% Gr

Mol% Py

Mol% Alm

DMg

DCa

DFe

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 KD

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 KD

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/SiO2

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/SiO2

−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

DMg

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

DCa

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

DFe

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

 

Details as in Table 3

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 r0 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 (R2) below 0.25. Plots of predicted versus measured values of E and r0 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 r0 shows, in our view, that it is essential to base r0 models on systematic experiments in simple systems, as discussed previously in Van Westrenen et al. (1999, 2000a, b). The crystal-chemical r0 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.

In contrast, correlation coefficients and coefficients of determination for models of D0 indicate potentially useful predictive expressions. Multiple linear regression results for the D0 expressions are summarized in Table 5. This table lists the variable subsets that we determined to be linearly independent of one another using the bivariate correlation calculations presented above. For each of the two versions of the dataset (one including the few hydrous experiments, the other excluding them), we tabulate R2 values for the four types of regression computed for each of these subsets. For the hydrous-included data, the best result is for an expression containing Fe–Mg KD, Al–Si KD, DMg, and garnet mol% almandine, with an R2 value of 0.731. However, an even stronger result emerges from the regressions performed on the hydrous-excluded data. As shown from bivariate correlation results given in Table 4, mol% almandine cannot be used in combination with Fe–Mg KD, Al–Si KD, and DMg in this case, because it shows a correlation with Al–Si KD with a significance <0.05. The results for the three remaining independent variables give an R2 of 0.867 (boldface entry in Table 5). Below, we examine this result in more detail.
Table 5

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 KD, Al–Si KD, melt FeO/SiO2

0.706

0.691

0.691

0.691

 10,000/T, Fe–Mg KD, Al–Si KD

0.682

0.651

0.674

0.651

 P (GPa), Fe–Mg KD, Al–Si KD, melt FeO/SiO2

0.471

0.429

0.470

0.429

 P (GPa), Fe–Mg KD, Al–Si KD

0.409

0.385

0.385

0.385

 Fe–Mg KD, Al–Si KD, melt MgO/SiO2

0.577

0.536

0.572

0.536

 Fe–Mg KD, Al–Si KD, garnet mol% Alm, DMg

0.731

0.731

0.731

0.731

 Fe–Mg KD, Al–Si KD, melt Mg#

0.489

0.487

0.487

0.487

 Fe–Mg KD, Al–Si KD, garnet mol% Py

0.520

0.498

0.498

0.498

 Fe–Mg KD, Al–Si KD, garnet mol% Alm

0.463

0.441

0.441

0.441

 Fe–Mg KD, Al–Si KD, DMg

0.609

0.608

0.608

0.608

 Fe–Mg KD, Al–Si KD, DCa

0.698

0.642

0.698

0.642

 Fe–Mg KD, melt ionic porosity, melt Mg#

0.641

0.637

0.637

0.637

 Fe–Mg KD, garnet mol% Gr, melt Mg#

0.555

0.555

0.555

0.555

 Fe–Mg KD, Al–Si KD, 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 KD, Al–Si KD

0.631

0.586

0.631

0.586

 P (GPa), Fe–Mg KD, Al–Si KD

0.416

0.371

0.416

0.371

 P (GPa), melt FeO/SiO2, DCa

0.696

0.696

0.696

0.696

 Fe–Mg KD, Al–Si KD, melt MgO/SiO2

0.549

0.548

0.548

0.548

 Fe–MgKD, Al–Si KD, DMg

0.867

0.867

0.867

0.867

 Fe–Mg KD, melt ionic porosity, garnet mol% Alm

0.680

0.659

0.659

0.659

 Fe–Mg KD, melt NBO/t, garnet mol% Alm

0.608

0.608

0.608

0.608

 Fe–Mg KD, Al–Si KD, Al per 12 oxygens

0.416

0.353

0.353

0.353

 Fe–Mg KD, Al–Si KD, Si per 12 oxygens

0.422

0.399

0.399

0.399

 Fe–Mg KD, Al–Si KD, DCa, Al per 12 oxygens

0.737

0.707

0.737

0.707

 Fe–Mg KD, DCa, Al per 12 oxygens

0.721

0.707

0.707

0.707

 Fe–Mg KD, Al–Si KD, DCa

0.619

0.54

0.619

0.54

 Al–Si KD, DCa, Al per 12 oxygens

0.714

0.707

0.707

0.707

 DCa, Al per 12 oxygens, melt FeO/SiO2

0.722

0.707

0.707

0.707

See text for discussion. All abbreviations as in Table 2. Most successful multiple linear regression indicated with boldface; details of that model given in Table 6

Successful D0 model

The apparently successful model listed in boldface in Table 5 relates D0 to the variable subset Fe–Mg KD, Al–Si KD, and DMg. The R2 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 KD 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 KD and DMg, as well as for the F statistic associated with the overall regression result. From these results, it is clear that Al–Si KD is not in fact required in predicting D0.

The resulting expression for predicting D0 (Table 6) is 3.24 ± 1.46 (Fe–Mg KD) + 3.01 ± 0.20 (DMg) – 2.59 ± 1.04 (errors are 1σ) The WvW01 model also included a DMg term. The addition of the Fe–Mg exchange term represents an additional effect of melt composition on predicted D0 that was hitherto not identified. In Fig. 2a we plot the predicted D0 values against those from fits of the experimental data to Eq. 1; compare to Fig. 9 in van Westrenen and Draper (this issue). The three outlying points at high predicted D0 are the three Sc-bearing hydrous experiments of Barth et al. (2002), suggesting the model will not reliably predict partitioning in hydrous systems; we return to this point more fully below. A final indication that the variable Al–Si KD is not required to predict D0 is that if the regression coefficients including that variable are used, the positions of the plotted points are virtually indistinguishable from those on Fig. 2. That is, removal of that variable results in the same predicted D0 values as when it is included. In Fig. 2b, we compare the D0 values predicted by the thermodynamic and statistical approaches to fitted values for the subset of data for which this direct comparison is possible (see above and van Westrenen and Draper, this issue, for these constraints). Here it is evident that the two models predict D0 equally well. There are no obvious conditions or compositions over which one model more successfully predicts D0 than does the other, for example at one range of temperature or pressure or over a particular compositional range.
Table 6

Details of successful statistical D0 model

Variable

Coefficient

t Statistic

P value

Fe–Mg KD

3.24 ± 1.46

2.22

0.033

DMg

3.01 ± 0.20

15.19

<0.001

Intercept

−2.59 ± 1.04

−2.48

0.018

Abbreviations as in Table 2. Multiple regression coefficient (R) = 0.931; coefficient of determination (R2) = 0.867. Model F value = 117 with significance <0.001

https://static-content.springer.com/image/art%3A10.1007%2Fs00410-007-0235-3/MediaObjects/410_2007_235_Fig2_HTML.gif
Fig. 2

a Plot of D0 fitted to Eq. 1 from Sc-bearing experimental datasets versus D0 predicted from multiple linear regressions (Tables 6, 7). Line illustrates one-to-one correspondence. Compare with Fig. 9 of van Westrenen and Draper (this issue). b Comparison of D0 predicted from thermodynamic (Thermo in legend) and statistical (Stats in legend) models to D0 fitted to Eq. 1 from Sc-bearing experimental datasets (error bars omitted); direct comparisons possible for only ∼25 experiments because of constraints on data useable in the two approaches as described in these two companion papers. Statistical D0 expression yields predictions as good as thermodynamic expression, although uncertainties are slightly greater. See text for discussion

Updated model and comparisons

New model equations

Our updated predictive model now consists of new expressions for E, r0, and D0. The former two were derived in the first paper of this companion pair, and prediction of D0 is possible using either the thermodynamic formulation in that paper or the statistical evaluations outlined here. The four expressions are written out in full in Table 7. An Excel spreadsheet set up to yield values for E, r0, and D0 from the relevant parameters, and to calculate through Eq. 1 predicted values for garnet-melt D values for the REE, Y and Sc, is available both by request from the authors and at the websites http://www.unm.edu/∼draper/geo.html and http://www.geo.vu.nl/∼wvwest/.
Table 7

Preferred predictive expressions

Equation 3 of van Westrenen et al. (this issue):

r0 =  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 + r0)−3 +  12.4(±13) (P) − 0.072(±7) (T) +  237(±8) (Al + Cr)apfu

Thermodynamic D0:

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 D0:

D0 =  (3.24 ± 1.46) Fe–Mg KD + (3.01 ± 0.20) DMg – (2.59 ± 1.04)

Abbreviations as in Table 2, with Spes, And, Uv =  mol % spessartine, andradite, uvovarite respectively; (Al + Cr)apfu = Al and Cr cations per 12-oxygen formula unit; DFe =  FeOgarnet/FeOmelt. R is the gas constant, P in GPa, T in Kelvin. Value calculated for r0 in first expression is used in that for E. γgarnetFe from Eq. 14 of van Westrenen et al. (this issue)

The crystal-chemical expression for r0 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 r0 values against temperature, that when propagated result in uncertainties on r0 of 0.01–0.001 Å, compared to typical absolute values of 0.88–0.99 Å. Model E values incorporate propagated uncertainties in r0, 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 D0 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 D0 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 D0 expressions. Advantages of the statistical D0 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 D0 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 KD dependence on P and T is required to use the statistical D0 expression in these cases.

At the completion of our statistical evaluations, we were unable to isolate a variable that can tie D0 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 KD, 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

In Table 8 and Fig. 3 we present comparisons between measured D values and those predicted from the updated model. Table 8 lists percent relative mismatches between model predictions and measurements for trivalent cation partitioning, defined as
$$ \frac{{|D_{{{\rm predicted}}} - D_{{{\rm measured}}} |}}{{D_{{{\rm measured}}}}} \times 100, $$
where Dpredicted is the D value for a given cation predicted from the model equations, Dmeasured is the measured value, and the absolute value of the difference is used in the numerator. Table 8 provides the minimum, maximum, and average mismatches for each element, for both cases where the thermodynamic and the statistical expressions for D0 are used (recall that E and r0 are always predicted using the thermodynamic relations). For most elements, the average mismatch is on the order of a few tens of percent relative. Note that for seven elements (Ce, Nd, Sm, Gd, Dy, Er, and Lu), the use of the statistical D0 expression yields lower or equal maximum and average mismatches, whereas for the thermodynamic D0 case, Sc, Y, Eu, and Yb have the lowest average mismatches. Values for Gd, Dy, and Er are essentially indistinguishable between the two cases.
Table 8

Percent relative mismatches between measured partitioning values and model predictions

 

Sc

Y

Ce

Nd

Sm

Eu

Gd

Dy

Er

Yb

Lu

Thermodynamic E, D0

 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 D0

 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

https://static-content.springer.com/image/art%3A10.1007%2Fs00410-007-0235-3/MediaObjects/410_2007_235_Fig3_HTML.gif
Fig. 3

Plots comparing experimental data with original WvW01 model (dashed curves; WvW01 in legends) and with updated version presented here (solid curves; New Model in legends). a Nominally anhydrous experiments in basaltic systems. Data from run MP169 of Pertermann et al. (2004) and run TM694-3 of Salters and Longhi (1999). Neither run included Sc data, so both were excluded from statistical database. Updated model improves upon WvW01 version for both basaltic (Salters) and eclogitic (Pertermann) compositional systems. b Nominally anhydrous experiments in Fe-rich systems. Data from run A140 on low-Ti lunar picrite from Draper et al. (2006) and runs 223 + 243 on Homestead L5 chondrite (5 GPa, small majorite content) from Draper et al. (2003). Both were included in statistical database. Updated model significantly improves upon WvW01 model. c Hydrous experiments in basaltic to intermediate systems. Data from run 31 of Barth et al. (2002) and run 4091 of Nicholls and Harris (1980). Although updated model is major improvement over WvW01, mismatch is still unacceptably large. Similar relations hold for hydrous experiments producing majoritic garnet at high pressure (Inoue et al. 2000) (not shown). d Nominally anhydrous experiments in ultramafic systems at high pressure, with substantial majorite content in garnet. Data from Corgne and Wood (2004), runs 233 + 244 (9 GPa, larger majorite content) of Draper et al. (2003), run 62 of Walter et al. (2004). Former two included in statistical database, latter one, lacking Sc data, was not. Model update is significant improvement over WvW01. Mismatch with largest ionic radius cations (lightest REE) in Corgne and Wood (2004) and Walter et al. (2004) experiments is attributed to minor incorporation of melt in garnet trace element analyses

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 D0 (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 H2O on garnet-melt partitioning using a thermodynamic approach. The D0 values used in their calibration were largely based on constrained fits to Sc-free experiments, using WvW01 model values for r0 and E as boundary conditions. Owing to the relatively low temperatures involved, our new crystal-chemical formulations predict significantly lower r0 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 D0 that are constrained by the new rather than old models for r0 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 r0 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.

Copyright information

© Springer-Verlag 2007