Quantifying garnet-melt trace element partitioning using lattice-strain theory: new crystal-chemical and thermodynamic constraints

A new model for r ₀

Based on observed variations in r ₀ in a limited number of experiments performed at pressures between 2.5 and 5 GPa, the pressure dependence of r ₀ was previously constrained to −0.005 Å/GPa (van Westrenen et al. 2000a). This pressure dependence leads to severe mismatches between observed and predicted r ₀ at pressures near the upper pressure stability limit of majorite (>20 GPa). For example, the van Westrenen et al. (2000a, 2001) calibration predicts \(r_0 = 0.82\,\hbox{\AA}\) for the Corgne and Wood (2004) and Walter et al. (2004) experiments at 23–25 GPa, compared to the observed values that range from 0.88 to 0.91 Å.

In the new model, the measured pressure dependence of the dimensions of the large dodecahedral garnet X-site, on which the REE are known to reside (e.g., Quartieri et al. 1999a, b, 2002, 2004), is used as an independent means to constrain the pressure dependence of r ₀. The decrease in average dodecahedral cation–oxygen distance <X–O> with increasing pressure was determined to high accuracy by Zhang et al. (1998) for end-member pyrope (Mg₃Al₂Si₃O₁₂) garnet (Fig. 4). These data were taken to be representative for the behaviour of r ₀ in all garnet and majorite compositions. As shown in Fig. 4, <X–O> varies linearly across the complete pressure range relevant to igneous garnet/majorite stability (i.e., between 2.5 and 25 GPa), leading to a predicted change in r ₀ of −0.0044 ± 0.0003 Å/GPa.

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The preponderance of experiments at 3 ± 0.1 GPa (Table 1, Fig. 3) allows for the unambiguous identification of a clear temperature dependence of r ₀, with r ₀ varying at a rate of 0.000058 ± 0.000007 Å/K (Fig. 5). Although intuitively it seems reasonable to expect an increase in r ₀ values with increasing temperature as crystal lattices expand, due to the small temperature ranges involved this effect could not be resolved previously in lattice-strain based REE mineral-melt partitioning models. The resulting predictive equation for r ₀ incorporating both compositional, P, and T effects (with compositions given in terms of mole fractions of the garnet end-members, P in GPa, and T in Kelvin, and taking the values of 3 GPa and 1,818 K mentioned above as anchor points) is given below (Eq. 3):

$$ \begin{aligned} r_0\,(\hbox{REE},\hbox{\AA}) &= 0.9302\,X_{\rm Py} + 0.993\,X_{\rm Gr}+ 0.916\,X_{\rm Alm} + 0.946\,X_{\rm Spes} + 1.05\,(X_{\rm And}+ X_{\rm Uv})\\ &\quad - 0.0044(\pm 0.0003)\,(P - 3) + 0.000058(\pm 0.000007)\,(T -1818). \end{aligned} $$

(3)

For majoritic garnets, mole fractions to be used in Eq. 3 were calculated as follows: First, all Ca, Fe²⁺, Mn²⁺, Na and K was assigned to the large X site. Mg was then divided between the X-site (leading to a total of three cations per formula unit) and the Y site. The pyrope fraction X _Py was then calculated looking at the X-site composition only. We opted to assume only Mg entered the smaller majorite Y-site, because it is the smallest of the major 1+/2+ cations in majorite. Alternative site assignments (e.g., partially assigning Fe²⁺, Mn and Mg to the Y-site) would lead to a slightly different fit parameter for the temperature dependence in Eq. 3.

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Equation 3 predicts r ₀ for all 57 experimental data sets to within 0.017 Å (1σ), compared to 0.032 Å (1σ) achievable with van Westrenen et al. (2000a, 2001). The 2001 model performed increasingly badly as pressure increased towards the upper majorite stability limit of approximately 25 GPa in terrestrial mantle compositions (Walter et al. 2004; Corgne and Wood 2004). Crucially, the new model shows no correlation between absolute and relative values of r ₀ misfit and experimental pressure.

Notably, this major improvement was achieved without introducing into Eq. 3 any explicit term dealing with variations in garnet octahedral Y site composition, related to the incorporation of excess Si (in the case of majorite-bearing garnets) or Ti (in the case of Ti-rich experiments). Ostensibly, within the accuracy required for interpreting element partitioning data, the effective radius r ₀ of the large dodecahedral X site in garnet and majorite appears unaffected by changes in the compositions of the Y (and tetrahedral Z) sites. Neither increasing garnet Ti content (Bennett et al. 2004; Pertermann et al. 2004; Dwarzski et al. 2006), nor increasing majorite content with increasing pressure (Draper et al. 2003, 2006; Corgne and Wood 2004; Walter et al. 2004) affect r ₀ for the REE. As discussed in the next section, this contrasts sharply with observed variations of the apparent Young’s modulus E of the garnet X site, predictions of which do require explicit incorporation of the effect of the presence of Si and/or Ti on the garnet Y site.

A new model for E

The observed apparent Young’s modulus E for trivalent trace elements entering the garnet X site is large, both in absolute terms, and in relative terms compared to E values for trivalent cations entering crystallographic sites in other important igneous minerals, such as the clinopyroxene M2 site (e.g., Wood and Blundy 1997). Fitted 3+ E values for the experiments listed in Table 1 range from 465 ± 40 GPa to 730 ± 32 GPa. These high values reflect the ability of garnet to fractionate heavy from light REE to a much larger extent than most other major rock-forming minerals, leading to steep REE patterns in melts produced in the presence of residual garnet (the so-called ‘garnet signature’ identified in many terrestrial mantle melts, e.g., Salters and Hart 1989; Shen and Forsyth 1995; Hellebrand et al. 2002).

Young’s moduli of garnet structures cover the range 245–275 GPa (depending on chemical composition), virtually identical to the range for reported majorite values (240–285 GPa) (Whitney et al. 2007; van Westrenen unpublished compilation). Corrected from the nominal charge of 2+ to a charge of 3+ using the Hazen and Finger (1979) formalism, this translates into ‘expected’ E values between 360 and 428 GPa. All apparent Young’s moduli derived from garnet-melt and majorite-melt partitioning datasets lie significantly above this range. The discrepancy becomes even larger when considering that the Young’s modulus of the garnet XO₈ polyhedron is lower than that of the garnet structure as a whole, as it is the most compressible polyhedron in the structure. For example, the measured Young’s modulus of the MgO₈ in pyrope is 160 ± 1 GPa, compared to 257 ± 3 GPa for the bulk mineral (Zhang et al. 1998).

As noted previously (van Westrenen et al. 2000a, b), the Hazen and Finger relation (Eq. 4) therefore cannot be used as a predictive model for E. Garnets and majorites are not unique in this respect: even larger deviations have been reported for the clinopyroxene M1 site (Hill et al. 2000) and the Zr site in zircon (e.g., Hanchar and van Westrenen 2007). Van Westrenen et al. (2000b) extensively discussed possible reasons for the mismatch between apparent Young’s modulus and ‘true’ modulus, and we refer the reader to that publication for more information.

van Westrenen et al. (2000a) took an alternative, empirical approach to predicting E. They found that, in accordance with data compilations by Anderson (1972) for oxide, halide and sulphide compounds, a power law dependence of E on r ₀ could reproduce observed E values:

$$E\,(\hbox{REE in garnet, GPa}) = 1.17 \times 10^{12}\,Z_{\rm c} (1.38 + r_0\,(\hbox{REE in garnet, \AA}))^{-26.7}.$$

(5)

Figure 6 shows how poorly Eq. 5 performs when applied to Si-rich (majorite-bearing) and Ti-rich experimental data. On average, Eq. 5 overestimates E values for all data obtained at P > 5 GPa by >300 GPa, far exceeding the average error on the fitted value of E in this pressure range (73 GPa). A fundamental shortcoming of Eq. 5 in light of majorite-melt partitioning data is that it predicts rapidly increasing E values as r ₀ decreases (crosses in Fig. 6). All studies of majorite-melt and Ti-rich garnet-melt partitioning of REE listed in Table 1 (Draper et al. 2003, 2006; Corgne and Wood 2004; Walter et al. 2004; Dwarzski et al. 2006) agree that fitted E values for Si/Ti-rich garnets are smaller than E values for garnets without a majorite/Ti component, while at the same time fitted (and predicted) values of r ₀ in majoritic garnets are smaller (squares in Fig. 6).

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A new model for D ₀: thermodynamic treatment

Entropy of fusion

The entropy of fusion of \(\hbox{JFe}_2\hbox{Al}_3\hbox{Si}_2\hbox{O}_{12}, \Delta S_{{{\rm f}(P,T_{{\rm f}})}}^{0},\) can be estimated by analysing a set of partitioning experiments performed at constant pressure over a wide range of temperatures. In this case, the pressure terms in Eq. 16 are constant, and the right hand side of the equation can be calculated from measured values of garnet and melt major element composition, and fitted values of \(D_0. \Delta S_{{{\rm f}(P.T_{{\rm f}})}}^{0}\) can then be obtained from the slope of a plot of apparent free energy of fusion versus temperature.

As was the case in 2001, the anhydrous garnet-melt partitioning database is dominated by experiments performed at 3 ± 0.2 GPa (n = 34, Table 1). Twenty-five of these are iron-bearing, a requirement to be able to use Eq. 16. Calculated apparent free energies of fusion for JFe₂Al₃Si₂O₁₂ in these 3 GPa experiments are plotted against temperature in Fig. 7. The slope of the resulting straight line gives \(\Delta S_{{{\rm f}({\rm{3GPa}},T_{{\rm f}})}}^{0} = 218 \pm 7\, \hbox{J}\hbox{mol}^{-1} \hbox{K}^{-1}.\) This value is similar to the best-fit value 226± 23 J mol⁻¹K⁻¹ previously derived for JMg₂Al₃Si₂O₁₂ (van Westrenen et al. 2001). The 1σ error of 7 J mol⁻¹ K⁻¹ derived from analysis of garnet-melt partitioning data approaches typical errors quoted for calorimetric entropy of fusion measurements for garnets (e.g., Newton et al. 1977; Téqui et al. 1991). The error is also significantly smaller than 1σ = 23 J mol⁻¹ K⁻¹ previously derived for JMg₂Al₃Si₂O₁₂, due to the much larger temperature range available (300 K compared to 150 K). The entropy of fusion is within error of an interpolation between calorimetric data for pure pyrope \((\Delta S_{{{\rm f}(0.1.T_{{\rm f}})}}^{0} = 162 \pm 5\, \hbox{J\,mol}^{-1}\hbox{K}^{-1}\) —Newton et al. 1977; \(\Delta S_{{{\rm f}(0.1.T_{{\rm f}})}}^{0} = 154\, \hbox{J\,mol}^{-1}\hbox{K}^{-1}\) —Téqui et al. 1991) and YAG (230 J mol⁻¹ K⁻¹—Lin et al. 1999). Replacement of either Mg or Fe by trivalent cations, charge-balanced by replacing Si by Al, thus leads to an increase in the entropy of fusion of the resulting garnet component.

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The effect of pressure on \(\Delta S_{{{\rm f}(P,T_{{\rm f}})}}^{0},\) calculated from average garnet thermal expansion data (Fei 1995) to be approximately 8 J mol⁻¹ K⁻¹ over the pressure range from 2.5 to 25 GPa, is similar to the 1σ error resulting from our fitting method (7 J mol⁻¹ K⁻¹). Following van Westrenen et al. (2001) we therefore applied our value of \(\Delta S_{{{\rm f}({\rm \,3GPa},T_{{\rm f}})}}^{0}\) to compensate for the temperature effect on all garnet-melt partitioning data irrespective of pressure. This allows us to estimate the enthalpy and volume of fusion for JFe₂Al₃Si₂O₁₂ using the partitioning data from Table 1. After rearranging Eq. 16, including our estimate for \(\Delta S_{{{\rm f}(P,T_{{\rm f}})}}^{0},\) we obtain:

$$RT\ln {\left({{\left({\gamma^{{\rm garnet}}_{{\rm Fe}} D_{{{\rm{Fe}}}}} \right)}^{2} D_{0} (3 +)} \right)} + 218(\pm 7)T = \Delta H_{{{\rm f}(0.1,T_{{\rm f}})}}^{0} + P\Delta V + \frac{1}{2}{\left(\frac{\partial \Delta V}{\partial P} \right)}P^{2}. $$

(17)

Enthalpy and volume of fusion

By plotting the left hand side of Eq. 16 against pressure, information is obtained about \(\Delta H_{{{\rm f}(0.1,T_{{\rm f}})}}^{0}\) and \(\Delta V_{{{\rm f}(0.1,T_{{\rm f}})}}^{0}\) (Fig. 8). Experiments are now available at 19 different pressures between 2.4 and 25 GPa (Table 1), compared to four pressures in the van Westrenen et al. (2001) model. The pressure dependence in Fig. 8 appears to be linear rather than parabolic. In contrast to the model of Wood and Blundy (2002) we argue that the pressure derivative of the volume change \((\frac{\partial \Delta V}{\partial P})\) in Eq. 17 is therefore negligible. We note that this conclusion could change once reliable majorite-melt partitioning data filling in the pressure gap between 9 and 23 GPa become available.

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The pressure range for which garnet-melt partitioning data are now available is very large. As a result, inferences about pressure and temperature dependencies on garnet-melt D values, especially at the highest pressures, could be affected by inter-laboratory differences in pressure calibration. To prevent unwarranted propagation of possible pressure calibration errors, the slope and intercept of the best-fit line in Fig. 8 were derived from data obtained between 2.4 and 9 GPa only, leading to \(\Delta H_{{{\rm f}(0.1,T_{{\rm f}})}}^{0}= 400.3 \pm 1.1\,\hbox{kJ\,mol}^{-1}\) and \(\Delta V_{{{\rm f}(0.1,T_{{\rm f}})}}^{0}= 4.59 \pm 0.15\, \hbox{cm}^3\hbox{\,mol}^{-1}.\) As seen in the inset of Fig. 8, these values are fully consistent with the 23–25 GPa data. We conclude that there are no systematic errors in pressure calibration between the lower and higher pressure data sets in Table 1.

The enthalpy of fusion of JFe₂Al₃Si₂O₁₂ is similar to the value of 418 ± 12 kJ mol⁻¹ previously obtained for JMg₂Al₃Si₂O₁₂ by van Westrenen et al. (2001). However, the volume of fusion of JFe₂Al₃Si₂O₁₂ is less than half the value of \(\Delta V_{{{\rm f}(0.1,T_{{\rm f}})}}^{0}= 10.4 \pm 1.0\,\hbox{cm}^3\,\hbox{mol}^{-1}\) derived for JMg₂Al₃Si₂O₁₂. Again, errors in the thermodynamic fusion properties of JFe₂Al₃Si₂O₁₂ are significantly reduced because of the growth of the data set covering a much wider pressure range. Calorimetric data on YAG (Lin et al. 1999) show that YAG has an estimated \(\Delta H_{{{\rm f}(0.1,T_{{\rm f}})}}^{0}\) of 516 kJ mol⁻¹. Our derived value of 400.3 ± 1.1 kJ mol⁻¹ is therefore what might be expected from a progression from pyrope to YAG. The estimated melting point of JFe₂Al₃Si₂O₁₂ is 1,835 ± 60 K, slightly lower than that of JMg₂Al₃Si₂O₁₂ (T _f = 1,850 ± 140 K, van Westrenen et al. 2001), in accordance with an observed decrease in melting temperature going from pyrope to almandine (Butvina et al. 2001). T _f is in between the known values for pyrope (1,570 ± 30 K, van Westrenen et al. 2001) and YAG (1,970 ± 30 K, Fratello and Brandle 1993).

Predicting D ₀

In summary, the entropy of fusion of hypothetical REE garnet compound JFe₂Al₃Si₂O₁₂, derived from garnet-melt REE partitioning data, is 218 ± 7 J mol⁻¹ K⁻¹, its enthalpy of fusion is 400.3 ± 1.1 kJ mol⁻¹, and its volume of fusion is 4.59 ± 0.15 cm³ mol⁻¹. These values can be used to predict D ₀, using Eq. 17 (where R is the gas constant, T in Kelvin, P in GPa, and γ _Fe ^garnet is taken from Eq. 14), to yield

$$ D_{0} (\rm REE) = \raise0.7ex\hbox{${\exp {\left(\frac{400290{\left({ \pm 1100} \right)} + 4586{\left({ \pm 150} \right)}P - 218{\left({ \pm 7} \right)}T} {RT} \right)}}$} \!\mathord{\left/ {\vphantom {{\exp {\left(\frac{400290{\left({ \pm 1100} \right)} + 4586{\left({ \pm 150} \right)}P - 218{\left({ \pm 7} \right)}T}{RT} \right)}} {{\left({\gamma ^{{garnet}}_{{Fe}} D_{{Fe}} } \right)}^{2} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${{\left({\gamma ^{{\rm garnet}}_{{Fe}} D_{{\rm Fe}} } \right)}^{2} }$}. $$

(18)

Equation 18 does not incorporate any explicit term to deal with effects on D ₀ of increasing Si (as pressure increases) or increasing Ti concentrations in the garnet Y-site. Si and Ti on the garnet Y-site to a first degree do not appear to influence the thermodynamics of REE exchange between garnet and melt. Eq. 18 does not take into account the possible influence of melt structure and/or composition on garnet-melt partitioning, apart from incorporating an explicit dependence on the Fe content of the melt through the D _Fe term. Whether melt composition and/or structure are important variables in mineral-melt partitioning continues to be hotly debated (e.g., Blundy and Wood 2003a; Gaetani 2004; Schmidt et al. 2006). In the companion paper (Draper and van Westrenen this issue) we will show that an alternative, statistical parameterisation of D ₀ variations requires incorporation of an an additional term involving Fe–Mg exchange between garnet and coexisting melt. Both D ₀ models reproduce experimental data equally well. We conclude that the currently available garnet-melt partitioning data base cannot be used to argue either for or against a melt compositional effect on garnet-melt D values.

A comparison between predicted and fitted D ₀ values for the REE, Y and Sc for the experiments from Table 1 is shown in Fig. 9. D ₀ values varying by over one order or magnitude can be predicted with confidence across the P–T range of garnet stability, including pressures of 25 GPa.

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Together, Eqs. 1, 3, 6 and 18 can be used to predict D _REE,Y,Sc as a function of pressure, temperature, garnet major element composition, and D _Fe. Three examples of this are shown in Fig. 10. Figure 10c shows a comparison between our predictive model and a recently published dataset (Tuff and Gibson 2007) which was not used for any aspect of model construction. To highlight the sensitivity of our model to temperature variations, dotted curves in Fig. 10c illustrate how model predictions vary if temperature is raised or lowered by 25 K compared to the experimental run temperature quoted in Tuff and Gibson (2007). Errors in experimental temperatures on the order of ±25 K are common at the high temperatures required to equilibrate garnet and melt at pressures >3 GPa, and this could well be a reason for some of the relatively small remaining discrepancies between model and data.

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Table 2 lists standard deviations of the difference between predicted and observed D _REE,Y,Sc (total n > 300) that stem from the propagation of errors in the prediction of the three individual lattice-strain parameters. These deviations average from 25% for Er to 70% for Ce. The maximum error in D prediction is 218% for one measurement of D _Dy. This is remarkably low considering the total spread in D _REE,Y,Sc values of over four orders of magnitude.

Table 2

Mismatches (in per cent) between measured and predicted values of garnet-melt partition coefficients from Table 1

Mismatch (per cent)	Ce	Nd	Sm	Eu	Gd	Dy	Er	Yb	Lu	Y	Sc
Minimum	25	10	0.2	3.0	2.1	5.0	1.7	0.0	2.2	1.0	0.8
Maximum	96	127	103	117	61	218	73	160	74	147	131
Average	70	53	42	37	26	50	25	37.1	33	38	33