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Eleanor C R Green, Tim J B Holland, Roger Powell, Owen M Weller, Nicolas Riel, Corrigendum to: Melting of Peridotites through to Granites: a Simple Thermodynamic Model in the System KNCFMASHTOCr, and, a Thermodynamic Model for the Subsolidus Evolution and Melting of Peridotite., Journal of Petrology, Volume 66, Issue 1, January 2025, egae079, https://doi.org/10.1093/petrology/egae079
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Abstract
We have identified and corrected an algebraic error originally made in the formulation of the thermodynamic model for silicate melt in Holland et al. (2018), J. Petrol., 59, 881–900, article egy048. The error was propagated subsequently in Tomlinson & Holland (2021), J. Petrol., 62, 1–23, article egab012. Below we explain the algebraic correction, and describe the scale of its impact on typical calculations, which in many systems is likely to be within model uncertainty at worst, and negligible at best. We provide a correctly formulated re-calibration of the Holland et al. (2018) silicate melt model. In presenting illustrative calculations with this melt model, we take the opportunity to summarise the changes to the accompanying thermodynamic models for minerals that we have implemented over several publications since 2018.
Corrections to: Journal of Petrology, 2018, doi: 10.1093/petrology/egy048, Journal of Petrology, 2021, doi: 10.1093/petrology/egab012
ALGEBRAIC CORRECTION
In the discussion below, we use the term “composition-dependent equation of state” (|$x$|-eos) to refer to the “thermodynamic models” of Holland et al. (2018) and Tomlinson & Holland (2021), sometimes known as “solution models”.
In both the Holland et al. (2018) and Tomlinson & Holland (2021) silicate melt |$x$|-eos, the error arises in the formulation of site fractions on the M site. In the simpler Holland et al. (2018) model, for the H|${}_2$|O-free melt, these site fractions were written in terms of the |$x$|-eos compositional variables as
Here, |${X}_{\mathrm{m}}^{\mathrm{n}}$| is the fraction of element |$m$| on site |$n$|; |$f\kern0em o$|, |$f\kern0em a$|, |$wo$| and |$sl$| are compositional variables; and sumM is the total number of cations on the M site, needed for normalisation, and given by |$\textrm{sumM}=\mathrm{4}f\kern0em o+4f\kern0em a+ sl+ wo$|. The compositional variables are defined as:
where |${p}_i$| is the proportion of end-member |$i$|, and the melt speciation variable |$y$| is given by |$y={p}_{\mathrm{ctL}}$|. The end-member ctL lies inside the compositional subspace created by the end-members woL, silL and q4L, and can be made from them via the internal reaction ctL = woL + silL + |$\frac{1}{4}$| q4L. The denominator in the composition variables arises because the number of mixing units present in the melt is a function of the extent of the internal reaction.
The correct formulation for the erroneous site fractions, expressed first in terms of end-member proportions and then in terms of the compositional and speciation variables, is:
with sumM defined as
We now explain why the first rendition of the site fractions is incorrect. In our approach to model development, the |$x$|-eos for phase |$\ell$| is formulated in terms of the chemical potential, |${\mu}_i$|, of each end-member |$i$| in an independent set of end-members of the phase:
with gas constant R, temperature |$\mathrm{T}$|, |${\mu}_i^0$| the chemical potential of the pure end-member, |${\gamma}_i$| the non-ideal activity coefficient and |${a}_i^{\mathrm{id}}$| the ideal activity of |$i$|. The expressions for the |${\mu}_i$| can be assembled into a single expression for the Gibbs energy of solution of phase |$\ell$|, |${G}_{\ell}^{\mathrm{sol}}$|, via:
although we have not routinely taken this step. For the intended |${\mu}_i$| expressions to be thermodynamically valid, they must be written such that they could be assembled into a |${G}_{\ell}^{\mathrm{sol} }$| expression that is fully separable into the contributions of its individual end-members, i.e. with |${\mu}_i={\mu}_i^{{\prime}}$| for
the standard definition of chemical potential. Here, |${n}_i$| is the number of moles of |$i$|, such that |${p}_i={n}_i/{\sum}_i{n}_i$|.
For the Holland et al. (2018) and Tomlinson & Holland (2021) melt models, when the correct site fraction expressions are used to form the ideal activities of melt end-members fo (Mg|${}_4$|Si|${}_2$|O|${}_8$|), fa (Fe|${}_4$|Si|${}_2$|O|${}_8$|), wo1L (CaSiO|${}_3$|) and sl1L (Al|${}_2$|SiO|${}_5$|), the resultant |${\mu}_i$| expressions are thermodynamically valid. When the incorrect site fractions were used, the expressions written for |${\mu}_i$| led to a |${G}_{\mathrm{liq}}^{\mathrm{sol}}$| function that was not fully separable into the contributions of its end-members, yielding |${\mu}_i\ne{\mu}_i^{{\prime}}$|.
IMPLICATIONS FOR PETROLOGICAL MODELLERS
Given that the |$x$|-eos discussed above have been used by other authors in petrological modelling work, we must examine the effects of the algebraic error on calculations made with the Holland et al. (2018) and Tomlinson & Holland (2021)|$x$|-eos , and offer alternatives for future modelling.
In a recent paper, Weller et al. (2024), we presented a set of |$x$|-eos for petrological modelling in dry subalkaline to alkaline magmatic systems. The melt |$x$|-eos in that set is correctly formulated. However, since the Weller et al. (2024),|$x$|-eos do not extend into volatile-bearing systems, we provide a replacement for the Holland et al. (2018) H2O-bearing melt |$x$|-eos in the Supplements to this Note of Correction. We do not however provide a replacement for the melt |$x$|-eos of Tomlinson & Holland (2021) here, since melting relations in peridotitic systems are the subject of ongoing model development.
In parameterizing the replacement |$x$|-eos, we used all of the constraints listed in Holland et al. (2018) and Tomlinson & Holland (2021), plus: (1) the experiments of Krawczynski & Grove (2012) on Ti-rich Apollo orange glass composition, at 10 kbar, 1435 |${}^{\circ }$|C and 15 kbar, 1400 |${}^{\circ }$|C; (2) the experiments of Toplis & Caroll (1995) for basalt composition SC1, under both oxiding and reducing conditions, at 1 atmosphere and 1110 or 1075 |${}^{\circ }$|C; (3) the 3 kbar experiments of Neave et al. (2019), on the maximum temperature of olivine, plagioclase and clinopyroxene stability in synthetic analogues of Icelandic basalts from Háleyjabunga and Stapafel. We also took the opportunity to replace several of the |$x$|-eos used for the mineral phases in Holland et al. (2018) with more recent versions, as summarised in Supplement 2.
We can make a first order estimate of the effect of the algebraic problem, by comparing calculations using the revised |$x$|-eos with the older calculations of Holand et al. (2018), Tomlinsin & Holland (2021), and Weller et al. (2024). The calculations shown in those papers represent anhydrous peridotitic, basaltic and pyroxenitic bulk compositions, and hydrous tonalitic bulk compositions. The comparison with the revised |$x$|-eos is given in Supplement 3, where reference pseudosections were calculated with the Gibbs energy minimization software MAGEMin, Riel et al. (2022). Green et al. (2016) and Weller et al. (2024) provided a scale for the uncertainties in calculated pseudosections by comparing calculations with experiments. Uncertainties of |$\pm$|1 kbar for a pressure-sensitive boundary, or |$\pm{50}^{\circ }$|C for a temperature-sensitive boundary, can be loosely be considered to represent a |$\sigma$| uncertainty in a boundary that introduces a phase that can reach significant proportions. This uncertainty estimation should be qualified with respect to phases such as ilmenite, which are only ever present as an accessory phases. The presence or absence of accessory phases at a given pressure–temperature coordinate carries much higher uncertainty, because they contribute so little to the Gibbs energy of the assemblage that precise prediction would make extraordinary demands of the |$x$|-eos calibration.
Using these guidelines for estimating the intrinsic uncertainty in the modelling, the combined changes in both the melt and mineral |$x$|-eos generally yield differences that are comparable with this scale, except in the case of boundaries involving the appearance of accessory phases, which should be seen as particularly imprecise due to their very small contribution to the Gibbs energy of the assemblage. The only notable deviation from these guidelines occurs in the calculation for wet tonalite 101 in Fig. S3-6, in which a very large, geologically unrealistic H2O-content has been imposed, sufficient to saturate the system with aqueous fluid at |$>\mathrm{500}$||${}^{\circ }$|C above the solidus (compare with Figure 10a of Holland et al. (2018); bulk composition is from Piwinskii (1968). In this calculation, the plagioclase-out boundary is significantly different in the older versus the newer calculation at 5 kbar and above.
We conclude that the effect of incorrect melt algebra is unlikely to have been harmful in a practical sense for geologically realistic calculations, although it is clearly unsatisfactory for phase equilibrium calculations to have involved a strictly non-thermodymamic formulation.
Data availability statement
Details of new equations of state associated with this Note of Correction are provided in Supplement 1. A complete description of all |$x$|-eos used in the calculations in Supplement 3, along with thermodynamic input files for thermocalc, may be downloaded from https://hpxeosandthermocalc.org.
Acknowledgements
O.W. acknowledges the support of UK Research and Innovation Future Leaders Fellowship grant MR/V02292X/1. N.R. was supported by the European Research Council through the MAGMA project, ERC Consolidator Grant 771143.
