liquid Mg partitioning

    Calcic amphibole thermobarometry in metamorphic and igneous rocks: New calibrations based on plagioclase/amphibole Al-Si partitioning and amphibole/liquid Mg partitioning J.F. Molina, J.A. Moreno, A. Castro, C. Rodr´ıguez, G.B. Fershtater PII: DOI: Reference:

S0024-4937(15)00236-4 doi: 10.1016/j.lithos.2015.06.027 LITHOS 3636

To appear in:

LITHOS

Received date: Accepted date:

19 December 2014 12 June 2015

Please cite this article as: Molina, J.F., Moreno, J.A., Castro, A., Rodr´ıguez, C., Fershtater, G.B., Calcic amphibole thermobarometry in metamorphic and igneous rocks: New calibrations based on plagioclase/amphibole Al-Si partitioning and amphibole/liquid Mg partitioning, LITHOS (2015), doi: 10.1016/j.lithos.2015.06.027

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ACCEPTED MANUSCRIPT Calcic amphibole thermobarometry in metamorphic and igneous rocks: new calibrations based on plagioclase/amphibole Al-Si partitioning and amphibole/liquid Mg partitioning

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J.F. Molina*1, J.A. Moreno1, 2, A. Castro3, C. Rodríguez3, G.B. Fershtater4 1: Departamento de Mineralogía y Petrología, Universidad de Granada, Campus de Fuentenueva

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18002 Granada, Spain

2: Department of Geology and Natural Resources, Institute of Geosciences, University of Campinas -

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UNICAMP, 13083-970 Campinas, SP, Brazil

3: UA Petrología Experimental, CSIC-Universidad de Huelva, Facultad de Ciencias Experimentales,

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21071 Huelva, Spain

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4: IGG Russian Academy of Science, Ekaterinburg, Russia

e-mail: [email protected].

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*: Corresponding author: José F. Molina

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Telephone: ++34958246611.

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Fax: ++34958243668.

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ACCEPTED MANUSCRIPT Abstract

Dependencies of plagioclase/amphibole Al-Si partitioning,

, on temperature, pressure and phase compositions are investigated employing robust

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partitioning,

, and amphibole/liquid Mg

regression methods based on MM-estimators. A database with 92 amphibole-plagioclase pairs —

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temperature range: 650-1050 ºC; amphibole compositional limits: > 0.02 apfu (23O) Ti and > 0.05 apfu Al — and 148 amphibole-glass pairs —temperature range: 800-1100 ºC; amphibole compositional limit: > 0.75 — compiled from experiments in the literature was used for the

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calculations (amphibole normalization scheme: 13-CNK method).

, and albite fraction in plagioclase,

on pressure, temperature, Al fraction in , leading to the barometric expression:

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amphibole T1-site,

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Statistical analysis reveals a significant dependence of

was found to have a significant dependence on temperature and the logarithm of

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The

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(precision: ±1.5 to ±2.3 kbar; expressed at 1s).

ratio (glass composition expressed as anhydrous mole fraction of cation components)

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that led to the thermometer:

(precision: ±35 to ±45 ºC).

Amphibole-liquid compositional relationships revealed that the temperature dependence of was mostly controlled by a positive correlation of

with temperature that led to the thermometer:

(precision: ±37 to ±42 ºC).

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ACCEPTED MANUSCRIPT The plagioclase-amphibole barometer is suitable for a large diversity of amphibole-plagioclasebearing assemblages from metamorphic (amphibolites and mafic granulites) and igneous (metaluminous granitoids to gabbros) rocks, whereas the liquid-only and the amphibole-liquid

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thermometers are applicable to alkaline and subalkaline igneous rocks.

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The calibrated expressions yield P-T estimates consistent with those obtained with widely used barometers and thermometers, and extend the use of the plagioclase-amphibole barometer to quartz-free and/or garnetfree assemblages.

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Keywords

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Plagioclase-amphibole barometry; amphibole-liquid thermometry; liquid-only thermometry; mineral/liquid major-element partitioning; mineral/mineral major-element partitioning

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1. Introduction

Empirically calibrated thermometers and barometers are specially well suited for estimating

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equilibrium P-T conditions in mineral assemblages that involve complex phases whose mixing

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properties are not fully understood (e.g., Schmidt, 1992; Patiño-Douce, 1993; McCarthy and PatiñoDouce, 1998). Recent empirical calibrations of clinopyroxene-melt and calcic amphibole-only

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thermobarometers (e.g., Ridolfi and Renzulli, 2012; Masotta et al., 2013; Mollo et al., 2013) document the interest of these methods for constraining P-T conditions of magma crystallization (see also review in Putirka, 2008).

Calcic amphibole can be an important constituent of a large diversity of hydrous, basic to intermediate, metamorphic and igneous rocks (e.g., Robinson et al., 1982; Wones and Gilbert, 1982; Spear, 1993; Martin, 2007; Molina et al., 2009; Bucher and Grapes, 2011). In the metamorphic environment, aluminous hornblende-bearing assemblages are stable over a wide P-T field that extends from amphibolite- to granulite- and high-T eclogite-facies conditions (e.g., Poli, 1991; Yaxley and Green, 1994; Ernst and Liu, 1998; Molina and Poli, 2000; Bucher and Grapes, 2011). At lower temperatures, the hornblendic amphibole is replaced by sodic-calcic amphibole at relatively high-P conditions (medium-T eclogite- and albite-epidote amphibolite-facies; e.g., Poli, 1991, 1993;

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ACCEPTED MANUSCRIPT Schimidt, 1992, 1993; Ernst and Liu, 1998; Molina and Poli, 1998; Bucher and Grapes, 2011) and by actinolite at lower-pressure greeschist-facies conditions (e.g., Spear, 1993; Bucher and Grapes, 2011). In alkaline igneous systems, kaersutite and high-Ti pargasite are common in highly silica-depleted

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magmatic rocks as foidites, basanites and camptonites, and in alkali basalts to hawaiites and mildly alkalic trachytoids (Martin, 2007; Molina et al., 2009); whereas in subalkaline compositions, the

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stability of calcic amphiboles, mostly lower-Ti hornblende, is restricted to more siliceous magmatic rocks that range from andesites to metaluminous rhyolites (e.g., Sisson and Grove, 1993; Barclay and

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Carmichael, 2004; Molina et al., 2009).

Because this widespread occurrence, numerous amphibole-based thermobarometric expressions have

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been proposed: garnet-amphibole-plagioclase-quartz barometer (Kohn and Spear, 1989, 1990; Dale et al., 2000); amphibole-plagioclase-epidote barometer (Plyusnina, 1982); amphibole-plagioclase-quartz barometer (Bhadra and Bhattacharya, 2007); plagioclase-amphibole barometer (Fershtater, 1990); Al-

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in-amphibole barometer for granitioids (e.g., Hammarstrom and Zen, 1986; Johnson and Rutherford,

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1989; Schmidt, 1992; Anderson and Smith, 1995); semiquantitative Al-in-amphibole thermobarometer for MORB compositions (Ernst and Liu, 1998); amphibole/liquid Na-K exchange thermometer (Helz,

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1979); edenite-quartz-tremolite-albite and edenite-albite-richterite-anorthite thermometers (Holland and Blundy, 1994); and calcic amphibole-only thermobarometers for igneous rocks (Ridolfi et al.,

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2010; Ridolfi and Renzulli, 2012). Despite this great effort, there is still some room for further improvement of amphibole thermobarometry using empirical approaches. First, an experimentally based calibration of the plagioclase /amphibole Al-Si partitioning barometer from Fershtater (1990) has not been done yet, although it may represent an interesting complement to garnet-amphibole-plagioclase-quartz and amphibole-plagioclase-quartz barometry as neither garnet- nor quartz-saturation is required. Second, even though numerous mineral/liquid Mg partitioning thermometers have been proposed (e.g., Roeder and Emslie, 1970; Beattie, 1993; Sugawara, 2000; Gudfinnsson and Presnall, 2001; Putirka, et al. 2007; Putirka, 2008), the capabilities for thermometric purposes of the partitioning of Mg between amphibole and liquid have not been explored yet.

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ACCEPTED MANUSCRIPT In this work, we explore pressure, temperature and phase composition dependencies of the Al-Si partitioning between plagioclase and amphibole, and the Mg partitioning between amphibole and liquid using robust regression methods based on MM-estimators and derive three new

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thermobarometric expressions. A compositional database with 92 amphibole-plagioclase pairs and 148 amphibole-glass pairs compiled from experiments in the literature is used for calibrating and testing

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the thermobarometric expressions. Applications to selected examples of metamorphic and igneous rocks are also discussed.

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2. Experimental database and data selection

For this study, an amphibole-plagioclase-glass compositional database has been created with

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experiments compiled from the literature (Appendixes A and B from the electronic supplementary material). It contains experimental runs with a duration > 240 h at 650-800 ºC, > 120 h at 830-890 ºC,

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> 40 h at 900-1000 ºC and > 20 h at > 1000 ºC. Temperature conditions range from 650 to 1050 ºC for

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amphibole-plagioclase pairs and from 800 to 1100 ºC for amphibole-liquid pairs. Amphibole-plagioclase pairs presenting absolute deviations in estimated amphibole-plagioclase

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temperatures (edenite-quartz-tremolite-albite (Eqn. HB94-A) and edenite-albite-richterite-anorthite (Eqn. HB94-B) thermometers; Holland and Blundy, 1994) < 2.5s (1s ≈ 40 ºC) were selected for fitting

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the barometric expression (Fig. 1). Data selection for amphibole-glass pairs was carried out using constraints imposed from

and

relationships (abbreviations in Table 1).

These parameters show poor correlation with temperature (Figs. 2A-B) and are well suited for the detection of analytical problems in glass analyses (e.g., Na loss during glass analyses and imprecise determination of Ti in high-silica glasses). We selected only those data for which

and

values range, respectively, from -9 to 4 kJ and from -12 to 6.1 kJ (Figs. 2A-C). Thus, we excluded 12 data (ca. 7.5% of the total) with values lying outside these ranges (Fig. 2C). The selected experimental database comprises 92 amphibole-plagioclase pairs and 148 amphiboleglass pairs (Tables 2 and 3). Around 70% of the data (60 amphibole-plagioclase pairs and 102 amphibole-glass pairs) were employed for fitting the thermobarometric expressions (calibration data

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ACCEPTED MANUSCRIPT set), whereas the rest of selected data was used for testing the derived expressions. The number of amphibole-glass pairs used in this work is appreciably higher than that employed recently for calibrating amphibole-only thermobarometric and chemometric expressions by Ridolfi et

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al. (2010) for subalkaline systems (9 to 45 data) and by Ridolfi and Renzulli (2012) for both alkaline

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and subalkaline systems (20 to 61 data). The larger number of data permits, as previously mentioned, to split the compositional database into two groups (calibration and test data sets) that allow us to perform a statistical test of the accuracy and precision of the calibrated expressions. On the other hand,

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the database also includes low-temperature amphibole-plagioclase pairs that make it possible to extend the application of the plagioclase-amphibole barometer derived in this work to subsolidus

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assemblages.

3. Thermodynamic formulation: thermobarometers based on element partitioning

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In this work, we present thermobarometric expressions based on empirical relationships between

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major-element partition coefficients, pressure, temperature, and phase compositions. Below, it is shown how the use of partition coefficients, instead of compositions of a single-phase as in

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amphibole-only thermobarometry (e.g., Ridolfi et al., 2010; Ridolfi and Renzulli, 2012), permits to get relatively simple thermobarometric expressions with a relatively small number of fitting parameters

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that account for compositional dependences of partition coefficients. Since the pioneering studies by Ramberg and De Vore (1951), numerous thermometers and barometers have been developed that are based on the partitioning of elements between coexisting phases (e.g., Ganguly and Saxena, 1987, and references therein). The expressions include both mineral/liquid partition coefficients —e.g., olivine/liquid Mg partitioning thermometer (Roeder and Emslie, 1970; Leeman, 1978; Beattie, 1993; Sugawara, 2000; Gudfinnsson and Presnall, 2001; Putirka et al., 2007); spinel/liquid thermometers (Ariskin and Nikolaev, 1996; Ariskin and Barmina, 1999); clinopyroxene/liquid Al partitioning barometer (Putirka, 2008) — and mineral/mineral partition coefficients —e.g., orthopyroxene/clinopyroxene Na partitioning thermometer (Brey and Köhler, 1990); many thermometers based on the Fe2+-Mg partitioning between coexisting ferromagnesian phases (e.g., Ganguly and Saxena, 1987, Krogh-Ravna and Paquin, 2003, and references therein);

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ACCEPTED MANUSCRIPT olivine/clinopyroxene Ca partitioning thermometer (Köhler and Brey, 1990); olivine/spinel Al partitioning thermometer (Wan et al., 2008); intersector Ca-Na and Ti partitioning in tourmaline (van Hinsberg and Schumacher, 2007). In addition, significant pressure dependence has been detected for

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clinopyroxene/plagioclase Ca-Na partitioning in ultramafic rocks (Borghini et al., 2011) and for clinopyroxene/liquid Na partitioning (Villiger et al., 2007). Two-element partition coefficients have

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also been used for the discussion of mineral-silicate liquid compositional relationships —e.g., amphibole/liquid Si-Al partitioning (Sisson and Grove, 1993); plagioclase/liquid Ca-Na partitioning (Sisson and Grove, 1993; Villiger et al., 2007); Si-Mg, Fe-Mg and Ca-Al partitioning between liquid

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and ferropericlase, MgSi-perovskite and majoritic garnet (Trønnes and Frost, 2002).

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The thermodynamic basis for element partitioning between two coexisting phases has been thoroughly discussed in the literature (e.g., Saxena, 1973; Perchuk, 1977; Hanson and Langmuir, 1981; Langmuir and Hanson, 1981; Ulmer, 1989). A thermodynamic expression for the partition coefficient of an

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element M between two phases  and ,

, can be derived from the equilibrium condition of

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chemical components between coexisting phases (e.g., Saxena, 1973):

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(1a)

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By the definition of chemical potential:

where

and

(2a)

are, respectively, the standard state chemical potential and the activity coefficient

of element M in phase .

Rearranging terms in Eqn. (2a), it is obtained:

(3a)

where, for crystalline phases, the values that can achieve the mole fraction of cations that are essential structural components are limited by stoichiometric constraints imposed by site occupancies and charge balances that preserve phase integrity (e.g., Hanson and Langmuir, 1981; Langmuir and

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ACCEPTED MANUSCRIPT Hanson, 1981). For a second element N, there is an additional relation of equilibrium between phases  and :

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(1b)

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Subtracting relation (1b) from (1a) and rearranging terms we find the expression between chemical

(2b)

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potentials:

that can be considered the equilibrium condition for an exchange reaction when the two element

site in each phase (e.g., Fe2+-Mg2+).

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components have similar ionic radius and the same valence state, occupying therefore the same single

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A relation similar to (3a) can be obtained for Eqn. (2b):

where

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(3b)

is the two-element partition coefficient and verifies that:

.

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When two cations with different valence states (e.g., Si4+-Al3+, Si4+-Mg2+, Ca2+-Al3+ or Ca2+-Na+) are considered, expression (3b) is still valid, even tough relation (2b) could not be related to a simple exchange reaction (see Trønnes and Frost, 2002, for further discussion). An equivalent thermodynamic treatment can be done using oxides, MO, and mineral end-members, MF (where F is an inert framework), as components because the three approaches are related to each other by appropriate formation reactions, i.e., by appropriate definitions of standard states as stated by Navrotsky (1978). Equilibrium relationships expressed in terms of M (or MO) and MF components can be easily transformed to each other in more simple instances. Thus, for example, the partitioning of MgO between olivine and liquid can also be treated in terms of the formation reaction of forsterite component in olivine:

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ACCEPTED MANUSCRIPT 2 MgO (liq) + SiO2 (liq) = Mg2SiO4 (ol) with equilibrium constant:

and

, and an expression relating

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where

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(4)

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can be derived from (4):

and

that evidences a liquid compositional dependence of

. Plagioclase/liquid Ca-Na partitioning can

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also be expressed in terms of the exchange reaction:

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NaSi3AlO8 (liq) + CaSi2Al2O8 (plg) = CaSi2Al2O8 (liq) + NaSi3AlO8 (plg) as a consequence of the simple coupled substitution of Ca2+ and Al3+ by Na+ and Si4+. However, for

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equilibria involving more complex phases, there are not simple relations between M and phase component treatments because various independent phase components contribute to the total content

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of component M. For instance, total Na content in clinopyroxene depends on the extent of acmite (NaFe3+Si2O6) and jadeite (NaAlSi2O6) substitutions and, therefore,

-pressure relationships are

significantly affected by oxygen fugacity at fO2 > FMQ (Villiger et al., 2007). When reactions based on the partitioning of elements between coexisting phases are used for thermobarometric purposes, the compositional dependence of partition coefficients must be empirically corrected. In some cases, relatively simple expressions relating partition coefficients to temperature, pressure and phase compositions can be deduced from constraints imposed by empirical relationships. For instance, Perchuk (1977) showed significant correlation of Mg partition coefficients for garnet and biotite and for garnet and cordierite with their corresponding Fe2+-Mg partition coefficients, whereas Ariskin and Nikolaev (1996) derived thermometric expressions based on

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ACCEPTED MANUSCRIPT spinel/liquid major-element partitioning equilibria that included melt structure-chemical parameters (MSCP) to correct for compositional dependencies —thermobarometric expressions with more complex compositional corrections are summarized in Putirka (2008). In some circumstances, there

be significantly correlated, as noted by Ulmer (1989) who derived with temperature.

-T and

-

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expressions because a strong correlation of

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can be no unique D-P-T-X relation because temperature, pressure and compositional parameters can

—where being r is a single-element or a two-element

In this work, the dependence of

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partition coefficient— on temperature, pressure and composition is given by:

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where the terms

(5)

are functions of the chemical composition of the involved phases and the

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superscript o denotes a standard state property.

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Once the model parameters have been determined by liner regression methods, thermobarometric

(6a)

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Ganguly, 2008):

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expressions can be derived rearranging terms in relation (5) (e.g., Ganguly and Saxena, 1987;

(6b)

Other approaches propose to perform the regression analysis using pressure or temperature as dependent variable (e.g., Putirka, 2008). Although the two methods can yield precise and accurate estimates, it is important to note that they are not equivalent. So, in the first case it is investigated the pressure, temperature and phase composition dependence of

, whereas in the second one,

emphasis is put on the dependence of the variable of interest, temperature or pressure, on the rest of variables. The first procedure is followed in this work as it was found to give more accurate results.

4. Regression methods

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ACCEPTED MANUSCRIPT 4.1. Robust regression methods Robust regression methods must be used when outlying observations are present (e.g., Maronna et al., 2006). In general, three types of outliers are distinguished (e.g., Rousseeuw and Leroy, 2003), shown in

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Figure 3A: 1) vertical outliers: outlying observations in the values of the dependent variable, thus having

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large residuals; 2) bad leverage points: outlying observations in the values of the independent variables and located far from the true regression line; and 3) good leverage points: outlying observations in the values of the independent variables, but located close to the regression line. Ordinary least square (OLS) estimations

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of intercept and slope are affected by the presence of bad leverage points, whereas vertical outliers affect, mostly, OLS estimations of the intercept. The presence of good leverage points does not affect OLS

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estimations, but it deflates the estimated standard errors impacting, therefore, on statistical inference. To alleviate the impact of outliers, Huber (1973) introduced robust regression methods based on

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maximum-likelihood estimators (M-estimators) that are robust to vertical outliers. However, they are not robust to bad leverage points and can fail when clusters of outliers occur due to a masking effect, namely

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one outlier can mask the presence of another (see Rousseeuw and van Zomeren, 1990; Verardi and Croux, 2009). These drawbacks have been properly solved by the MM-estimators introduced by Yohai (1987) —

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the MM refers to the fact that multiple M-estimations are performed during computation. These estimators

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combine high efficiency1 that guarantees comparatively low variance and high breakdown point2 that guarantees a large resistance to outliers. The superiority of MM-estimators over OLS and M-estimators can be illustrated by means of a simple numerical example. Let be an X-Y population consisting of 20 data that approximately obey the expression:

, and 10 outliers (i.e., 33% of the total), and calculate X-Y relationships

using OLS and robust M- and MM-estimators. The results displayed in Figure 3A show that only the MMestimator gets the solution, whereas estimations based on OLS and robust M-estimators fail because the presence of multiple leverage points and vertical outliers. 1

An estimator T2 of a parameter θ is said to be more efficient than a second estimator of the same parameter, T1, if and it is usually measured by the relative efficiency of T2 with respect to T1, (e.g., Dekking et al., 2005, Chapter 20). 2 The breakdown point is the smallest fraction of outliers that can be admitted in a population without substantial impact on the estimate (Andersen, 2008); e.g., an estimator that yields correct parameter estimates in populations contaminated with up to 50% of outliers has a breakdown point of 50%.

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ACCEPTED MANUSCRIPT 4.2. Robust methods for outlier detection When a population is contaminated by a set of outliers, it is also necessary to use robust methods to detect them. As noted by Rousseeuw and van Zomeren (1990), the classical method for outlier detection can fail

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to find outliers when there are several outliers because it is based on the sample mean and covariance

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matrix, which are, in turn, affected by the outliers. In order to detect the outliers, these authors proposed a graphical tool (Fig. 3B) that plots the robust standardized residuals3 on the vertical axis fort identifying vertical outliers and, on the horizontal axis, a measure of the outlyingness of the independent variables

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given by a robust Mahalanobis distance4.

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4.3. Calibration procedure of the thermobarometric expressions Robust regression methods along with regression diagnostics for outlier detection must be used for calibrating thermometric and barometric expressions (e.g., Powell, 1985). Outliers can be caused by

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experimental errors that include errors in reproducing P-T conditions in the laboratory, lack of attainment

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of equilibrium during a run or errors related to compositional measurements (Putirka, 2008). However, they can also appear simply because intrinsic limitations of empirical, linear in many cases, models

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employed for deriving the thermobarometric expressions that fail to accommodate complex underlying

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relationships between variables.

Linear expressions of

with pressure, temperature and phase compositions were derived by

robust regression methods based on MM-estimators using an algorithm implemented in Stata software by Verardi and Croux (2009). The algorithm is called in Stata using the command "mmregress". The algorithm computes, invoking "initial" option, a preliminary S-estimator, i.e., a set of regression parameters that minimizes the scale parameter

. Once the S-estimator is obtained, a MM-estimator is computed by

applying an iteratively reweighted OLS algorithm up to convergence using the 3

value obtained in the

The robust standardized residual is the residual divided by the scale parameter, , a robust estimator of the standard error, (e.g. Verardi and Croux, 2009). 4 The Mahalanobis distance is a measure of the distance of an individual data point to the centre of a multivariate distribution. For a population with n data and p independent variables, the Mahalanobis distance is defined as where is the ith row vector of matrix , is the vector of the centroid of the population and is the covariance matrix of the independent variables; in order to get a robust estimation of , the two latter parameters must be estimated robustly (e.g., Verardi and Croux, 2009).

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ACCEPTED MANUSCRIPT previous stage (see Verardi and Croux, 2009, for details). Calculations were performed with an efficiency fixed at 0.7, which is a good compromise between robustness and efficiency (Verardi and Croux, 2009). Forward selection and backward elimination procedures were employed for selection of independent

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variables; variables with t-values > |5|, what implies for all cases that P>|t| = 0, were included in the model.

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Vertical outliers and leverage points were detected using the robust graphical tool of Rousseeuw and van Zomeren (1990). This is performed invoking the graph option in the mmregress command (Verardi and Croux, 2009). Thus, a plot of robust standardized residuals against Mahalanobis distances is drawn setting (where p is the

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the limits for considering data as outliers to −2.25 and +2.25 for the former and to

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number of regression parameters) for the latter (Verardi and Croux, 2009). In order to develop explicit geo-barometers and geo-thermometers, the fitted expressions were rearranged in the form of the Eqns. (6a) and (6b). Precision and systematic errors in pressure and temperature

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estimates were evaluated by fitting the relationships between their calculated and experimental values using

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both the calibration and the test data sets. OLS regression methods were employed, as, in this stage of the analysis, the main concern is to evaluate how the expressions predict all experimental data from the two

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data sets rather than to search for optimal expressions. Accordingly, the presence of systematic errors was , and experimental,

, values:

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checked evaluating whether the general linear relation between estimated,

is significantly indistinguishable from the one-to-one relation:

This is carried out assessing the closeness of the slope to 1 and performing a t-test on the constant term. As a measure of the model error, i.e., the precision of the expression (Putirka, 2008), we have considered the root-mean-square errors (RMSE) obtained by OLS regression on the calibration and the test data sets. In summary, we have selected those thermobarometric expressions that meet the following conditions: 1) low number of outliers (in fact, one or zero); 2) a precision comparable to that of other thermobarometric

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ACCEPTED MANUSCRIPT expressions published in the literature; 3) relationships between calculated and measured dependent variable significantly indistinguishable from the one-to-one relation; and (4) t-values > |5| for all selected independent variables.

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5. Phase compositions

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5.1. Amphibole

Amphibole formulas and Fe3+/Fe2+ ratios were determined assuming 13 cations exclusive of Ca, Na and K (13-CNK method; Robinson et al., 1982) on a basis of 23 oxygen atoms. This is the IMA-

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favoured procedure because it reproduces Fe3+ and Fe2+ values reasonably close to true determined

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values for calcic (excluding kaersutite), sodic–calcic and sodic amphiboles (Leake et al., 1997). Amphibole OH contents were calculated considering F + Cl + OH = 2.

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Amphibole compositions are described using the following linearly independent set of arbitrary phase components in the seven-component condensed system: Si4+ - Ti4+ - R3+ (Al3+ + Fe3+) - R2+ (Fe2+ +

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Mn2+ + Mg2+) - Ca2+ - R+ (Na+ + K+) - W- (OH- + F- + Cl-); tremolite: (☐Ca2[R2+]5Si8O22[W-]2), Al+Fetschermakite (☐Ca2[R2+]3[R3+]2Si6Al2O22[W-]2), Ti-tschermakite (☐Ca2([R2+]3Ti2Si4Al4O22[W-]2),

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Al+Fe-glaucophane (☐Na2[R2+]3[R3+]2Si8O22[W-]2) and edenite ([R+]Ca2[R2+]5Si7Al1O22[W-]2). When

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necessary, the condensed system is expanded for considering Al-glaucophane (☐Na2[R2+]3Al2Si8O22[W-]2), Al-tschermakite (☐Ca2[R2+]3Al2Si6Al2O22[W-]2) and Fe-tschermakite (☐Ca2[R2+]3[Fe3+]2Si6Al2O22[W-]2). The proportions of phase components are given by expressions reported in Table 1. Amphibole compositions in amphibole-plagioclase assemblages are mostly tschermakite (52% of the data), magnesiohornblende (28%) and magnesiohastingsite (12%), whereas kaersutite, ferrohornblende and sodic-calcic amphibole represent, altogether, only ca. 7.6% of the total (Figs. 4AB; classification after Leake et al., 1997). In amphibole-liquid assemblages, amphibole composition is also dominated by tschermakite that represents 39% of the data, with magnesiohastingsite being slightly more abundant than magnesiohornblende (26 and 17% of the data, respectively); kearsutite is also relatively abundant reaching ca. 13% of the data, whereas pargasite and ferropargasite account for

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ACCEPTED MANUSCRIPT only ca. 5.4% of the total (Figs. 4A-B). 5.2. Plagioclase

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Plagioclase composition varies from oligoclase to bytownite (Ab11-84). Orthoclase contents are

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relatively low reaching up to 8 mol.%. 5.3. Glass

For the amphibole-liquid expression, glass composition is expressed in terms of anhydrous mole

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fractions of cation components (Table 1) following the approach suggested by Putirka (2008).

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Glass encompasses a wide compositional range, presenting ca. 60% of the data subalkaline compositions (Fig. 5A). Around 73% of the data show magnesian compositions (Fig. 5B). Alkaline glasses range from foidites, basanites and alkali basalts to tephrites and phonolites, and from

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trachybasalts to mildly alkalic trachytoids (Fig. 5A). Subalkaline glasses range from basaltic andesites

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and andesites to dacites and rhyolites (Fig. 5A).

estimators

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6. Results: calibration of calcic amphibole-based thermometers and barometers using MM-

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6.1. Plagioclase/amphibole Al-Si partitioning barometer There is compelling experimental evidence that Al-tschermakite substitution in amphibole shows positive correlation with pressure in a large diversity of plagioclase-bearing assemblages (e.g., Poli, 1993, Schmidt, 1993, Anderson and Smith, 1995, and references therein): the nine-phase critical assemblage for granitiods: amphibole + biotite + plagioclase + K-feldspar + quartz + melt + Fe-Ti oxide + sphene + fluid; amphibole + plagioclase + quartz + epidote + garnet; amphibole + plagioclase + epidote + calcite + quartz; amphibole + anorthite + clinopyroxene + quartz; amphibole + anorthite + orthopyroxene + spinel; amphibole + anorthite + clinopyroxene + melt; amphibole + anorthite + corundum + chlorite. For amphibole coexisting with plagioclase from the experimental database synthesized at T > 650 ºC with > 0.02 apfu (23O) Ti and > 0.05 apfu AlVI, NaA + K versus AlIV relationships (Fig. 6A) show a

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ACCEPTED MANUSCRIPT compositional trend with a slope close to 0.5 suggesting an important contribution of Al+Fetschermakite + Ti-tschermakite components —note an average excess of AlIV of 0.8 apfu (see also discussion in Poli and Schmidt, 1992). On the other hand, AlVI + Fe3+ + 2Ti + A versus AlIV

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relationships (Fig. 6B) suggest that, on average, ca. 0.2 apfu AlVI + Fe3+ are present in amphibole as Al+Fe-glaucophane component (see also Fig. 6C). Because the importance Al-tschermakite (+Al-

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glacucophane), the occupancy of AlVI increases with pressure reaching the highest values in high-P, near-solidus experiments from Schmidt (1992, 1993) (Fig. 7A). Total Al contests also show a positive correlation with pressure (Fig. 7B), but with more scattering due to the influence of edenite, Ti-

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tschermkite and Fe-tschermakite components.

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Numerous barometers based on reactions buffering Al-tschermakite component in amphiboleplagioclase-bearing assemblages have been proposed. Thus, simple empirical expressions that relate the total Al content in amphibole with pressure have been calibrated for granitoids presenting during

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crystallization the critical nine-phase assemblage previously mentioned (e.g. Hammarstrom and Zen,

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1986; Johnson and Rutherford, 1989; Schmidt, 1992; Anderson and Smith, 1995). Fershtater (1990) proposed an empirical barometer based on a partition coefficient defined as the quotient of Al/Si ratio , that could be extended to a wider

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in plagioclase divided by the Al/Si ratio in amphibole,

range of amphibole-plagioclase assemblages from both mafic metamorphic rocks and felsic to mafic

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igneous rocks. This author indicated that temperature and compositional dependences cancel due to normalization of plagioclase Al/Si ratio by that of amphibole resulting that

is mostly

pressure dependent. The plagioclase Al/Si ratio is directly related to the anorthite content by stoichiometric constraints:

where

; whereas amphibole Al/Si ratio depends on the contents in edenite,

Al-tscheramakite, Fe-tschermakite, Ti-tscheramakite and Al-glaucophane components:

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ACCEPTED MANUSCRIPT

Plagioclase Al/Si ratio presents an overall positive correlation with temperature (Fig. 8A), whereas

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amphibole Al/Si ratio increases with pressure (Fig. 8B) as a consequence of the involvement of Altschermakite (+Al-glaucophane) substitution. In accordance with Fershtater (1990),

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shows a negative correlation with pressure (Fig. 8C); however, in order to get precise barometric expressions, compositional and temperature corrections must be introduced to account for complex

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phase-component substitutions that control amphibole composition.

Accordingly, for amphibole-plagioclase pairs from the calibration data set synthesized at T > 650 ºC

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with amphibole having > 0.02 apfu Ti and > 0.05 apfu AlVI,

fits to a linear relation in

pressure and temperature variables with two significant compositional parameters,

in plagioclase, that has a scale parameter of ca. 296.8 J (Table 4); it accommodates well to the

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and

in amphibole

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calibration data being detected no outliers (Fig. 9A).

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Solving for pressure, the plagioclase-amphibole barometer is obtained:

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where T is in K and R is the gas constant (8.3144 J K-1 mol-1). The expression has a Clapeyron slope of ca. 32 bar K-1 and a precision that ranges from ±1.5 kbar (uncertainties reported at 1s level, unless indicated otherwise) for the calibration data set to ±2.3 kbar for the test data set (Fig. 10A; Table 5).

can be considered as the ideal part of the equilibrium constant for a relation between chemical potentials similar to expression (2b):

. As discussed in

section 3, this relation cannot be considered an equilibrium condition for a simple exchange reaction because Si and Al contents in amphibole are function of the substitution extent of various phase components as previously indicated. 6.2. Amphibole/liquid Mg partitioning thermometer

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ACCEPTED MANUSCRIPT For amphibole-glass pairs from the calibration data set synthesized at T > 800 ºC with

> 0.75,

shows a linear negative correlation with temperature (Fig. 11A). In order to analyse versus temperature relationships various models including melt structure-chemical

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parameters (MSCP), as proposed by Ariskin and Nikolaev (1996), were checked using MM-estimators.

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Fitted coefficients for pressure were not statistically significant. Thus, we found a reliable expression with a logarithmic term in

with a scale parameter of 2204.4 J (Table 6) that accommodates

well compositional pairs from the calibration data set (Fig. 9B; note absence of outlying data).

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Solving for temperature, it is obtained the amphibole-liquid thermometer:

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test data set (Fig. 10B; Table 7).

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The estimated precision for this expression ranges from ±35 ºC for the calibration data set to ±45 ºC for the

This treatment based on the equilibrium condition for the partitioning of Mg between amphibole and liquid

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is preferable to reactions between phase components as, for example:

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☐Ca2Mg5Si8O22(OH)2 (amp) = 8 SiO2 (liq) + 5 MgO (liq) + 2 CaO (liq) + H2O (liq) which considers tremolite as the Mg-phase component, because its equilibrium constant:

requires an estimation of liquid H2O activity that is, in general, unknown in both natural and synthetic silicate glasses.

The temperature dependence of

is mostly controlled by a positive correlation of

with

temperature (Fig. 11B). This type of relation is well-known in olivine + other mineral phase-saturated liquids and represents the basis for a number of empirical calibrations of liquid-only thermometers (e.g., Ulmer, 1989, Sugawara, 2000, Putirka 2008, and references therein). For the multiply-saturated liquids

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ACCEPTED MANUSCRIPT from the experimental database, a liquid-only thermometer with the same MSCP as that for the amphibole-

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liquid thermometer is found with one outlaying value (Fig. 9C; scale parameter: 34 ºC, Table 8):

that yields a precision ranging from ±37 ºC for the calibration data set to ±42 ºC for the test data set (Fig.

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10C; Table 9), which is similar to that for the amphibole-liquid thermometer. The two expressions can be used together to verify the extent of compositional equilibrium between amphibole and matrix glass in volcanic rocks as well as to search for model compositions of amphibole-saturated liquids in geochemical

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modelling of volcanic and plutonic rocks.

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7. Discussion

7.1. Accuracy and precision of thermobarometric expressions calibrated using MM-estimators

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The plagioclase/amphibole Al-Si partitioning barometer and the Mg-in-liquid and the

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amphibole/liquid Mg partitioning thermometers calibrated in this work perform reasonably well (Figs. 9 and 10) and yield R2 values > 0.937 with fitted slopes for relationships between calculated and

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measured pressures and temperatures indistinguishable from the one-to-one line within uncertainties (Tables 5, 7 and 9). Remarkably, the barometric expression reproduces well pressure conditions of

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high-Al amphibole-bearing assemblages from near-solidus experiments from Schmidt (1992, 1993), not included in the calibration data set (Fig. 10A). The magnitudes of precision for the calibrated expressions (±1.5 to ±2.3 kbar and ±35 to ±45 ºC) are close to the values reported in widely used thermobarometric expressions that typically range from ±1 to ±4 kbar and from ±30 to ±50 ºC (e.g., McCarthy and Patiño-Douce, 1998; Krogh-Ravna and Paquin, 2003; Putirka, 2008; Nimis and Grütter, 2010). The uncertainties for the plagioclase-amphibole barometer are similar to, or better than, these for the amphibole-plagioclase-quartz barometer from Bhadra and Bhattacharya (2007) that are close to ±2.3 kbar (Fig. 12A). For the liquid-only and the amphibole-liquid thermometers, the uncertainties are similar to these for the edenite-quartz-tremolite-albite and the edenite-albite-richterite-anorthite

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ACCEPTED MANUSCRIPT thermometers (ca. ±40 ºC; Holland and Blundy, 1994). These thermometers also reproduce well experimental temperatures for most amphibole-glass pairs from an experimental database compiled from the literature by Erdmann et al. (2014), yielding a precision of respectively ±41 ºC and ±34 ºC

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(Fig. 12B) —runs 47 and 50-26 (Table 2 from the same authors) present larger discrepancies.

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7.2. Comparison to amphibole-only thermobarometers

A test of the amphibole-only expressions from Ridolfi et al. (2010) and Ridolfi and Renzulli (2012) carried out recently by Erdmann et al. (2014) led to the conclusion that the thermometers work

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reasonably well, but the barometers are inaccurate and give untenable estimates.

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Accordingly, when testing these expressions with the experimental database used in this work, the amphibole-only thermometric expressions perform well with precisions ranging form ±32 to ±36 ºC (Figs. 13A-B) that are similar to these for the liquid-only and the amphibole-liquid thermometers

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derived in this work. By contrast, the amphibole-only barometric expressions present significant

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systematic errors, but expressions 1d and 1e that yield precisions close, respectively, to ±2.4 and ±3.5 kbar (Figs. 13C-H). Therefore, expression 1d can still be used for pressure estimations, but it is

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important to note that it was calibrated in the pressure range 4 to 15 kbar.

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7.3. Applications to high-grade metamorphic and igneous rocks In this section, the plagioclase/amphibole Al-Si partitioning barometer and the Mg-in-liquid and the amphibole/liquid Mg partitioning thermometers are applied to natural assemblages from metamorphic and igneous rocks to evaluate the consistency of the P-T estimates with other thermobarometric expressions (Tables 10 and 11). Two Excel spreadsheets downloadable from the electronic supplementary material (Appendixes C and D) have been programmed to make it easy to compute the P-T estimates. The barometer can be applied to mineral assemblages containing amphibole with > 0.02 apfu Ti and > 0.05 apfu AlVI (13-CNK method) equilibrated at T > 650 ºC, whereas the use of the thermometers is limited to assemblages crystallized at T > 800 ºC with amphibole presenting > 0.75

values

(13-CNK method). In addition, more likely pressure and temperature estimates would be expected if

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ACCEPTED MANUSCRIPT phase compositions lie within the compositional fields defined by the experimental data in the diagrams from Figure 14. 7.3.1. Application of the plagioclase/amphibole Al-Si partitioning barometer to high-grade metamorphic

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rocks and mafic-ultramafic igneous rocks

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The plagioclase/amphibole Al-Si partitioning barometer has been applied to 15 samples (Table 10): 1) one sample of garnet amphibolite from Aravalli-Delhi Mobile Belt in Northwestern India (Bhowmik et al., 2010); 2) one sample of feldspathic garnet-hornblende granulite from Doubtful Sound in New

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Zealand (Oliver, 1977); 3) 7 samples of olivine hornblendites from the hornblende gabbro sill complex from Onion Valley, California (Sisson et al., 1996); 4) 5 samples of garnet amphibolites from the

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Paleoproterozoic Rudall Complex in central Western Australia (Smithies and Bagas, 1997); 5) and one sample of garnet amphibolite from Fiskenaesset in South Western Greenland (Weaver et al., 1982).

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The composition of all amphibole-plagioclase pairs lies inside or next to the compositional fields

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defined by the experimental data (Figs. 14A-E). Pressure conditions assumed for the mineral assemblages (Po column in Table 10) are those reported

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by the authors, except for sample 36511 from Doubtful Sound that is taken from Kohn and Spear (1990). Smithies and Bagas (1997) calculated pressure conditions for garnet amphibolites from the

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Paleoproterozoic Rudall Complex using the clinopyroxene-plagioclase-quartz barometer from Ellis (1980), the garnet-clinopyroxene-plagioclase-quartz barometer from Newtown and Perkins (1982) and the two calibrations of the garnet-amphibole-plagioclase-quartz barometer from Kohn and Spear (1990). For the garnet amphibolite R30A1 from Aravalli-Delhi Mobile Belt, Bhowmik et al. (2010) determined pressure conditions by amphibole-plagioclase-quartz barometry (calibration from Bhadra and Bhattacharya, 2007) and equilibrium calculation methods (average THERMOCALC P-T estimates). Pressure conditions for feldspathic garnet-hornblende granulite 36511 were estimated by Kohn and Spear (1990) using garnet-plagioclase-orthopyroxene-quartz barometry (Bohlen et al., 1983; Newton and Perkins, 1982; Powell and Holland, 1988) and by garnet-plagioclase-clinopyroxenequartz barometry (Newton and Perkins, 1982; Powell and Holland, 1988). For garnet amphibolite 121655 from Fiskenaesset, Weaver et al. (1982) estimated pressure conditions using the garnet-

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ACCEPTED MANUSCRIPT plagioclase-pyroxene-quartz barometer from Perkins and Newton (1981). Sisson et al. (1996) determined emplacement pressures for olivine hornblendites from Onion Valley using constraints from phase relationships in metamorphic pendant rocks —pressure < 4.2 kbar inferred from presence

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of andalusite or sillimanite and absence of kyanite— and from experimental liquidus relationships — pressure > 1 kbar as a consequence of liquidus or near-liquidus crystallization of hornblende from

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high-alumina basalt liquids.

Pressure conditions calculated with the plagioclase/amphibole Al-Si partitioning barometer at

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temperature conditions recalculated with the amphibole-plagioclase thermometer (Eqn. HB94-B; Holland and Blundy, 1994; Table 10) are consistent with those reported in the literature (Fig. 15A).

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For 12 samples, the agreement between the calculated and published pressure estimates is, on the average, better than ±2 kbar.

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7.3.2. Application of the Mg-in-liquid and the amphibole/liquid Mg partitioning thermometers to volcanic

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and plutonic rocks

In volcanic rocks, an assessment of the degree of attainment of compositional equilibrium between

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amphibole phenocrysts and matrix glass can be carried out comparing temperatures estimated with the amphibole/liquid Mg partitioning thermometer to those given by other thermometric expressions as

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the Mg-in-liquid thermometer calibrated in this work, the amphibole-only thermometers from Ridolfi et al. (2010) and Ridolfi and Renzulli (2012) or the amphibole-plagioclase thermometers from Holland and Blundy (1994).

As a second goal, the liquid-only and the amphibole/liquid Mg partitioning thermometers can be used in plutonic rocks for constraining model compositions of the liquids in equilibrium with amphibole, as proposed in Putirka et al. (1996) and Putirka (2008) for other mineral phases. So, a set of model liquid compositions that may include whole-rock analyses as well as corrected whole-rock compositions for mineral accumulation and fractionation could be used for testing mineral-liquid equilibrium compositions (e.g., Putirka, 2008). For the first purpose three samples have been selected: 1) four amphibole analyses and two glass

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ACCEPTED MANUSCRIPT compositions (matrix and inclusions in plagioclase; average values of at least 10 analyses) from sample 540 from the Plinian dacitic Chiltepe Tephra from Apoyeque volcano in west-central Nicaragua (Kutterolf et al., 2011); and 2) two glass compositions (average values of 10 analyses) and

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four amphibole compositions (grain rim and core analyses) from samples LGM9 and 661Mc of partially crystalline hornblende gabbro inclusions from Little Glass Mountain Rhyolite in California

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(Brophy et al., 2011). Temperatures were recalculated using the amphibole-only thermometer from Ridolfi and Renzulli (2012) at pressure ranges indicated by the authors (Table 11).

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Temperature estimates for all amphibole-glass pairs, including those from samples LGM9 and 661Mc that plot outside the compositional field from Figure 14F, given by the liquid-only and the amphibole-

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liquid thermometers are consistent with those obtained by the amphibole-only thermometer (agreement better than ±70 ºC). For four pairs (LGM9 (only rim composition), 540-hbl-03, 540-hbl-33 and 540-hbl-41) consistency between calculated and published temperatures is better with deviations

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lower than ±40 ºC (Table 11; Figs. 15B-C).

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For the second goal, one sample of quartz monzonite from the Katerina Ring Complex in southern Sinai has been used (sample KA-29; Moreno et al., 2014). It contains scarce green grains of

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magnesiohornblende (modal abundance: ca. 5vol.%) with rare grain cores of brown, high-Ti

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hastingsitic amphibole (Moreno et al., 2014). Given the low-modal abundance of the brown amphibole, the bulk composition of the quartz monzonite could represent that of a liquid saturated in the high-Ti hastingsitic amphibole if this latter is primocrystic. The liquid-only and the amphiboleliquid thermometers yield temperature estimates that lie within the range of variation of those obtained with the amphibole-plagioclase and the amphibole-only thermometers, with absolute deviations close to the uncertainties of the expression derived in this work (Table 11; Figs. 15B-C). Therefore, this result is consistent with the brown, high-Ti hastingsitic amphibole representing primocrysts in close equilibrium with liquids with compositions similar to that of the quartz monzonite.

8. Conclusions Three empirical thermobarometric expressions —plagioclase/amphibole Al-Si partitioning barometer, Mg-in-liquid thermometer and amphibole/liquid Mg partitioning thermometer — are calibrated using

23

ACCEPTED MANUSCRIPT robust regression methods based on MM-estimators that determine P-T conditions with uncertainties of ±1.5 to ±2.3 kbar and ±35 to ±45 ºC. The derived expressions can only be used for amphibole compositions admitting normalization to 13-

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CNK, with > 0.02 apfu Ti and > 0.05 apfu AlVI in the temperatures range 650-1050 ºC for the in the temperature range 800-1100 ºC for the two

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barometer, and with > 0.75 thermometers.

The plagioclase-amphibole barometer is suitable for a large variety of amphibole-plagioclase-bearing

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— even quartz-free and/or garnet-free — assemblages from amphibolites and mafic granulites, as well as from igneous rocks ranging from metaluminous granitoids to gabbros, whereas the liquid-only and

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the amphibole-liquid thermometers are applicable to alkaline and subalkaline igneous rocks. The empirical expressions perform so well as the amphibole-plagioclase-quartz barometer from

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Bhadra and Bhattacharya (2007) and the edenite-quartz-tremolite-albite and the edenite-albite-

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richterite-anorthite thermometers from Holland and Blundy (1994) that use more formal thermodynamic approaches.

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For natural assemblages, the calibrated expressions yield P-T estimates consistent with widely used

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barometers and thermometers at 2s level or better. Acknowledgements

We thank Jiba Ganguly and an anonymous reviewer for the thorough revision of the manuscript and for their insightful comments. We are grateful to Marco Scambelluri for his efficient and helpful editorial handling. We thank Keith D. Putirka for helpful comments on a previous version of this paper. This work has been financially supported by the Spanish Grants CGL2008-02864 (JFM and JAM), CGL2010-22022C02-01 (AC and CR) and CGL2013-40785-P (JFM), and the Andalusian Grants RNM2163 (JFM) and P09-RNM-05378 (AC and CR). Financial support to JAM from the Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), Brazil (Grant 2014/04920-0), is acknowledged with thanks.

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Moreno, J.A., Molina, J.F., Montero, P., Abu Anbar, M., Scarrow, J.H., Cambeses, A., Bea, F., 2014. Unraveling sources of A-type magmas in juvenile continental crust: constraints from

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thermometry, O2 and H2O barometries, and consequences for biotite stability. Chemical

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Perkins, D., Newton, R.C., 1981. Chamockite geobarometers based on coexisting garnet-pyoxene-

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Poli, S., 1993. The amphibolite-eclogite transformation: an experimental study on basalt. American

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Ramberg, H, De Vore, D.G.W., 1951. The distribution of Fe2+ and Mg in coexisting olivines and pyroxenes. Journal of Geology 59,193–210.

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Ridolfi, F., Renzulli, A., 2012. Calcic amphiboles in calc-alkaline and alkaline magmas: thermobarometric and chemometric empirical equations valid up to 1130 °C and 2.2 GPa.

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Contributions to Mineralogy and Petrology 163, 877–895.

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Ridolfi, F., Renzulli, A., Puerini, M., 2010. Stability and chemical equilibrium of amphibole in calcalkaline magmas: an overview, new thermobarometric formulations and application to

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subduction-related volcanoes. Contributions to Mineralogy and Petrology 160, 45–66. Robinson, P., Spear, F.S., Schumacher, J.C., Laird, J., Klein, C., Evans, B.W., Doolan, B.L., 1982. Phase relations of metamorphic amphiboles: natural occurrence and theory. Reviews in Mineralogy 9B, 1-227. Roeder, P.L., Emslie, R.F., 1970. Olivine-liquid equilibrium. Contributions to Mineralogy and Petrology 29, 275–289. Rousseeuw, P.J., Leroy, A.M., 2003. Robust regression and outlier detection. New York, Wiley. Rousseeuw, P.J., van Zomeren, B.C., 1990. Unmasking multivariate outliers and leverage points. Journal of the American Statistical Association 85, 633–639.

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calibration of the Al-in-hornblende barometer. Contributions to Mineralogy and Petrology 110,

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Sisson, T.W., Grove, T.L., 1993. Experimental investigations of the role of H2O in calc-alkaline

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ACCEPTED MANUSCRIPT van Hinsberg, V.J., Shumacher, J.C., 2007. Intersector element partitioning in tourmaline: a potentially powerful single crystal thermometer. Contributions to Mineralogy and Petrology 153, 289–301.

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Wan, Z., Coogan, L.A., Canil, D., 2008. Experimental calibration of aluminium partitioning between

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olivine and spinel as a geothermometer. American Mineralogist 93, 1142–1147. Weaver, B.L., Tarney, J., Windley, B., Leake, B.E., 1982. Geochemistry and petrogenesis of Archean

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metavolcanic amphibolites from Fiskenaesset, S.W. Greenland. Geochimica et Cosmochimica

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9B, 355–390.

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Wones, D.R., Gilbert, M.C., 1982. Amphiboles in the igneous environment. Reviews in Mineralogy

Yaxley, G.M., Green, D.H., 1994. Experimental demonstration of refractory carbonate-bearing

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eclogite and siliceous melt in the subduction regime. Earth and Planetary Science Letters 128, 313-325.

Yohai, V.J., 1987. High breakdown point and high efficiency robust estimates for regression. Annals of Statistics 15, 642–656.

33

ACCEPTED MANUSCRIPT Figure captions Fig. 1. Selection of experimental runs. Calculated vs. measured temperatures for selected amphibole-

anorthite (Eqn. HB94-B) thermometers (Holland and Blundy, 1994).

temperature. C)

vs. temperature. B)

vs.

.

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Fig. 2. Selection of experimental runs. A)

PT

plagioclase pairs using the edenite-quartz-tremolite-albite (Eqn. HB94-A) and the edenite-albite-richterite-

vs.

NU

Fig. 3. Comparison of regression methods using a model data set with 33% of outliers. A) Dependent variable vs. independent variable with fitted lines using OLS, M-estimators and MM-estimators. B)

MA

Diagnostic plot of standardized robust residuals versus robust Mahalanobis distances for MM-estimates. Fig. 4. Composition of amphibole from the experimental database. A) Si vs. NaA + K. B) Si vs. Ti.

ED

Abbreviations: Act, actinolite; Ed, edenite; Gl, glaucophane; Hbl, hornblende; Hst, hastingsite; Krs,

PT

kaersutite; Prg, pargasite; Tr, tremolite; Ts, tschermakite. Fig. 5. Composition of glass from the experimental database. A) Total alkalis vs. silica (TAS) diagram. B)

CE

Fe-number vs. silica diagram after Frost et al. (2001). Data expressed on an anhydrous basis.

AC

Fig. 6. Composition of amphibole coexisting with plagioclase from the experimental database. A) NaA + K vs. AlIV. B) AlVI + Fe3+ + 2Ti + A vs. AlIV. C) AlVI + Fe3+ vs. NaM4. Abbreviations in Table 1. Fig. 7. Composition of amphibole coexisting with plagioclase from the experimental database. A) AlVI vs. pressure. B) Total Al vs. pressure.

Fig. 8. P-T-Al/Si-

relationships in amphibole and plagioclase from the experimental database. A)

Al/Si ratio in plagioclase vs. temperature. B) Al/Si ratio in amphibole coexisting with plagioclase vs. pressure. C)

vs. pressure.

Fig. 9. Diagnostic plot of standardized robust residuals vs. robust Mahalanobis distance for the expressions derived in this work using MM-estimators. A)

. B)

34

. C) Temperature, Mg-in-

ACCEPTED MANUSCRIPT liquid expression. Fig. 10. Precision and accuracy of the empirical thermobarometric expressions derived in this work. A) Plagioclase/amphibole Al-Si partitioning barometer. B) Amphibole/liquid Mg partitioning thermometer. C)

-

relationships in amphibole and glass from the experimental database. A)

vs. temperature. B)

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Fig. 11. T-

PT

Mg-in-liquid thermometer.

vs. temperature.

NU

Fig. 12. A) Two calibrations of the amphibole-plagioclase-quartz barometer from Bhadra and Bhattacharya (2007) tested using data reported in Table 8 from the same authors. B) Mg-in-liquid and amphibole/liquid

MA

Mg partitioning thermometers derived in this work tested using data from Erdmann et al. (2014). Fig. 13. Precision and accuracy of empirical thermobarometric expressions derived by Ridolfi et al. (2010)

ED

and Ridolfi and Renzulli (2012). A) Equation 1 (Ridolfi et al., 2010). B) Equation 2 (Ridolfi and Renzulli, 2012). C) Equation 4 (Ridolfi et al., 2010). D) Equation 1a (Ridolfi and Renzulli, 2012). E) Equation 1b

PT

(Ridolfi and Renzulli, 2012). F) Equation 1c (Ridolfi and Renzulli, 2012). G) Equation 1d (Ridolfi and

CE

Renzulli, 2012). H) Equation 1e (Ridolfi and Renzulli, 2012). Calculations performed using only amphibole compositions from the experimental database that meet the requirements for application of the

AC

thermobarometric expressions.

Fig. 14. Compositional fields to test the compatibility of mineral and glass compositions from natural assemblages with those from the experimental database. A)

vs.

in plagioclase. B)

vs. AlIV in amphibole. C) AlIV vs. Ti in amphibole. D) AlIV vs. NaA in amphibole. E) AlIV vs. AlVI in amphibole. F)

vs.

. Compositions of amphibole-plagioclase

and amphibole-glass pairs from natural assemblages used for testing the empirical expressions derived in this work are also shown. Compositions of amphibole-glass pairs from samples LGM9 and 661Mc plot outside the compositional field from diagram F, but temperature estimates are consistent with those obtained by amphibole-only thermometry (see text for discussion). Solid lines, boundaries for compositional fields of selected glass and mineral compositions from the experimental database. Data

35

ACCEPTED MANUSCRIPT source for amphibole-plagioclase and amphibole-glass pairs from natural assemblages reported in Tables 10 and 11. Fig. 15. Applications of the thermobarometric expressions derived in this work to natural assemblages

PT

from metamorphic and igneous rocks. A) Pressures calculated with the plagioclase/amphibole Al-Si

SC RI

partitioning barometer compared to those estimated with other published barometers. B) Temperatures calculated with the Mg-in-liquid thermometer compared to those estimated with the amphibole-only (RR12-2; Ridolfi and Renzulli, 2012) and the amphibole-plagioclase (HB94-B; Holland and Blundy, 1994)

NU

thermometers. C) Temperatures calculated with the amphibole/liquid Mg partitioning thermometer compared to those estimated with the amphibole-only and the amphibole-plagioclase thermometers. Data

AC

CE

PT

ED

MA

source reported in Tables 10 and 11.

36

Figure 1

AC

CE

PT

ED

MA

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PT

ACCEPTED MANUSCRIPT

37

AC

CE

PT

ED

MA

NU

SC RI

PT

ACCEPTED MANUSCRIPT

Figure 2

38

ED

MA

NU

SC RI

PT

ACCEPTED MANUSCRIPT

AC

CE

PT

Figure 3

39

ED

MA

NU

SC RI

PT

ACCEPTED MANUSCRIPT

AC

CE

PT

Figure 4

40

Figure 5

AC

CE

PT

ED

MA

NU

SC RI

PT

ACCEPTED MANUSCRIPT

41

AC

CE

PT

ED

MA

NU

SC RI

PT

ACCEPTED MANUSCRIPT

Figure 6

42

Figure 7

AC

CE

PT

ED

MA

NU

SC RI

PT

ACCEPTED MANUSCRIPT

43

AC

CE

PT

ED

MA

NU

SC RI

PT

ACCEPTED MANUSCRIPT

Figure 8

44

AC

CE

PT

ED

MA

NU

SC RI

PT

ACCEPTED MANUSCRIPT

Figure 9

45

AC

CE

PT

ED

MA

NU

SC RI

PT

ACCEPTED MANUSCRIPT

Figure 10

46

AC

CE

PT

ED

MA

NU

SC RI

PT

ACCEPTED MANUSCRIPT

Figure 11

47

Figure 12

AC

CE

PT

ED

MA

NU

SC RI

PT

ACCEPTED MANUSCRIPT

48

AC

CE

PT

ED

MA

NU

SC RI

PT

ACCEPTED MANUSCRIPT

Figure 13

49

AC

CE

PT

ED

MA

NU

SC RI

PT

ACCEPTED MANUSCRIPT

Figure 14

50

AC

CE

PT

ED

MA

NU

SC RI

PT

ACCEPTED MANUSCRIPT

Figure 15

51

ACCEPTED MANUSCRIPT Table 1. Symbols used in this paper

PT

: occupancy of cation N in amphibole; formula normalized to 23 oxygen atoms

SC RI

: Al fraction in T1-site in amphibole : A-site vacancy in amphibole

: total cation content in amphibole

: fraction of cation N in amphibole

: fraction of cation N in liquid

MA

: albite fraction in plagioclase

NU

total cation content in liquid

: molar plagioclase/amphibole Al-Si partition coefficient

ED

: molar amphibole/liquid Mg partition coefficient : molar amphibole/liquid Na-Si partition coefficient

AC

CE

PT

: molar amphibole/liquid Ca-Ti partition coefficient

52

ACCEPTED MANUSCRIPT Table 2. Summary of the selected amphibole-plagioclase pairs 92 observations

Mean

Std. Dev.

Min

Max

T(ºC)

845.3043

114.3024

650

1050

P(kbar)

5.24337

3.620677

1

6.353904

.2892756

5.791408

7.062472

1.646096

.2892756

.937528

2.208592

.2066681

.1047181

.0479317

.5940331

.4108032

.3032541

.0842389

1.371138

.7809191

.2905819

.0699333

1.546639

.8536659

.4845413

.0094633

1.841986

2.715129

.5160586

1.478165

3.64063

1.677849

.1076789

1.252082

1.884307

.1352711

.0335072

.643003

.050928

0

.2233318

.0538654

.6260409

.9421533

.1329628

.4988247

.9972618

.4115239

.0723189

.234382

.5521481

.4307541

.1837119

.1083147

.84

6551.951

2937.052

-2065.716

10974.73

AC

CE

.760676

SC RI

NU

PT

.0943434 .8388991

15

MA

ED

.2667872

PT

Variable

53

ACCEPTED MANUSCRIPT Table 3. Summary of the selected amphibole-glass pairs 148 observations

Mean

Std. Dev.

Min

Max

T(ºC)

944.9189

73.91285

830

1100

P(kbar)

6.602047

4.99648

.8

6.206193

.2801342

5.7192

1.793807

.2801342

.2909841

.1566438

.3024295

.1435991

.7443152 .7678566

SC RI

.6999382

.0065147

.8303934

.3372016

.0125842

1.365908

.4639461

.0017145

2.255144

.4388777

1.714141

3.826318

.0918365

1.501247

1.932602

NU

0

.1660024

.0098384

.8073161

.1254659

.0800431

.0093435

.4381233

.8433028

.0459493

.7506233

.9663011

.7906964

.1196917

.4668248

.9995146

.580561

.0759404

.3705989

.7014155

.0065017

.0071599

0

.0415639

.1920987

.0175092

.1465827

.2305199

.040056

.0257754

.0053661

.1233747

.0267392

.0246931

.0009773

.1411817

.0488079

.02361

.0142063

.1465581

.0755578

.0226443

.0286687

.1480818

.0274542

.0131522

.0007573

.0620114

-1.690795

.3405029

-2.523202

-.7690551

22583.71

7781.743

4922.113

44731.89

PT

.3508644

CE AC

6.871886 2.2808

ED

1.686768

20

1.128114

MA

2.869046

PT

Variable

54

ACCEPTED MANUSCRIPT Table 4. Robust regression statistics for the Al-Si partitioning between plagioclase and amphibole in assemblages with >0.02 and >0.05 synthesized at T > 650 ºC. Units: J, K, kbar Calibration data set: 60 observations

t

P>|t|

T

8.741383

.7463331

11.71

0.000

P

-274.3004

19.26982

-14.23

0.000

-23376.61

1418.076

-16.48

-7578.912

439.0553

-17.26

Constant

11302.26

1240.185

9.11

Scale parameter

296.8307

7.245698 |10.23707 -312.918 |-235.6828

0.000

-26218.5 |-20534.72

0.000

-8458.798 |-6699.025

0.000

8816.869 |13787.64

NU MA ED PT

CE AC

55

95% Conf. Interval

PT

Std. Err.

SC RI

Coefficients

ACCEPTED MANUSCRIPT Table 5. Test for the plagioclase/amphibole Al-Si partitioning barometer Calibration data set (60 observations) OLS: Coefficients

Std. Err.

t

P>|t|

95% Conf. Interval

P(kbar)Exp

.9111099

.0582353

15.65

0.000

.7945393 |1.027681

R2 = 0.8084

Constant

.9327577

.3148974

2.96

0.004

.3024222 |1.563093

RMSE = 1.3947

SC RI

OLS:

PT

P(kbar)Cal

Coefficients

Std. Err.

t

P>|t|

95% Conf. Interval

P(kbar)Exp

1.052626

.0354254

29.71

0.000

.9817397 | 1.123512

NU

P(kbar)Cal

R2 = 0.9374 RMSE = 1.4838

Test data set (32 observations)

MA

OLS: Coefficients

Std. Err.

t

P(kbar)Exp

.9447232

.0975223

9.69

Constant

1.504282

.7647785

ED

P(kbar)Cal

1.97

P>|t|

95% Conf. Interval

0.000

.745556 | 1.14389

R2 = 0.7497

0.058

-.0576039 | 3.066168

RMSE = 2.1957

Coefficients

P(kbar)Exp

1.110003

Std. Err.

t

P>|t|

95% Conf. Interval

.0517353

21.46

0.000

1.004488 | 1.215518

CE

P(kbar)Cal

PT

OLS:

R2 = 0.9369

AC

RMSE = 2.2951

56

ACCEPTED MANUSCRIPT Table 6. Robust regression statistics for the Mg partitioning between amphibole and liquid in assemblages with > 0.75 synthesized in the temperature range 800-1100 ºC. Units: J, K, kbar

Std. Err.

t

P>|t|

-57.69794

9.076876

-6.36

0.000

-11896.15

1652.52

-7.20

Constant

71974.99

12035.32

5.98

Scale parameter

2204.43

MA ED PT CE AC

57

-75.70843 | -39.68745

0.000

-15175.11 | -8617.192

0.000

48094.3 | 95855.68

NU

T

95% Conf. Interval

SC RI

Coefficients

PT

Calibration data set: 102 observations

ACCEPTED MANUSCRIPT Table 7. Test for the amphibole/liquid Mg-partitioning thermometer Calibration data set (102 observations) OLS: Coefficients

Std. Err.

t

P>|t|

95% Conf. Interval

T (ºC)Exp

1.003798

.0477828

21.01

0.000

.9089982 | 1.098598

R2 = 0.8153

Constant

-8.793057

45.45529

-0.19

0.847

-98.97506 | 81.38895

RMSE = 35.314

SC RI

OLS:

PT

T (ºC)Cal

Coefficients

Std. Err.

t

P>|t|

95% Conf. Interval

T (ºC)Exp

.994582

.0036581

271.9

0.000

.9873254 | 1.001839

NU

T (ºC)Cal

R2 = 0.9986 RMSE = 35.145

Test data set (46 observations)

MA

OLS: Coefficients

Std. Err.

t

T (ºC)Exp

1.056691

.0891272

11.86

Constant

-70.2198

83.77698

ED

T (ºC)Cal

-0.84

P>|t|

95% Conf. Interval

0.000

.8770673 | 1.236316

R2 = 0.7616

0.406

-239.0612 | 98.62161

RMSE = 44.807

Coefficients

T (ºC)Exp

.9822198

Std. Err.

t

P>|t|

95% Conf. Interval

.0070051

140.2

0.000

.9681108 | .9963289

R2 = 0.9977 RMSE = 44.659

AC

CE

T (ºC)Cal

PT

OLS:

58

ACCEPTED MANUSCRIPT Table 8. Robust regression statistics for the Mg-in-liquid thermometer in assemblages with > 0.75 synthesized in the temperature range 800-1100 ºC. Units: ºC Calibration data set: 102 observations

t

P>|t|

107.1836

10.06836

10.65

0.000

-108.4329

15.46568

-7.01

Constant

1184.2

39.70258

29.83

Scale parameter

33.69322

MA ED PT CE AC

59

95% Conf. Interval

PT

Std. Err.

87.20581 | 127.1614

SC RI

Coefficients

0.000

-139.1202 | -77.74565

0.000

1105.421 | 1262.978

NU

T

ACCEPTED MANUSCRIPT Table 9. Test for the Mg-in-liquid thermometer Calibration data set (102 observations) OLS: Coefficients

Std. Err.

t

P>|t|

95% Conf. Interval

T (ºC)Exp

.8490997

.0482247

17.61

0.000

.7534233 | .9447762

R2 = 0.7537

Constant

138.8331

45.87564

3.03

0.003

47.8171 | 229.849

RMSE = 35.64

SC RI

OLS:

PT

T (ºC)Cal

Coefficients

Std. Err.

t

P>|t|

95% Conf. Interval

T (ºC)Exp

.9946093

.0038565

257.9

0.000

.9869589 | 1.00226

NU

T (ºC)Cal

R2 = 0.9985 RMSE = 37.052

Test data set (46 observations)

MA

OLS: Coefficients

Std. Err.

t

T (ºC)Exp

.9563064

.0845141

11.32

Constant

28.18706

79.44076

ED

T (ºC)Cal

0.35

P>|t|

95% Conf. Interval

0.000

.7859794 | 1.126633

R2 = 0.7384

0.724

-131.9153 | 188.2894

RMSE = 42.488

Coefficients

T (ºC)Exp

.9862001

Std. Err.

t

P>|t|

95% Conf. Interval

.0065996

149.3

0.000

.9729079 | .9994923

R2 = 0.9979 RMSE = 42.073

AC

CE

T (ºC)Cal

PT

OLS:

60

ACCEPTED MANUSCRIPT

9.80 | 10.50

86S22 86S56 87S1 87S21a 87S33b 87S34 87S42

3.00 3.00 3.00 3.00 3.00 3.00 3.00

2.00 | 4.00 2.00 | 4.00 2.00 | 4.00 2.00 | 4.00 2.00 | 4.00 2.00 | 4.00 2.00 | 4.00

1012 1019 1010 986 1008 1004 972

112980 113088 117703 117762 119109

11.95 11.10 11.50 11.75 10.60

11.60 | 12.30 10.70 | 11.50 11.30 | 11.70 11.50 | 12.00 10.20 | 11.00

721 725 730 703 770

13.00

849

0.46 Sisson et al. (1996) 1007 | 1018 0.49 1015 | 1023 0.48 1006 | 1014 0.48 981 | 991 0.50 1005 | 1012 0.49 999 | 1008 0.51 968 | 977 0.49 Smithies & Bagas(1997) 717 | 724 0.39 722 | 727 0.41 729 | 731 0.46 700 | 705 0.40 767 | 772 0.38 Weaver et al. (1982) 0.43

AC CE

121655 7.00 746 a: pressure reported by the authors; various barometers (see text for details) b: Holland and Blundy (1994)

61

0.65

Av.

Range

426

423 | 428

SC RI

10.15

NU

R30A1-Hbl40-Plg42 Oliver (1977) 36511

MA

Av.

T(ºC) Eqn. HB94-Bb Av. Range Bhowmik et al. (2010) 656 650 | 661 0.47

PT ED

ID

Po(kbar)a Range

PT

Table 10. Application of the plagioclase/amphibole Al-Si partitioning barometer P(kbar) Plg-amp Bar Av. Range 11.41

11.25 | 11.57

12.59

ΔP(kbar) 1.26

0.58

2373

0.10 0.13 0.13 0.12 0.14 0.12 0.13

9519 10081 9942 8991 9656 8801 9059

9478 | 9560 10054 | 10109 9912 | 9972 8959 | 9024 9629 | 9682 8770 | 8831 9024 | 9093

2.91 1.05 1.38 2.69 1.40 2.67 2.69

2.88 | 2.94 1.04 | 1.06 1.36 | 1.39 2.67 | 2.72 1.39 | 1.42 2.64 | 2.70 2.67 | 2.72

-0.09 -1.95 -1.62 -0.31 -1.60 -0.33 -0.31

0.72 0.69 0.71 0.71 0.60

2682 3041 1836 2268 5330

2674 | 2690 3033 | 3049 1834 | 1839 2262 | 2274 5318 | 5342

9.65 7.93 7.93 10.08 6.35

9.59 | 9.72 7.88 | 7.99 7.90 | 7.97 10.02 | 10.14 6.32 | 6.38

-2.30 -3.17 -3.57 -1.67 -4.25

0.74

2582

7.21

-0.41

0.21

ACCEPTED MANUSCRIPT

Table 11. Application of the Mg-in-liquid and the amphibole/liquid Mg partitioning thermometers

ID

Av.

Range

T(ºC) Eqn. RR12-2e Av.

T(ºC) Calculated Liq-only Ther

Range

938

937 | 940

-3.33

4.62

661Mc-rim

940

939 | 942

-3.33

4.58

LGM9-core

962

961 | 964

-3.02

LGM9-rim

947

946 | 949

-3.02

Amp-liq Ther

ΔT(ºC) Eqn. HB94-B Liq-only Ther

Amp-liq Ther

ΔT(ºC) Eqn. RR12-2 Liq-only Ther

Amp-liq Ther

868

887

-70

-51

868

892

-72

-49

3.91

909

922

-53

-40

3.88

909

926

-38

-22

MA

NU

661Mc-core

SC RI

Brophy et al. (2011)a

PT

T(ºC) Eqn. HB94-Bd

837

829 | 846

-1.95

3.27

862

848

25

11

540-hbl-03-plg. incl.

837

829 | 846

-1.96

3.19

872

859

35

22

540-hbl-32-mxt

909

900 | 917

-1.95

3.28

862

847

-47

-61

540-hbl-32-plg. incl.

909

900 | 917

-1.96

3.20

872

858

-37

-51

540-hbl-33-mxt

902

893 | 910

-1.95

3.21

862

856

-40

-46

540-hbl-33-plg. incl.

902

893 | 910

-1.96

3.12

872

867

-30

-35

540-hbl-41-mxt

876

868 | 885

-1.95

3.22

862

854

-14

-22

540-hbl-41-plg. incl.

876

868 | 885

-1.96

3.14

872

865

-4

-11

AC CE

540-hbl-03-mxt

PT ED

Kutterolf et al. (2011)b

Moreno et al. (2014)c KA29-0.42Ti-Ab52

918

916 | 921

990

979 | 1001

-1.95

2.37

957

957

39

39

-33

-33

KA29-0.43Ti-Ab52

929

926 | 931

998

987 | 1010

-1.95

2.37

957

956

28

28

-41

-42

62

ACCEPTED MANUSCRIPT

KA29-0.36Ti-Ab52

912

910 | 914

995

984 | 1007

-1.95

2.33

AC CE

PT ED

MA

NU

SC RI

PT

a: temperature calculated at 2.8 and 3.2 kbar b: temperature calculated at 2 and 4 kbar c: temperature calculated at 2 and 5 kbar d: Holland and Blundy (1994) e: Ridolfi and Renzulli (2012)

957

63

962

45

49

-38

-33