A new activity model for Fe–Mg–Al biotites: II—Applications in the K₂O–FeO–MgO–Al₂O₃–SiO₂–H₂O (KFMASH) system

Abstract

The new biotite activity model and standard-state thermodynamic properties of Ann, Phl, and Eas presented in part-I were used to make pseudosections of bulk compositions representing experimental Fe–Mg exchange equilibria and (model) pelitic bulk rock compositions in the system K₂O–FeO–MgO–Al₂O₃–SiO₂–H₂O (KFMASH), using mainly the software Perple_X. These pseudosection calculations (termed ‘our calculation(s)’ in the following) were compared to analogous ones performed with the solution model of biotite and thermodynamic data cited in White et al. (J Metamorph Geol 32:261–286, 2014, 10.1111/jmg.12071), termed ‘W14 calculation’. Our calculations with the experimental bulk composition used by Zhou (Ti–Mg–Fe biotites: formation, substitution, and thermodynamic properties at 650 to 900 °C and 1.1 Kb with fO₂ defined by the CH₄–graphite buffer. PhD thesis, State University of New York, 1994) in his experimental study of the Fe–Mg exchange between biotite (Bt) and olivine (Ol) confirm that biotite had no or only minimal octahedral Al (Al^VI) in these experiments. The experimental data of Ferry and Spear (—FS78, Contrib Mineral Petrol 66:113–117, 1978, 10.1007/BF00372150) on the Fe–Mg distribution between biotite and garnet (Grt) are well reproduced by our calculations. The computed composition of biotite (X_Fe) in equilibrium with garnet of Alm₉₀Py₁₀ composition and the resulting lnK_D values as a function of temperature are in good agreement with the experimental brackets. An analogous W14 calculation on the same Fe-rich bulk composition predicts too high X_Fe^Bt in order of 0.1 mol fraction. The Al^VI contents of biotite of about 0.3–0.45 apfu, as measured by Gessman et al. (Am Mineral 82:1225–1240, 1997, 10.2138/am-1997-11-1218) in similar biotite–garnet exchange experiments performed with Alm₈₀Py₂₀ and Alm₇₀Py₃₀ garnets, are well reproduced by our, as well as by W14 calculations. The extent of Tschermak substitution in biotite in the FS78 experiments, which had Fe-richer bulk compositions, has not been measured. Comparing the FS78 biotites with the ones from Gessman et al. (Am Mineral 82:1225–1240, 1997, 10.2138/am-1997-11-1218), it is very likely that the biotites reported in FS78 contained Al^VI in the same order of ca. 0.3–0.4 apfu. A T–X_Fe (= molar FeO/(FeO + MgO) pseudosection demonstrates the bulk composition dependence of lnK_D of the Mg/Fe^Grt/Bt exchange reaction in high-variance fields. Further comparisons, demonstrating the application of the new biotite solution model in the KFMASH system, are presented in pseudosections constructed for an average model pelite, as well as for a natural high-T/low-P and a natural high-P metapelite. The pseudosections show that biotite according to our biotite model breaks down at lower temperatures and pressures than predicted from the W14 biotite model in the KFMASH system. This means that KFMASH biotite can break down before the wet solidus is reached, which can explain the existence of dry high-T/low-P metapelites. At higher pressures, biotite according to our calculations breaks down at lower pressures than computed with the W14 biotite model. Before biotite breaks down, however, its Al^VI content based on our calculations could potentially be used for pseudosection barometry, similarly as the Si-in-phengite barometer. These trends need to be confirmed by a future extension of our model which incorporates Ti, Fe³⁺ and a di–tri-octahedral substitution.

Introduction

In part-I, a new KFMASH activity model was presented for biotite, an important rock-forming mineral. Mixing parameters and standard-state properties were derived through an integrated approach combining results from relaxation calorimetry, from powder XRD and line-broadening in powder absorption infrared (IR) spectra of well-characterized biotites with the thermodynamic analysis of existing phase-equilibrium data, complemented by density functional theory (DFT) calculations.

The aim of this part-II is to demonstrate the impact of this new biotite model on computed phase relations in the KFMASH system and comparing these phase relationships with those obtained with the biotite solution model of White et al. (2014).

Computational methods

As a general remark, we use in all applications the activity–composition (a–X) relations for biotite as outlined in part-I with mixing properties given in Table 8 and standard-state thermodynamic properties for annite (Ann), phlogopite (Phl), and eastonite (Eas) given in Table 9 of the companion paper. For alternative versions of our biotite model with symmetric W^G_AnnPhl and symmetric/asymmetric W^G_AnnPhl obtained from thermodynamic analysis of the olivine–biotite exchange equilibrium using a W^G_FaFo of 10 kJ/mol instead of 5 kJ/mol (one-site), we used the modified thermodynamic parameters as given in the discussion section of part-I (in the first paragraph entitled ‘standard enthalpy of formation values of biotite endmembers’). All other thermodynamic properties are those from the endmember database of Holland and Powell (2011), version ‘ds62’, and the a–X relationships of White et al. (2014) for solid solutions. These solid solutions were calibrated specifically for metapelite rocks. For certain pseudosections, we used, in addition to the solid solutions mentioned in White et al. (2014), the talc model of Holland and Powell (1998) and the olivine model of Holland and Powell (2011) with the W^G_FaFo corrected to 5 kJ/mol (one-site). We compare the phase diagram calculations using our biotite model to predictions made with the biotite model included in the a–X relationships of White et al. (2014, W14), keeping everything else the same.

Self-written programs in Mathematica® were used for computing the equilibrium Fe–Mg distribution between biotite and olivine (Fig. 1) and biotite–orthopyroxene (Opx) (Fig. 2). For this purpose, self-written routines were generated to calculate the thermodynamic functions describing the exchange. The a–X relations of olivine were based on a symmetric W^G_FaFo = 5 kJ/mol (one site, see discussion in part-I), and those of orthopyroxene were calculated according to White et al. (2014) (Figs. 3, 4, 5).

Fig. 1

Roozeboom plot of the experimental data of Zhou (1994) on the Fe–Mg exchange between biotite and olivine (P = 1.1 kbar) in terms of X_Fe^Ol vs. X_Fe^Bt at T = 800 °C (a) and T = 700 °C (b). Starting compositions are shown as black dots, which are connected to the final compositions (red dots) with a line. Recalculated curve using the biotite activity model and thermodynamic data from this study (Tables 7, 8 of part-I) is shown in red, and for comparison in blue using the thermodynamic data and activity models cited in White et al. (2014)—W14

Fig. 2

Roozeboom plot of the experimental data of Fonarev and Konilov (1986) on the Fe–Mg exchange between biotite and orthopyroxene (P = 4.9 kbar) in terms of X_Fe^Opx vs. X_Fe^Bt at T = 700 °C. Starting compositions are shown as black dots, connected to the final compositions (red dots) with a line. The recalculated curve resulting from this study is shown in red and in blue from a W14 calculation

Fig. 3

Plot of the experimental data of Ferry and Spear (1978, FS78) on the Fe–Mg exchange between biotite and garnet (P = 2.07 kbar). The experimentally determined molar Fe/(Fe + Mg) content of biotite (X_Fe^Bt) in equilibrium with garnet of Alm90Py10 composition is shown in a as open and filled squares (depending if the equilibrium was approximated from oversaturated or undersaturated conditions) as a function of temperature. Perple_X modeled X_Fe^Bt from our calculation (experimental bulk composition Alm90–Ann50 given in Table 1) is drawn as solid line. Broken line is an analogous W14 calculation: predicted Al^VI (apfu) of biotite as a function of temperature is also shown. The modeled temperature dependence of lnK_D of the garnet–biotite Fe–Mg exchange and computed with Perple_X is plotted in b for both cases of a this study (solid line) and a W14 calculation (broken line). Error bars of experimental lnK_D values are ± 2σ

Fig. 4

a X_Fe^Bt in equilibrium with garnet of Alm90Py10 composition (red; Ferry and Spear 1978, FS78), or of Alm80Py20 (blue) and Alm70Py30 composition (dark yellow; Gessmann et al. 1997, G97), i.e., color codes indicate subsequent lower X_Mg bulk compositions, as a function of temperature. Corresponding experimental lnK_Ds are shown in b with error bars (2σ) for the Alm70Py30 experiments. The Al^VI content of biotite, measured by G97 in their Alm80Py20 and Alm70Py30 experiments with a microprobe, is shown as black dots in the lower part of a. Solid lines are Perple_X-computed X_Fe^Bt, Al^VI, or lnK_D from our calculations using bulk compositions of experiments, as given in Table 1. The colors of the lines indicate to which experimental data the fit corresponds to

Fig. 5

Same caption as in Fig. 4, except that solid lines represent W14 calculations

Phase relations shown in Figs. 6, 7, 8, 10, 11, 13 and 14 and data given in Table 2 were computed with Perple_X suite of programs version 6.9.0. We used the program vertex for making pseudosections (Connolly 2005, 2009) using the bulk compositions, as given in Table 1. The equation of state for H₂O was that of Pitzer and Sterner (1995). In our calculations, the thermodynamic data file was hp62ver.dat with data for the biotite endmembers annite, phlogopite, and eastonite substituted by corresponding ones from Table 9 of part-I. This dataset in Perple_X format is given in Supplementary Table S1. In a W14 calculation, the original annite, phlogopite, and eastonite data from hp62ver.dat were used. The biotite activity model presented in part-I was coded for Perple_X and incorporated in solution_model.dat named Bio(D). It is also given in Supplementary Table S1.

Fig. 6

T–X_{FeO/(FeO + MgO)} pseudosection computed at P = 2.07 kbar with Perple_X (this study calculation in the KFMASH system, H₂O in excess, bulk given in Table 1), relevant to illustrate phase relations in the Fe–Mg exchange experiments between biotite and garnet of Ferry and Spear (1978, FS78) and Gessmann et al. (1997, G97). Purple vertical lines indicate the experimental bulk compositions. Note that cordierite bearing assemblages would be stable in G97 bulks. The color of phase assemblage fields indicates the thermodynamic variance of the equilibria. White = divariant fields, light gray = trivariant, and dark gray = quadrivariant. Adjacent fields always differ by at least one phase. In other words, adjacent fields always differ in their variance by one degree of freedom. Therefore, adjacent fields, which have the same color, have thin fields in between them containing both phases which differ in the adjacent fields. For example, in the case of adjacent divariant fields, a univariant reaction line separated the two fields. Fields separated by a point differ by two phases. These phase diagram rules apply for all subsequent pseudosections (Figs. 7, 8, 10, 11, 13, 14)

Fig. 7

T–X_{FeO/(FeO + MgO)} pseudosection computed at P = 2.07 kbar with Perple_X in the KFMASH system, with H₂O in excess (bulk given in Table 1). In contrast to Fig. 6, cordierite, chlorite, orthoamphibole and staurolite were excluded in this calculation. Isopleths are predicted Al^VI in biotite (colored lines) and X_Fe^Bt = Fe/Fe + Mg) (broken lines). Bulk compositions in the Fe–Mg exchange experiments between biotite and garnet of Ferry and Spear (1978) and of Gessmann et al. (1997) are shown as vertical lines labeled FS78 and G97. For FS78 bulk, computed X_Fe^Bt isopleths can be compared to experimentally determined X_Fe^Bt (purple numbers). Pseudosections shown in a and b are calculated with Fe–Mg mixing properties of biotite as obtained from the thermodynamic analysis of Zhou (1994)’s olivine–biotite Fe–Mg exchange experiments, using a W^G_FaFo of 5 kJ/mol, those in c and d of 10 kJ/mol (one-site basis). Our preferred calculation is pseudosection a with asymmetric Fe–Mg mixing in biotite (W^G_PhlAnn = − 8.8 kJ/mol, W^G_AnnPhl = 14.3 kJ/mol, associated ∆H^o_f,Ann = − 5131.6 kJ/mol, Tables 7, 8, part-I). Pseudosection b was computed with symmetric (W^G_AnnPhl = 9.3 kJ/mol, associated ∆H^o_f,Ann = − 5136.7 kJ/mol), c with asymmetric (W^G_PhlAnn = 0.9 kJ/mol, W^G_AnnPhl = 36.4 kJ/mol, associated ∆H^o_f,Ann = − 5136.2 kJ/mol), and d with symmetric Fe–Mg mixing in biotite (W^G_PhlAnn = 26.0 ± 1.7 kJ/mol, associated ∆H^o_f,Ann = − 5142.7 kJ/mol). In contrast to the examples in the other subfigures, the agreement between experimental and model X_Fe^Bt is excellent in a

Fig. 8

T–X_{FeO/(FeO + MgO)} pseudosection as in Fig. 7, with isopleths of lnK_D of the Fe–Mg exchange between biotite and garnet, illustrating the bulk composition dependence of lnK_D. Solid isopleths represent our calculation, broken lines a W14 calculation

Fig. 9

Plot of the experimental data of Perchuk and Lavrent’eva (1983, PL83) on the Fe–Mg exchange between biotite and garnet (P = 6 kbar): lnK_D vs. T is shown with error bars of 2σ in a, Al^VI of biotite (microprobe data from Aranovich et al. 1988) vs. T in b. Perple_X-computed lnK_D and Al^VI based on our calculation are drawn as solid lines and as broken lines resulting from a W14 calculation (bulk composition used is given in Table 1)

Fig. 10

P–T pseudosection in the KFMASH system for an ‘average metapelite’ (Table 1) resulting from a our calculation, compared to b a W14 calculation (White et al. 2014, their Fig. 4). Subsolidus equilibria are calculated under water-saturated conditions, while suprasolidus equilibria were calculated with a fixed bulk H₂O of 6.65 mol%

Fig. 11

T–X_{FeO/(FeO + MgO)} pseudosection in the KFMASH system for the ‘average metapelite’ shown in Fig. 11 under water-saturated conditions. Isopleths are Al^VI (apfu) in biotite. a is based on our calculation; b on a W14 calculation

Fig. 12

Plot of octahedral Al of biotite as a function of its X_Mg = Mg/(Mg + Fe) content for 450 biotites from low- to high-grade metapelitic samples from northwest Maine. Al^VI is the total octahedral Al (black); Al^IV—1 is the Tschermak substitution-balanced octahedral Al (red), and excess Al^VI is the difference between the two (blue, all in apfu). Solid red lines define a band of tetrahedral rotation angles between 7° and 9°, respectively (Benisek et al. 1999). The extent of the Tschermak substitution in natural biotites seems to concentrate in this field. See text for further discussion

Fig. 13

P–T pseudosection in the KFMASH system for high T-low P (granulite facies) rock X567, studied by Pitra and de Waal (2001), resulting a from our calculation, compared to b a W14 calculation. Blue line indicates the final breakdown of biotite, and the red line is the wet solidus. Subsolidus equilibria in the P–T pseudosection are calculated under water-saturated conditions, while suprasolidus equilibria were calculated with a fixed bulk H₂O of 7.00 mol%. Phase assemblage names in bold indicate the two peak metamorphic assemblages. Below the P–T pseudosection, a log₁₀($a_{\text{H}_{2}\text{O}}$)-T pseudosection is given. Purple lines in the pseudosections above and below indicate the same thermodynamic states at 2.6 kbar (a) and 3.2 kbar (b) and water-saturated conditions with an H₂O activity of 1 (log₁₀(1) = 0). Going down in the log₁₀($a_{\text{H}_{2}\text{O}}$)-T pseudosection shows the phase relationships that would establish if the activity of water was lowered at pressures as indicated by the isobaric purple line. Isopleths are Al^VI in biotite (apfu) and are drawn in color in both the P–T and the log₁₀($a_{\text{H}_{2}\text{O}}$)-T pseudosections

Fig. 14

P–T pseudosection in the KFMASH system for a high-pressure metapelite (sample M4), studied by Wei and Powell (2003), resulting a from our calculation, compared to b a W14 calculation. The phase equilibria are calculated assuming water and quartz saturated conditions (i.e., the activity of water and quartz are set to 1). A blue line indicates the breakdown of biotite. The red line is the wet solidus. Black broken lines show the Al^VI (apfu) in biotite. Colored lines show the Si-in-phengite (white mica), which has been used as a barometer in earlier studies (e.g., Wei and Powell 2003)

Table 1 Bulk compositions (molar) used in the construction of pseudosections with Perple_X and in the calculation of mineral–chemical data given in Table 2

Table 2 Compositions of KFMASH-biotite and -garnet computed with the software Perple_X (Connolly 2005, 2009—Perp) and Theriak-Domino (de Capitani and Brown 1987—TD) for bulk compositions used by Ferry and Spear (1978—FS78) in their experimental study of the Fe/Mg exchange between garnet and biotite at P = 2.07 kbar (molar garnet/biotite ratio was 98/2, and garnet Fe/Mg ratio was 9/1 in the experiments and assumed to remained unchanged)

Mineral–chemical quantities like Al^VI or mole fraction X_Fe = Fe/(Fe + Mg) in biotite and in other minerals were computed with the Perple_X program werami and appear as isopleths in the various pseudosections. The lnK_D for the garnet–biotite Fe–Mg exchange reaction (Figs. 3, 4, 5, 9) and contours of lnK_D (Fig. 8) were computed by self-written Mathematica functions processing data files containing X_Fe and X_Mg of biotite and garnet as a function of temperature and/or pressure/bulk composition, produced with werami.

Supplementary calculations were done with Theriak (de Capitani and Brown 1987; de Capitani and Petrakakis 2010) in case of the garnet–biotite Mg/Fe exchange reaction (Table 2) using the Theriak version 09. 03. 2019 and the database tcds62c. The new standard-state data, and our new biotite activity model in Theriak readable format, are given in Supplementary Table S2. Mineral abbreviations used in the text and figures are according to Whitney and Evans (2010).

Applications of the new biotite activity model in the KFMASH system

Experimental Fe–Mg exchange equilibria involving biotite

Notice that the experimental olivine–biotite equilibrium was used for the calibration of our model. In contrast, biotite–orthopyroxene, biotite–garnet, and biotite–garnet–cordierite equilibria were not and could, therefore, be used for validation purposes.

Biotite—olivine

The experimental data of Zhou (1994) on the Fe–Mg exchange between biotite and olivine have been used in part-I, combined with other experimental data to calibrate Fe–Mg biotite mixing properties and to derive enthalpy of formation values for annite, phlogopite, and eastonite. They are presented in Fig. 1a, b, together with recalculated our-study Roozeboom plots, made using the program Mathematica^®, based on the reaction:

$${1}/{2}\;{\text{Forsterite}} + {1}/{3}\;{\text{Annite}} = {1}/{2}\;{\text{Fayalite}} + {1}/{3}\;{\text{Phlogopite}},$$

(1)

and compared to a W14 calculation (details of the experimental setup, starting materials, and biotite composition were given in part-I). Although our-study and W14 Roozeboom plots were calculated using considerably different biotite activity models and thermodynamic data for annite and phlogopite, the computed X_Fe^Ol vs. X_Fe^Bt curves are quite similar, satisfying the compositional brackets within error (Fig. 1a (at 800 °C), b (at 700 °C), the other isothermal Roozeboom plots are given in Supplementary Fig. S1). The only exception is at Fe-rich conditions, where our calculation predicts a somewhat lower X_Fe^Bt in better agreement with the experimental record. This is, of course, not surprising, as Zhou’s data have been used in part-I to calibrate Fe–Mg biotite mixing properties based on the assumption that these experiments indeed represent equilibrium Fe–Mg distributions between olivine and biotite.

Zhou (1994) used biotite:olivine weight ratios of 1:1, 2:1, or 1:2 in his experiments; however, it is not clear from his tables which ratio was used in which run. We constructed an appropriate bulk composition for his experiments at 800 °C, given in Table 1, by combining an olivine with X_Fe^Ol = 0.5 with an Fe–Mg biotite with molar Si:Al = 3:1 and X_Fe^Bt = 0.3, which is the equilibrium composition of biotite co-existing with olivine at this temperature (Fig. 1a). The resulting P–T pseudosection (not shown), computed with Perple_X and our biotite model, shows biotite + olivine + kalsilite as stable phases in roughly equal molar amounts above 730 °C up to 900 °C (at P = 1.1 kbar). The predicted Al^VI level in biotite is zero confirming the in part-I discussed evidence of no Tschermak components in biotites in the exchange experiments of Zhou (1994). The use of the olivine–biotite exchange reaction to calibrate the Fe–Mg part of the model has a considerable benefit compared to, e.g., the garnet–biotite exchange experiments, where the Mg–Fe interaction in biotite is affected by a considerable amount of octahedral Al (see below).

Biotite–orthopyroxene

The calculation of the Fe–Mg distribution between biotite and orthopyroxene and comparison with experimentally determined distribution coefficients (K_Ds) constitutes an independent test on the validity of biotite properties derived in the companion paper. This exchange, given by the reaction:

$${\text{Annite}} + {3}/{\text{2 Enstatite}} = {\text{Phlogopite}} + {3}/{\text{2 Ferrosilite}},$$

(2)

which was experimentally studied by Fonarev and Konilov (1986) at P = 4.9 kbar, T’s of 700, 750, and 800 °C and log₁₀f_O2 conditions dictated by the quartz–fayalite–magnetite (QFM) oxygen fugacity (f_O2) buffer. Whereas biotite from the run products was found to be homogeneous, orthopyroxene was notably inhomogeneous, comprising unreacted starting compositions, intermediate metastable, and evolved possible equilibrium compositions. Fonarev and Konilov (1986) applied a complex statistical treatment to the whole set of their experimental data to establish equilibrium relations and to define mineral compositions that likely represent equilibrium. If their experiments indeed successfully bracketed equilibrium Fe–Mg distributions remains an open question. Their data at T = 700 °C are shown in Fig. 2 including calculated distribution curves based on our and W14 calculations. Both types of calculations represent the experimental data within error. For a given X_Fe^Opx, however, our biotite model yields a 0.05–0.1 lower mole fraction X_Fe^Bt. This also holds for the other isothermal Roozeboom plots in the Supplementary Materials.

Biotite–garnet

The biotite–garnet exchange reaction constitutes yet another independent test on the validity of biotite properties derived in the companion paper.

The experiments of Ferry and Spear (1978)

Using synthetic starting materials, Ferry and Spear (1978–FS78) undertook the first experimental study of the partitioning of Fe and Mg between biotite and garnet, given by the reaction:

$${\text{Phlogopite}} + {\text{Almandine}} = {\text{Annite}} + {\text{Pyrope}},$$

(3)

at P = 2.07 kbar, T’s between 550 and 800 °C (spaced in ~ 50 °C intervals) and oxygen fugacity buffered by the graphite–methane buffer. Because their exchange experiments were conducted with a very high molar garnet:biotite ratio (98:2), garnet composition (Alm₉₀Py₁₀ and Alm₈₀Py₂₀) remained unchanged during the experiments, which was also proven by microprobe analysis from the run products. Equilibrium was demonstrated by performing two experiments at each T, in which starting biotite had either a more Mg-rich or a more Fe-rich composition than biotite in equilibrium with garnet at this T (X_Fe^Bt in these starting biotites was 0.25, 0.5, 0.75, or 1.0). After the experiment, run products were dispersed on polished diamond surfaces, and biotite compositions were measured with a microprobe using a linear relationship between Fe and Mg X-ray intensities and concentrations (see FS78 for more details). Thus, only the Fe and Mg contents of reacted biotite grains were measured, whereas there is no information on the content of possible Al^VI in biotite. Six compositional brackets were determined in such a way by FS78 through experiments with Alm₉₀Py₁₀ garnet. The equilibrated biotites were found to differ not more than 0.04 in their final X_Fe^Bt mole fractions (FS78 give an accuracy of ± 0.01 for X_Fe^Bt). As a result of increasing experimental temperatures from 550 to 800 °C, their X_Fe^Bt rose from values around 0.61–0.75, which is a change of ca. 0.15 in mole fraction X_Fe^Bt for this temperature interval of 250 °C. Two wider brackets were obtained by FS78 in experiments using Alm₈₀Py₂₀.

We have reconstructed bulk compositions for the FS78 experiments, based on the molar 98:2 garnet:biotite ratio they have used (Alm90–Ann50, Alm90–Ann75, and Alm90–Ann100, Table 1) and have calculated the equilibrium compositions of biotite and garnet at the experimental temperatures and a pressure of 2.07 kbar using Perple_X and Theriak software and our and W14 biotite thermodynamic data and solution models (note that for garnet W_AlmPy = 2.5 kJ/mol in all calculations). The results are given in Table 2 and are plotted in Fig. 3 for the case of bulk composition Alm90–Ann50. From Fig. 3a, which shows X_Fe^Bt and Al^VI in biotite as a function of temperature, it can be seen that the our calculations reproduce the experimentally determined X_Fe^Bt of FS78 well within error. The W14 calculation fails to do so, it predicts a ca. 0.1 mol fraction too large X_Fe^Bt. As a consequence, lnK_D^this−study, where K_D = (Mg/Fe)^Grt/(Mg/Fe)^Bt, is consistent with all FS78 experimental lnK_D-brackets, whereas lnK_D^W14 is systematically lower outside the 2σ error range of the data (Fig. 3b). Both biotite models indicate an Al^VI content in biotite in the order of 0.2–0.4 apfu. However, Al^VI is predicted to decrease with T by an our calculation and to increase with T by a W14 calculation (note that computed Al^VI corresponds to the Tschermak substitution-balanced octahedral Al, i.e., Al^VI = Al^IV-1).

The volume % garnet and biotite computed by Perple_X and Theriak for the experimental bulk compositions are quite similar. They amount to 1.4–2.7 vol% biotite and 96.5–96.9 vol% garnet. Additional phases predicted to be stable with biotite + garnet are orthopyroxene, olivine, quartz, and sanidine/spinel in the high-T runs. If present indeed, they were undetectable due to their predicted small amounts < 1 vol% (Table 2).

The experiments of Gessmann et al. (1997)

Gessmann et al. (1997—G97) extended the FS78 work to more Mg-rich bulk compositions, by conducting the same type of exchange experiments (high garnet:biotite weight ratio of 95:5) with Alm₈₀Py₂₀ and Alm₇₀Py₃₀ garnet starting compositions, X_Fe^Bt starting compositions of 0.25, 0.4, 0.5, and 0.65, at the same pressure of P = 2.07 kbar and T’s ranging between 600 and 800 °C. They characterized garnet and biotite in their run products more comprehensively than FS78 by not only tabulating X_Fe^Bt, but also its Al^VI content and X_Fe^Grt. Similar to FS78, garnet composition remained nearly unchanged to within 0.02 mol fraction after the conclusion of an exchange experiment. The composition of biotite in G97 experiments in terms of X_Fe^Bt, equilibrated with either Alm₈₀Py₂₀ or Alm₇₀Py₃₀ garnets, is shown in Fig. 4a as a function of temperature. For comparison, the FS78 data from Table 1 are also plotted. As discussed by G97, their high-T Alm₇₀Py₃₀ data bear inconsistencies, which is obvious from the fact that, e.g., at 800 °C, X_Fe^Bt is larger than that of biotite equilibrated with Alm₈₀Py₂₀. At a given T, X_Fe^Bt in equilibrium with Alm₇₀Py₃₀ should be, however, systematically lower compared to X_Fe^Bt in equilibrium with Alm₈₀Py₂₀ (see G97 for further discussion and possible explanations for the deviation of their high-T Alm₇₀Py₃₀ runs). The Al^VI content of biotite is shown in Fig. 4a (lower part), and experimental lnK_D values of G97, and FS78 for exchange reaction (3) are plotted in Fig. 4b as a function of temperature. The experimental record shows that lnK_D gets less negative, i.e., K_D increases with increasing bulk-X_Mg content.

Similar as in the case of FS78 described above, bulk compositions were computed for the G97 experiments (Alm80–Ann40 and Alm70–Ann25, Table 1) and phase relationships determined with Perple_X using our and W14 calculations. The predicted mineral compositions and lnK_D from our calculation are shown as curves in Fig. 4 to allow a comparison to the experimental values, those for the W14 calculation in Fig. 5. In both figures, the results from Alm90–Ann50 (bulk in FS78 experiments) are also included.

The following features are noticeable from the results of our calculations: For bulk Alm80–Ann40, X_Fe^Bt is larger by ca. 0.05 mol fractions than most experimental values, and for Alm70–Ann25, the agreement is good (except for the high-T data as mentioned above, Fig. 4a). The predicted Al^VI content in biotite matches the measured ones above 700 °C and is somewhat larger at lower Ts. It decreases slightly with rising temperatures and increases with increasing X_Mg of the bulk composition, i.e., experiments with Alm70 have higher Al^VI than experiments with Alm80. Computed lnK_D^this−study values agree within error with experimental ones for bulk Alm80–Ann40, whereas for bulk Alm70–Ann25, this is only the case at 600 °C. A slight dependence of lnK_D^this−study on bulk composition is visible in the form that it gets less negative (i.e., K_D increases with increasing bulk X_Mg).

The most obvious differences of W14-calculated to experimental mineral–chemical data of biotite (Fig. 5) are the following: The agreement of X_Fe^Bt with experimental values is good only for bulk Alm70–Ann25 (Fig. 5a), but gets increasingly poor for Alm80–Ann40 and especially for Alm90–Ann50 (FS78 experiments, as already discussed above). The same applies accordingly for lnK_D^W14, and its dependence on bulk-X_Mg is the reverse of that noticed above for our calculations (Fig. 5b). The predicted Al^VI content is in good agreement with the measured ones.

T-X _Fe pseudosection for FS78 and G97 experiments

In Fig. 6, we present a T-X_Fe pseudosection at P = 2.07 kbar (our calculation with Perple_X, no phases excluded except orthoamphibole), with molar X_Fe^bulk = FeO/(FeO + MgO) ranging between 0.5 and 1.0 and temperatures from 550 to 800 °C to illustrate computed phase relations relevant for the FS78 and G97 exchange experiments (X_Fe^bulk in FS78 and G97 is shown as vertical purple lines). The pseudosection shows that FS78 experimented in a range where biotite + garnet are stable at most T’s co-existing with orthopyroxene (in small amounts < 1 vol%, Table 2). At lowest T, chlorite (Chl) and at temperatures above ~ 750 °C, cordierite (Crd), olivine, sanidine, and spinel would be additionally stable. For the bulk compositions used by G97, on the other hand, cordierite is predicted at most T’s, + chlorite up to ca. 610 °C, to coexist with garnet + biotite. Because both minerals were not reported—the computed vol% of cordierite between 20 and 50 vol% in the high-T experiments, and 10–5 vol% chlorite for the low-T experiments should have been detected by XRPD—it can be assumed that kinetic factors inhibited the growth of both minerals in the experimental charges within the time-scale of the experiments. Similarly, Berman et al. (2007), based on phase-stability calculations with Theriak-Domino, noted that their experimental mineral assemblage biotite + sillimanite + sanidine + quartz is metastable with respect to cordierite + sanidine.

If cordierite, chlorite, staurolite, and orthoamphibole are excluded, a T-X_Fe pseudosection results as shown in Fig. 7a (our calculation). The biotite + garnet + orthopyroxene field extends to lower bulk-X_Mg, covering now also G97 experiments with Alm₈₀Py₂₀ (bulk Alm80–Ann40 in Table 1). For bulk Alm70–Ann25, low-variance 5-phase assemblages are predicted with garnet + biotite + orthopyroxene ± additional spinel, andalusite, or quartz. Noteworthy, biotite is not stable for this bulk composition at 800 °C, which might explain the deviation of experimental high-T lnK_D values from the lnK_D vs. T trend as defined by the lower temperature experimental data (Fig. 5b, G97, their Fig. 11). Compared to the high-variance 3-phase garnet + biotite + orthopyroxene field, where vol% orthopyroxene is very low (< 1 vol%) and thus experimentally not detectable, the amount of orthopyroxene for the Alm70–Ann25 bulk composition is much larger (20–40 vol%), which is consistent with the observation of G97 that orthopyroxene needles appear in their high-T runs for this bulk composition.

Isopleths for X_Fe^Bt and Al^VI in biotite are also plotted in the T-X_Fe pseudosection of Fig. 7a. X_Fe^Bt isopleths are relatively steep, so that, as observed in the FS78 experiments, X_Fe^Bt increases by only ~ 0.13 mol fractions over the whole temperature range from 550 to 800 °C. The mole fractions X_Fe^Bt as measured in the FS78 experiments are indicated as purple numbers next to the purple vertical line, which indicates the bulk composition of the experiments. These X_Fe^Bt are the mean of the equilibrium brackets of FS78 (exact experimental X_Fe^Bt values are also mentioned in Table 2). Comparison of the calculated X_Fe^Bt isopleths with these experimental values provides a direct test for internal consistency with the FS78 experiments. Figure 7a shows the near perfect agreement of our calculation in terms of X_Fe^Bt.

There exists a considerable debate on the likely Al^VI content of biotite in FS78 experiments (see Holdaway 2000, and discussion therein). FS78 themselves suppose that approximately 6% of octahedral cations in their synthetic Ann100 could be Al. The Al^VI isopleths have a negative slope in the garnet + biotite + orthopyroxene 3-phase field, and Al^VI is predicted to decrease with increasing T for a given bulk composition. This means that FS78 biotites were very likely (a) not as Al-rich as those analyzed by G97, (b) not Al^VI-free nor contained only minor Al^VI, as assumed in various works analyzing the garnet–biotite exchange data (e.g., Kleemann and Reinhardt 1994; Powell and Holland 1999; Holdaway 2000), but incorporated octahedral Al in the order of 0.25–0.35 apfu (Fig. 4a, Table 1). This conclusion holds also based on a W14 calculation for which the predicted Al^VI level would be around 0.2 apfu for FS78 biotites (Fig. 5a, Table 1).

We have also calculated the equivalent pseudosection of Fig. 7a with an alternative version of our biotite model, which uses a single symmetric interaction parameter to describe the enthalpy of mixing behavior along the Ann–Phl binary instead of two interaction parameters as in our original asymmetric model. As shown in part-I, assuming a symmetric interaction parameter in the single inversion least-squares procedure to derive calibrated parameters for our biotite model also leads to changes in ∆H^o_f,Ann. The pseudosection calculated with this ∆H^o_f,Ann = − 5136.7 ± 2.1 kJ/mol and W^G_AnnPhl = 9.3 ± 1.8 kJ/mol (3-site cation basis) is shown in Fig. 7b and indicates that a symmetric model has X_Fe^Bt isopleths, which disagree with the experimental X_Fe^Bt in the order of 0.1 molar fraction. To achieve consistency with the experimental data of FS78 on the Fe–Mg distribution between biotite and garnet, one needs to apply a rather unrealistically large non-ideality for Fe–Mg mixing in garnet (W^G_AlmPy around 12 kJ/mol, 3-site cation basis). Therefore, this symmetric model is not internally consistent with FS78 and not further considered in this paper.

We have used a W^G_FaFo = 5 kJ/mol (one-site basis) in our single inversion least-squares procedure to derive parameters for biotite from the thermodynamic analysis of the biotite–olivine Fe–Mg exchange data of Zhou (1994). However, the studies of Sack (1980) and Sack and Ghiorso (1989) point to a larger Fe–Mg non-ideality in olivine of W^G_FoFa around 10 kJ/mol (one-site basis). In part-I, we have used this mixing parameter of olivine to derive two alternate biotite models, one with symmetric and the other with asymmetric Fe–Mg mixing in biotite. The symmetric model has a W^G_{PhlAnn, sym} = 26.0 ± 1.7 kJ/mol and an associated different ∆H^o_f,Ann = − 5142.7 ± 2.10 kJ/mol. For the asymmetric case, W^G_PhlAnn = 0.9 ± 6.1 kJ/mol, W^G_AnnPhl = 36.4 ± 2.4 kJ/mol, and the associated ∆H^o_f,Ann = − 5136.2 ± 1.7 kJ/mol.

Figure 7c, d shows the pseudosections, equivalent to Fig. 7a, computed with these alternate biotite models (calibrated with W^G_FoFa = 10 kJ/mol, one-site basis). The X_Fe^Bt isopleths of both these models show considerable disagreement with the experimental X_Fe^Bt and are therefore not considered to be consistent with the FS78 experiments. Even when one changes W^G_AlmPy, no agreement closer than 0.1 molar fraction to the experimental value can be found. These biotite models, based on W^G_FoFa around 10 kJ/mol, are thus not further considered in this paper.

Finally, we compared our biotite model with the W14 biotite model. The dependence of lnK_D of the garnet–biotite Fe–Mg exchange reaction (Eq. 3) on bulk composition is shown in Fig. 8, which is the same T-X_Fe pseudosection as drawn in Fig. 7a, but with contours of lnK_D, computed for both cases of an our and a W14 calculation. In the low-variance fields, lnK_D is constant as a function of bulk composition and only temperature-dependent (X_Fe^bulk < ca. 0.75). In the high-variance fields toward more Fe-rich bulk compositions, lnK_D contours become increasingly convex in case of our calculations which means that lnK_D values get less negative with increasing bulk X_Mg at constant temperature, in accordance with the experimental evidence. For the FS78 bulk composition, the experimentally determined lnK_D values of FS78 (Table 2, Fig. 3b) can be directly read from the lnK_D isopleths in Fig. 8. From the computed bulk composition dependence of lnK_D, which can be seen in Fig. 8, it is also clear that the calibration of FS78 will overestimate temperatures when applied to more Mg-rich bulks compared to the experimental one when such a bulk compositional effect is not considered. A lnK_D of − 1.4, for example, means a temperature of ca. 675 °C for the FS78 bulk composition (at 2.07 kbar), but corresponds to a by ca. 23 °C lower temperature when bulk X_Mg increases by ca. 0.1 mol fractions (Fig. 8). It is thus not surprising that it was noted in the literature that the FS78 calibration appears to overestimate metamorphic temperatures (e.g., Chipera and Perkins 1988; Kleemann and Reinhardt 1994). The effect of retrograde net transfer equilibria on garnet–biotite geothermometry could also play a role, but was not considered in these papers.

The lnK_D isopleths from a W14 calculation are nearly similar to those obtained from our calculations in the low-variance fields at X_Fe^bulk < ca. 0.75. At more Fe-rich bulk compositions, however, lnK_D^W14 contours take a concave form and predict lnK_D’s inconsistent with the experimental data of FS78 and the bulk composition dependence of the distribution coefficient (Fig. 4b).

The experiments of Perchuk and Lavrent’eva (1983)

Perchuk and Lavrent’eva (1983—PL83) studied the Fe–Mg exchange equilibria between cordierite, garnet, and biotite experimentally using mostly natural starting materials at P = 6 kbar and T’s from 575 to 950 °C and f_O2 as defined by the Nickel–Nickel-oxide buffer. The large number of experiments over a wide range of Fe–Mg mineral compositions (X_Mg^Grt = 0.05–0.76, X_Mg^Bt = 0.22–0.83) is a major benefit of their work. A drawback is, however, that their natural run products, with grain sizes of up to 300 µm, were strongly zoned. Equilibrium between biotite and garnet was approached by experiments with starting compositions on opposite sides of the Fe–Mg distribution curve, and the most shifted compositions were assumed to represent equilibrium values (see PL83 for more details on their method to estimate equilibrium compositions). A further complication in their experiments is the fact the natural minerals used for the study contained CaO and MnO in the order of up to 4 wt% in the case of garnet and TiO₂ (1–2 wt%) in the case of biotite. The Al content of equilibrated biotites were not measured by PL83, but were so at a later stage by Aranovich et al. (1989) for a subset of the PL83 biotites (their Table 2, 27 data). The Al level in these biotites is rather uniform, with Al^VI in the range 0.32–0.42 apfu (Fig. 9b). The resulting lnK_D of the garnet–biotite equilibrium is plotted as a function of temperature in Fig. 9a, including 2σ-error bars, calculated based on the assumption that measured mole fractions of garnet and biotite have a 2σ-error of 0.02.

In their experiments conducted at 700 °C, PL83 used a weight ratio of biotite:garnet:cordierite = 2:1:1. From the data given in Tables 2 and 6 on the chemistry of starting materials, we have constructed a bulk composition that should represent at least part of their numerous 700 °C runs. With an X_Fe^bulk = 0.4, it is more Fe-poor than the exchange experiments of FS78 and G97 (Table 1). At P = 6 kbar, the computed P–T pseudosection our calculation with Perple_X, excluding chlorite, staurolite, orthoamphibole  and chloritoid) shows a garnet–biotite–cordierite 3-phase field between 660 and 800 °C. Additionally, minor amounts of spinel and quartz (< 1 vol%) are predicted at higher and lower T’s, respectively, indicating that the experimentally studied garnet + biotite + cordierite assemblage is stable at the P–T conditions applied by PL83. The predicted Al^VI content of biotite in the biotite–garnet–cordierite assemblage from our calculations is in good agreement with the measured values up to 800 °C and somewhat lower for the highest temperature data (Fig. 9b). Modeled lnK_D for both our and a W14 calculation is very similar when plotted against temperature (Fig. 9a). The lnK_D vs. T slope is, however, flatter than the experimental trend, so that agreement with the experimental K_D’s is poor above ~ 800 °C. We also tested another bulk composition, namely that in the 900 °C runs of PL83, where the biotite:garnet:cordierite weight ratio was 3:1:1 and found the same result, i.e., their highest-T brackets could not be reproduced by our nor by a W14 calculation.

Pseudosection of an average metapelite

White et al. (2014) have constructed a P–T pseudosection for an average metapelite bulk composition in the KFMASH system. We have adopted their model pelite bulk composition (Table 1) and compared the P–T pseudosection resulting from our with an analogous W14 calculation in Fig. 10a, b (see also W14, their Fig. 4). As already outlined above, the difference in topologies is thus merely the effect of using our biotite activity model (Table 8, part-I), as well as thermodynamic data of biotite endmembers from part-I (Table 9), all other data being the same. Generally, the change in topology is not very pronounced. Noteworthy is that high-pressure stability fields are shifted to somewhat lower pressures based on our calculations (e.g., chlorite + phengite + quartz, biotite + garnet + phengite + quartz) and the high-temperature fields (e.g., cordierite + biotite + sanidine + quartz, biotite + sanidine + sillimanite + quartz) to lower temperatures. This makes the low-variance fields larger. Low-variance fields are good for thermobarometry, as the chemical potentials of the chemical components are fixed by the presence of the many co-existing phases. This means that the current biotite model can be used to more precisely estimate pressures and temperatures over a more extensive P, T range. Using the same model pelite composition, we also made a T-X_Fe pseudosection (our calculation at P = 6 kbar, Fig. 11a). Low-variance fields involving garnet come in at X_Fe^bulk of ca. 0.6–0.7 above ~ 570 °C, whereas more Fe-rich bulks (X_Fe^bulk around 0.8–0.9) are required to stabilize these assemblages according to a W14 calculation (Fig. 11b). Note that Al^VI isopleths all become flat in the T-X pseudosection in the low-variance fields showing their independence on bulk composition. Considering isopleths of the Al^VI content in biotite from our biotite model (note that Al^VI = Al^IV-1), these show an exponential drop at very Mg-rich compositions as a function of X_Fe^bulk (in the range 0–0.2), i.e., Al^VI is extremely bulk X_Fe dependent, merging into flat, slightly negative slopes toward more Fe-rich bulks (with X_Fe^bulk > ca. 0.4). The isopleths of Al^VI increase with temperature (from ca. 0.2 apfu at low T’s), reach a value of ca. 0.43 at ca. 650 °C at maximum, and then decrease again. The trend of the Al^VI contours also predicts that the extent of the Tschermak substitution gets slightly larger going from Mg to Fe-rich biotites (as discussed in part-I, this seems to be related to the fact that biotites then have their ‘optimal’ tetrahedral rotation angles of ca. 8°). This is in accordance with the findings of, e.g., Foster (1960), who postulated an increase of octahedral Al with increasing Fe²⁺ in biotite. Guidotti et al. (1975), studying the crystal-chemistry of natural biotites from northwestern Maine stemming from various metamorphic grades, questioned Foster’s results and found no clear correlation between their Al^VI content and Mg/Fe ratio. In Fig. 12, we have plotted Al^VI, Al^IV − 1 (i.e., Tschermak balanced Al^VI), and excess Al^VI of 450 biotites from low- to high-grade metamorphic pelitic schists from northwestern Maine (data were provided by D. J. Henry) as a function of their magnesium number X_Mg. Considering Al^VI (black dots), data cluster at intermediate X_Mg and levels of Al^VI between ca. 0.35 and 0.55 apfu without a clear mutual dependence in accordance with the findings of Guidotti et al. (1975). If we plot Al^IV – 1 vs. X_Mg, a similar picture emerges as found by Benisek et al. (1999, their Fig. 10), i.e., Mg-rich biotites tend to have a lesser degree of Tschermak substitution than more Fe-rich ones (a similar trend was noted for calcic amphiboles by Robinson et al. 1982). This is in agreement with what can be read off from the Al^VI isopleths, as shown in Fig. 11a. The predicted range of ~ 0.15 to ~ 0.4 apfu for the Tschermak substitution-balanced octahedral Al content in biotites is in good agreement with that appearing in Fig. 12 (red dots) for the biotites of Maine, which nearly all lie in a band characterized by tetrahedral rotation angles between 7° and 9° (see also Benisek et al. 1999).

Compared to our calculations, isopleths from a W14 calculation indicate systematically lower Al^VI contents in biotite for a given bulk composition and temperature, too low to be consistent with evidence from, e.g., the natural Maine biotites shown in Fig. 12 (red dots), especially at low-grade temperatures, where Al^VI < 0.1 apfu is predicted (Fig. 11b). Toward higher metamorphic grades (temperature), this difference of ca. 0.2 Al^VI apfu between W14 and in this study computed Al^VI decreases.

A remarkable feature of almost all of the 450 natural biotites from Maine (and probably for metamorphic biotite in general) is the fact that these contain appreciable amounts of excess Al^VI, i.e., Al^VI in excess to that balanced by the Tschermaks substitution (Fig. 12, blue dots). The data for this excess Al^VI cluster around 0.1–0.25 apfu at roughly 1:1 Fe:Mg ratios, and its values seem to increase in Mg-rich bulk compositions. It is obvious from Fig. 12 that biotite activity models should account for this presence of excess Al^VI by introducing, e.g., a muscovite- or a K-vacancy component. We will do this in a forthcoming paper that presents a comprehensive biotite activity model extending that of this study to include excess Al^VI, Ti, and Fe³⁺ biotite endmembers.

Pseudosections of natural rocks in the KFMASH system

In this section, we use our KFMASH-biotite model to model two natural rocks, one who has experienced high-T, low-P conditions, and another one that has experienced high-P, low-T conditions. These two rocks have been modeled in the past in the KFMASH system using the predecessors of the W14 biotite model, i.e., the high-T, low-P rock by Pitra and de Waal (2001) and Tajčmanová et al. (2009) and the high-P, low-T rock by Wei and Powell (2003). These examples, therefore, merely show how the relationships between phases change with our model compared to the W14 biotite model in the KFMASH system. They cannot be used for validation of our biotite model, because the addition of Fe³⁺, Ti, di–tri-octahedral Al substitutions will modify the results and will increase the number of phases in the mineral assemblage. As mentioned above, this extended chemical system will be explored in a forthcoming paper. Therefore, the results discussed in this paragraph should be seen as resulting from a model in progress.

Pseudosection of a high-T-low-P metapelite

Pitra and de Waal (2001) (PdW) studied phase relations in high-T, low-P metapelites enclosed in granites of the Bushveld Complex using the simplified KFMASH system. They described an early paragenesis with andalusite (chiastolitic) + cordierite + biotite + quartz ± garnet recording P–T conditions of 550–600 °C and 2 kbar as was calculated by KFMASH pseudosection thermobarometry. This assemblage was transformed into a peak paragenesis containing garnet + sanidine + cordierite + biotite ± quartz and subsequently into spinel–cordierite symplectites (replacing earlier chiastolitic and) in a 2nd high-T event. This peak event was estimated to have occurred at 750–800 °C, again, based on the interpretation of KFMASH pseudosections in their original study. This heating event is attributed to the emplacement of nearby dry and hot A-type granites. According to PdW, Al^VI in biotite should range between 0.31 and 0.52 in the sanidine + cordierite + biotite + quartz (± garnet) stability fields. Tajčmanova et al. (2009) recalculated in their Fig. 2a KFMASH P–T pseudosection for sample X567 of PdW. Because they found that Al^VI (= X_Al^M1) in biotite was too high in these fields compared to that expected, they changed the enthalpy of the Fe–Mg ordering reaction 2/3 phlogopite + 1/3 annite = ordered Fe–Mg biotite (Obi, see part-I) to − 6.8 kJ/mol to achieve a better agreement with the natural observations. Here, we demonstrate that such a change in the ordering enthalpy, which they had used as a fitting parameter, is not necessary and that the observations are readily explained by either our new model or the W14 model which both use an enthalpy of ordering of − 2 kJ/mol.

Performing our and W14 calculations to sample X567 under water-saturated conditions (bulk given in Table 1) yields the P–T pseudosections shown in Fig. 13a, b. Both are consistent with the petrographic evidence of the first low-T metamorphic event, i.e., show a stability field for the early paragenesis (andalusite + cordierite + biotite + quartz ± garnet) followed by biotite + cordierite + sanidine + quartz with rising temperatures. However, in the W14 calculation, the wet solidus is already reached before the peak metamorphic assemblages could be established, i.e., garnet + cordierite + biotite + sanidine and finally cordierite + spinel parageneses at the highest T. In contrast, in our calculation, biotite breaks down at lower T (blue line in Fig. 13a). As biotite is no longer part of the assemblage, the melting temperature of the remaining mostly ‘dry’ minerals at water-saturated conditions is increased, and melt forms at about 50° higher. Because of this, the garnet + cordierite + biotite + sanidine and finally cordierite + spinel parageneses are established without any melt involvement. This example shows that the choice of biotite model in the KFMASH system can have a large effect on when partial melting would take place.

In the W14 (and PdW) calculation, the peak assemblages are metastable with respect to melt. Large-scale melting would form leucosomes, and these were not observed by PdW. Suprasolidus relationships resulting from the W14 calculation using 7 wt% H₂O (the amount that was needed to replicate wet solidus) gives unrealistic estimates between 5 and 10 vol% melt for the garnet–cordierite–biotite–sanidine–melt assemblages (Fig. 13b). Therefore, the W14 (and PdW) calculation requires a lowering of the H₂O activity to get to the right-phase assemblages. Dry conditions were also inferred by PdW due to the preservation of contrasting parageneses, by the presence of symplectites (rapid nucleation, slow growth), and by the absence of fibrous sill. To simulate dry conditions, a log₁₀(a_H2O)-T pseudosection was constructed (Fig. 13a) at 2.6 kbar (the peak P inferred by PdW) using our calculations and at 3.2 kbar using a W14 calculation. The purple isobaric lines in the pseudosection in Fig. 13b and the one in the log₁₀(a_H2O)-T pseudosection below are the same states at their respective pressure and water-saturated conditions with an H₂O activity of 1 (log₁₀(1) = 0). As is well known, the lowering of water activity (a) decreases the T of dehydration reactions and (b) increases the T of melting reactions. However, the magnitude of these effects does depend on the details of both biotite solution models as shown by the log₁₀(a_H2O)-T pseudosection.

PdW report that biotite in the garnet–cordierite–biotite–sanidine assemblage has X_Fe^Bt = 0.75–0.81, X_Fe^Crd = 0.64–0.69 and X_Fe^Grt = 0.90–0.92. The microprobe analysis of matrix biotite from sample X567 gives Al^VI = 0.48 apfu and X_Fe^Bt = 0.78 (PdW, Table 2). In both the P–T and log₁₀(a_H2O)-T plots of Fig. 13a, the Al^VI isopleths of biotite are given. At 2.6 kbar, T = 667 °C, and log(a_H2O) = − 0.133, the computed compositions (X_Fe^Bt = 0.73, Al^VI = 0.43 apfu, X_Fe^Crd = 0.65, X_Fe^Grt = 0.92) come nearest to the observed ones and only X_Fe^Bt and Al^VI are lower by ca. 0.02 and 0.04 mol fractions compared to measured compositions, respectively.

The W14 calculations show that under lower water activities, the solidus is moved to higher temperatures, and the garnet–cordierite–biotite–sanidine and finally cordierite + spinel parageneses form before melting occurs. For a W14 calculation, the best agreement with mineral-chemical data for sample X567 is obtained at a higher pressure of 3.2 kbar, T = 727 °C, and log(a_H2O) = − 0.1: X_Fe^Bt = 0.80, Al^VI = 0.40 apfu, X_Fe^Crd = 0.65, and X_Fe^Grt = 0.9.

At this moment, we have no means to say which of the models performs better as we have only worked in the simplified pelitic KFMASH system and tried to apply this to a chemical complex natural system. Validation of either model is not possible. However, the example does clearly show how the biotite model parameters of either model can result in quite different P–T-aH₂O pathways the rock could have undertaken. A typical question for many high-T, low-P studies is if dry conditions were reached before (partial) melting takes place, i.e., if water-saturated conditions prevailed, as is often assumed for prograde metamorphism. The earlier breakdown of biotite, which was also demonstrated by the model pelite system in KFMASH from the last paragraph, as shown by this example will influence at which temperature partial melting will start. It shows that biotite influences when dry conditions are reached as no dehydration is expected after the biotite-out curve. This example, therefore, shows that the important role biotite has in solving this question. Our new biotite model predicts, in the KFMASH system, in contrast to the W14 model, that biotite breaks down at lower temperatures before partial melting starts, which would move the (now dry) solidus to higher temperatures. It cannot be repeated enough that the results from this paragraph are for the KFMASH system only. The addition of Fe³⁺, Ti, di–tri-octahedral Al substitutions could potentially modify these results.

Pseudosection for a high-P low-T metapelite

As an example of our calculations on a high-P, low-T metapelite, we show phase relations for a garnet–phengite–biotite gneiss (sample M4, from the Dabie Shan area, central China), studied by Wei and Powell (2003), WP03, their Fig. 10 in Fig. 14a and compare it to an analogous W14 computation (Fig. 14b) in the KFMASH system. Both P–T pseudosections, constructed with the KFMASH bulk composition, estimated by WP03 for sample M4 (Table 1) appear almost identical concerning the distribution of stability fields of the mineral assemblages. The biotite-out (blue) line in our calculation occurs at about 1–2 kbar lower pressures. With regard to biotite, in KFMASH, isopleths of Al^VI show decreasing values with increasing pressure in both types of calculations. The parallelism between the Si content of phengite, a known barometer (Wei and Powell 2003), and biotite’s Al^VI content in biotite is striking and raises the hypothesis that Al^VI is a pressure indicator in KFMASH, especially in low-variance assemblages, such as garnet–chlorite–biotite–phengite–quartz, where it becomes independent of bulk composition, as many of the compositional degrees of freedom are fixed by the mineral assemblage. However, before we can make any generalizations and before we start testing the hypothesis that Al^VI is a pressure indicator also in more complex chemical systems, we first need to extend our biotite model to incorporate Ti, Fe³⁺, and the di–tri-octahedral substitution.

Conclusions

The application of our new biotite activity model and standard-state thermodynamic properties of Ann, Phl, and Eas presented in part-I to experimental phase equilibria and natural pelitic rocks, using mainly the software Perple_X, allows the following conclusions to be drawn:

• Biotite in the experimental Fe–Mg exchange data between biotite and olivine (Zhou 1994) had no or only minimal Al^VI, which is confirmed by pseudosection calculations with the experimental bulk composition. To extract, e.g., annites’ standard enthalpy of formation value, Zhou’s data are superior to existing biotite–orthopyroxene or biotite–garnet exchange data, because the extent of Tschermak substitution in biotite in the latter is not negligible.

• The Fe–Mg distribution between biotite–garnet, biotite–garnet–cordierite, and biotite–orthopyroxene are independent tests of the validity of our calculations presented herein. The biotite–orthopyroxene and Fe–Mg exchange equilibria between cordierite, garnet, and biotite are reasonable modeled by either our and W14 biotite models. However, the new biotite a–X relations in combination with the new enthalpy of formation value of biotite endmembers, as derived in part-I (Tables 7, 8), unlike the W14 biotite, successfully reproduce the FS78 experimental data. The resulting lnK_D values as a function of temperature are in good agreement with the experimental brackets of FS78s bulk composition of Alm₉₀Py₁₀ experiments (Figs. 3, 7, 8). The alternate biotite models, calibrated with symmetric W^G_PhlAnn and/or models that used W^G_FoFa around 10 kJ/mol (one-site basis) in the thermodynamic analysis of olivine–biotite Fe–Mg exchange data from Zhou (1994) to derive W^G_PhlAnn, are not compatible with FS78 experiments.

• Our calculations pertinent to the garnet–biotite reaction with more Mg-rich bulk compositions yield less negative lnK_D’s in accordance with experimental evidence [experiments of Gessman et al. (1997), with Alm₈₀Py₂₀ and Alm₇₀Py₃₀ as exchange partner of biotite, Fig. 4]. W14 calculations show the opposite trend (Fig. 5). The dependence of lnK_D on bulk-X_Mg is shown in a T–X_Fe pseudosection, which makes clear that the calibration of FS78 only applies to Fe-rich bulk compositions as used in the experiments and will yield too high temperatures when applied to more Mg-rich bulks far outside the experimental range.

• The predicted Al^VI contents for our as well as W14 calculations in biotite for G97 experimental bulk compositions ranges between ca. 0.3 and 0.45 apfu and are both in good agreement with measured values of G97 (microprobe data). Analogous computations for the somewhat Fe-richer FS78 bulk compositions indicate that also biotite in the FS78 exchange experiments contained Al^VI in the same order of ca. 0.3–0.4 apfu. Various speculations and assumptions on the extent of the Tschermak substitution in FS78 biotites (e.g., Holdaway 2000) can now be refuted based on this computational evidence. This is further supported by calculations with a more Mg-rich bulk composition as was used by PL83 in part of their biotite–garnet(–cordierite) Fe–Mg exchange experiments (X_Mg^bulk = 0.4). The predicted Al content of biotite is also in accordance with the measured one in this case for both our and W14 calculations (Fig. 9).

• Low variance fields in KFMASH pseudosection for an average metapelitic rock (Fig. 10) are shifted by ca. 50 K to lower T’s at high-T, low-P conditions and to lower P at higher P conditions as a result of the new biotite activity model and annite standard state properties. The relative increase of low-variance fields over high-variance fields containing biotite shows its increased potential for (pseudosection) thermobarometry, as, in these fields, the exchange relationships between the phases are mainly dependent on P, T and not on the bulk composition.

• As shown for the Maine biotites (Fig. 12), Tschermak substitution-balanced octahedral Al (i.e., Al^IV − 1) in natural biotites increases slightly with increasing X_Fe^Bt, a trend already noted by Benisek et al. (1999). This trend and the amount of this ‘Tschermak’-Al^VI are correctly predicted for an average pelite bulk composition using the new biotite activity model and annite standard state properties (Fig. 11). It seems to be related to the fact that tetrahedral rotation angles between 7° and 9° are energetically favored in the biotite structure.

• The upper thermal stability of biotite is reduced using our biotite model in KFMASH. This means that biotite can already break down before the wet solidus is reached. Partial melting would then depend on the dry solidus, which occurs at higher temperatures. It is often assumed that biotite dehydration reactions occur until the wet solidus is reached and, therefore, that water-saturated conditions can be assumed in the prograde path. Our biotite model questions that assumption in the KFMASH system.

• The upper pressure stability of biotite is reduced resulting from our calculations. Biotite Al^VI shows some potential as a barometer in the KFMASH system, especially in low-variance assemblages.

• There is a high need to extend the current model to incorporate Ti, Fe³⁺, and the di–tri-octahedral substitution to see if the relationships that we have found for our natural rocks in the KFMASH system hold in more complex chemical systems.