Introduction

For phase-equilibrium calculations involving biotite, an important phase especially in metamorphic (metapelite) assemblages, a thermodynamic model is a prerequisite in order to compute the activity-composition relationships for its constituent end-members. In petrological software, like Perple_X (Connolly 2005) and Thermocalc (Powell et al. 1998; Powell and Holland 2008) two activity models for biotite are currently in use, named Bi(W) and Bio(TCC) in Perple_X. The former, Bi(W), is a model for the K2O-FeO-MgO-Al2O3-SiO2-H2O-TiO2-O2 (KFMASHTO) system, as outlined in White et al. (2000, 2007, 2014), with KFMASH mixing parameters as originally calibrated by Powell and Holland (1999) and revised in Holland and Powell (2006). Tajčmanova et al. (2009 – T09) undertook a reformulation and reparameterization of this Bi(W)-model resulting in Bio(TCC). Both models are identical with regard to their KFMASH-parts, except that the enthalpy of the ordering reaction relating the biotite components annite (ann, KFe3[(OH)2AlSi3O10]), phlogopite (phl, KMg3[(OH)2AlSi3O10]) and ordered-Fe–Mg biotite (obi, KFeMg2[(OH)2AlSi3O10]), i.e., 2/3 phl + 1/3 ann = obi, was lowered by several kJ/mol in T09, affecting the octahedral Al-content predicted for KFMASH-biotite compared to Bi(W). Whereas in both models the ferric biotite end-member fbio is formulated identically, the recalibration of T09 is based on a differently defined Ti-biotite end-member tbio, with Ti replacing Mg on the M2 position and deprotonation of the hydroxyl site to achieve charge balance. Additionally, both models differ in their data extraction strategy and the underlying data set of ferric Fe- and Ti-contents of biotite from natural and experimental sources. Both models have been tested by Gervais et al. (2021), in their ability to reproduce the results of published partial melting experiments. According to their assessment, Bio(TCC) is the more favourable model and should be used in phase-equilibrium calculations.

Dachs and Benisek (2019, 2021a, b) recently presented a new biotite activity model for the KFMASH system. By applying an integrated approach combining experimental results from calorimetry, IR-spectroscopy and from phase-equilibrium studies with DFT calculations, they derived revised values for the thermodynamic standard state properties of the main biotite end-members ann, phl and eastonite (eas, KMg2Al[(OH)2Al2Si2O10]), as well as their mutual mixing behaviour. According to their analysis, near ideal activity-composition relationships are indicated for the phl-ann binary. Dachs et al. (2021b) constructed pseudosections for various KFMASH metapelites, where the performance of this model is compared to results obtained with Bi(W). They demonstrated that excess octahedral Al (AlVIex) is present in non-negligible amounts in natural metapelite biotites (Dachs et al. 2021b, Fig. 12), and emphasized the necessity to account for this by incorporating an adequate component into existing biotite activity models. Bi(W) and Bio(TCC) are not able to model AlVIex, because they are built of Tschermak substitution related biotite end-members only.

The aim of the present contribution is to

  • provide a new biotite activity model for use in phase-equilibrium calculations by extending the KFMASH-model of Dachs and Benisek (2021a) to a more general Fe3+- and Ti-bearing metapelite system (KFMASHTO). This is achieved by a) introducing a new di-octahedral biotite end-member that allows modelling of AlVIex (pyp), b) incorporating a Fe3+- and a Ti-biotite end-member into this model (see Table 1 for definition of end-members and their site populations). This includes to determine the thermodynamic standard state properties and to parameterize relevant mixing properties for these new end-members, by performing DFT calculations combined with the evaluation of experimental data,

    Table 1 Independent end-members and site distributions for KFMASHTO biotite. Note that the interlayer(A)-site in most endmembers is occupied by K (except pyp with a vacancy on A), and that the two T2-sites in all end-members are occupied by Si
    Full size table

  • perform test calculations with Perple_X demonstrating the performance of the new model, Bio(D), compared to using the published models Bi(W) and Bio(TCC).

Computational methods

Density functional theory (DFT)

Quantum–mechanical calculations were based on the DFT plane-wave pseudopotential approach implemented in the Castep code (Clark et al. 2005) included in the Materials Studio software from Biovia®. The calculations used the local density approximation (LDA) for the exchange–correlation functional (Ceperley and Alder 1980). To describe the core-valence interactions, ultrasoft pseudopotentials were used with the 1s1, 2s22p4, 2p63s2, 3s23p1, 3s23p2, 3s23p64s1, 3s23p63d24s2 and 3d64s2 electrons explicitly treated as valence electrons for H, O, Mg, Al, Si, K, Ti and Fe, respectively. The calculations on Fe-containing minerals used the LDA + U approach (Zhou et al. 2004), with U = 2.0 – 4.0 eV applied to the d orbitals of Fe. The k-point sampling used a Monkhorst–Pack grid (Monkhorst and Pack 1976) with a spacing of 0.02 Å−1 for the energy calculations. Convergence was tested by performing calculations using a denser k-point grid. The structural relaxation was calculated by applying the BFGS algorithm (Pfrommer et al. 1997), where the maximum force on the atom was within 0.01 eV/Å.

The enthalpy of mixing was simulated by the single defect method (Sluiter and Kawazoe 2002), which investigates supercells with almost end-member composition having only a single substitutional defect. The energy calculations of the end-members and such supercells provide the interaction parameters, because the results can easily be transformed into the slopes of the heat of mixing function (Li et al. 2014). The transformation of Castep energies (in eV) into enthalpies (in kJ/mol) was done as outlined in Benisek and Dachs (2018, 2020).

Phonon calculations with Castep, with methods exemplified in Benisek and Dachs (2020), were used to compute the heat-capacity functions and standard entropies So for tbio and fbio end-members.

Phase-equilibrium and pseudosection calculations

In order to extract ΔHof of tbio from experimental and in the case of fbio from natural data, self-written Mathematica programs were used to compute all required thermodynamic functions, based on biotite solution- and end-member properties given in Table 2 and Table 3. The standard state properties of all other phases, as well as their heat capacity, volume, thermal expansion and bulk modulus parameters were taken from Holland and Powell (2011) (Thermocalc file ‘tc-ds62.txt’). Solid solution properties, except that for biotite, were taken from White et al. (2014).

Table 2 Macroscopic mixing parameters W for KFMASHTO biotite (values in kJ/mol). Except for ann-phl, W’s are for symmetrical mixing. Mixing involving fbio is treated ideal (except mixing towards pyp with Wfbiopyp = 120 kJ/mol)
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Table 3 Standard state (1 bar, 298.15 K) thermodynamic properties of biotite end-members. Values for ann, phl and eas are from Dachs and Benisek (2021a), those for tbio and fbio are from this study (pyp data are not given but are thermodynamic dataset values from Perple_X-file hp62ver.dat). Cp = k0 + k1∙T −0.5 + k2∙T −2 + k3∙T −3 in J/(mol·K)
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The standard state data of biotite end-members were then implemented into the Perple_X data file ‘hp62ver.dat’, the solution properties as biotite activity model ‘Bio(D)’ into the Perple_X file ‘solution_model.dat’ (see supplementary Tables 1a and 1b). The other Perple_X solution models used were: Chl(W), Opx(W), Gt(W) Mica(W), St(W), Crd(W), Ilm(WPH), Mt(W), Sp(WPC), ‘feldspar’, melt(W), for chlorite, orthopyroxene, garnet, mica, staurolite, cordierite, ilmenite, magnetite, spinel, feldspar solid-solutions and melt, respectively.

Compositional data of biotite in various assemblages (like AlIV-, AlVI-, AlVIex-, Ti- contents and XFe), were computed with the Perple_X program Werami combined with self-written Mathematica® functions to extract mineral-chemical parameters of biotite for predefined values of P and T from data files generated with Werami. For comparison with results obtained by using Bio(D), calculations were repeated with the same set of solution models but with Bi(W) or Bio(TCC) as activity model for biotite. In order to simulate logfO2-conditions defined by oxygen-buffers like NNO (nickel—nickel oxide), QFM (quartz-fayalite-magnetite), CCO (cobalt-cobalt oxide) or CCH4 (graphite-methane), the file ‘NNO_PerpleX_fO2.xls’ was used (provided by St. Roozen and V. van Hinsberg, Montreal). By adding Ni and O2 to the list of components in a Perple_X calculation and changing the entropy of Ni to an appropriate value, as computed with this Excel sheet, logfO2-conditions, prevailing either during experimentation or during natural metamorphic events, could be simulated.

Pseudosections for the test-samples were also computed with Perple_X and the above-mentioned data files and solution models, raw versions are given in Online Resource 7. Isopleths of mineral-chemical parameters were generated with the program PyWerami (written by O. Lexa, Czech Geological Survey, see https://pypi.org/project/pywerami/).

Results

A new activity model for KFMASHTO biotite

The activity model of biotite presented in this study is an extension of the KFMASH model of Dachs and Benisek (2021a – DB21). The same logic as described there is applied here to extend it for the new components tbio, fbio and pyp (Table 1). As a simplification, disordered eastonite is no longer considered, as it was shown in DB21 that Mg–Al mixing on M2 is negligible at metamorphic P–T conditions.

Titanium-biotite end-member (tbio)

To calibrate the tbio end-member, we followed the strategy to first compute Cp and the standard-entropy, Sotbio, from Castep-calculations with methods exemplified in Benisek and Dachs (2020). Fitting the so obtained super-ambient Cp to a Berman-Brown-type polynomial (Berman and Brown 1985) gave:

$$C_{{\text{p}}}^{{{\text{tbio}}}} ({\text{J}}/\left( {{\text{mol}}\cdot{\text{K}}} \right))\, = \,{594}.{1} - {269}0 \cdot T^{{ - 0.{5}}} - {1}.{8837} \cdot {1}0^{{7}} \cdot T^{{ - {2}}} \, + \,{2}.{7692} \cdot {1}0^{{9}} \cdot T^{{ - {3}}}$$
(1)

By numerically integrating the Castep-computed Cp/T of tbio over the range 0 to 298.15 K, its So was constrained as Sotbio = 328.06 J/(mol·K) (Table 3).

Following Tajčmanova et al. (2009), we then extracted ΔHof values of the tbio end-member based on the equilibrium (mineral abbreviations not yet defined are: ilm = ilmenite, qtz = quartz, py = pyrope, alm = almandine):

$${\text{phl}}\, + \,{\text{3 eas}}\, + \,{\text{6 ilm}}\, + \,{\text{12 qtz}}\, = \,{\text{1 py}}\, + \,{\text{2 alm}}\, + \,{\text{6 tbio}}\, + \,{\text{6 H}}_{{2}} {\text{O}}$$
(2)

from experimental data of Patino-Douce et al. (1993 – PD93, see compilation in supplementary Table 2). The experimental data of Vielzeuf and Montel (1994 – VM94), and of Patino-Douce and Beard (1995 – PDB95), as well as the natural data of Williams and Grambing (1990 – WG90) and Holdaway et al. (1997 – H97) were also considered, but not used for extraction purposes for reasons discussed below.

In terms of chemical potentials μi, which are given by \({\mu }_{i}={\mu }_{i}^{o}+RTln{a}_{i}\), where μio is the chemical standard potential (i.e., the Gibbs free energy of pure i at P and T), R the gas constant and ai the activity of component i in a solid solution, it follows for equilibrium (2) that:

$${\mu }_{tbio}=\left(\frac{1}{6}\right)\left(3{\mu }_{phl}+3{\mu }_{eas}+6{\mu }_{ilm}+12{\mu }_{qtz}-1{\mu }_{py}-2{\mu }_{alm}-6{\mu }_{H2O}\right)$$
(3)

Writing \({\mu }_{tbio}^{o}={\mu }_{tbio}-RTln{a}_{tbio}\) at P and T in terms of standard thermodynamic properties, i.e. enthalpy of formation, \(\Delta {H}_{f,tbio}^{o}\), entropy, \({S}_{tbio}^{o}\), volume, Vtbio and heat capacity, Cp:

$${\mu }_{tbio}^{o}={G}_{tbio}^{P,T}={\Delta {H}_{f,tbio}^{o}+\underset{298.15}{\overset{T}{\int }}{C}_{P,tbio}dT}-T\left({S}_{tbio}^{o}+\underset{298.15}{\overset{T}{\int }}\frac{{C}_{P,tbio}}{T}dT\right)+\underset{1}{\overset{P}{\int }}{V}_{tbio}dP$$
(4)

and rearranging gives:

$${\mu }_{tbio}^{o}-\underset{298.15}{\overset{T}{\int }}{C}_{P,tbio}dT+T\left({S}_{tbio}^{o}+\underset{298.15}{\overset{T}{\int }}\frac{{C}_{P,tbio}}{T}dT\right)-\underset{1}{\overset{P}{\int }}{V}_{tbio}dP= \Delta {H}_{f,tbio}^{o}+{T(-S}_{tbio}^{o})$$
(5)

Equation (5) yields a straight-line relationship (Fig. 1), if its left-hand-side (LHS) is plotted against temperature with a slope of \({-S}_{tbio}^{o}\). By using the Castep-derived \({S}_{tbio}^{o}\), \(\Delta {H}_{f,tbio}^{o}\) can be directly extracted from available data using Eq. (5). Quartz and H2O were treated as pure phases and the experimentally measured compositions of garnet, ilmenite and biotite were used to compute the required activities of the garnet, ilmenite and biotite end-members (activity models used in Perple_X and Mathematica calculations were Grt(W) for garnet, Ilm(WPH) for ilmenite and Bio(D), the new biotite mixing model from this study with mixing parameters given in Table 2). The macroscopic W’s describing mixing between Ti and either Mg or Fe were constrained from Castep-calculations using the single-defect method and are Wphltbio =  ~ 0 kJ/mol, i.e., close to ideality, and Wanntbio = -30 kJ/mol. The less important W’s, namely Weastbio and Wobitbio were set to zero.

Fig. 1
figure 1

Left hand side (LHS)*10–6 of Eq. (5) plotted against temperature in (°C) showing experimental (T > 800 °C) and natural data (T < 650 °C) relevant for the tbio end-member. Magenta: Experimental data (with ilmenite as Fe-Ti oxide phase) of Patino Douce et al. (1993 – PD93) using a natural peraluminious metapelite bulk composition, blue: Vielzeuf and Montel (1994 – VM94) using a quartz-rich meta-greywacke bulk and brown: Patino Douce and Beard (1995 – PDB95) using a synthetic biotite gneiss bulk composition. Green: Natural metapelite data of Williams and Grambling (1990 – WG90) and red: Holdaway et al. (1997 – H97). The solid line was computed with the Castep-derived \({S}_{tbio}^{o}\) = 328.06 J/(mol·K) and \(\Delta {H}_{f,tbio}^{o}\) = −6131.11 kJ/mol, as extracted from PD93-experiments, broken line was computed using the same \({S}_{tbio}^{o}\) and \(\Delta {H}_{f,tbio}^{o}\) = −6149.44 kJ/mol, as would result from PDB95-experiments

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Using the experimental PD93 data, performed on a peraluminious metapelite bulk-composition, the derived ΔHof of tbio from Eq. (5) is -6131.11 ± 3.33 kJ/mol (Table 3). The PD93 data have been preferred over that from VM94 and PDB95, because a) they comprise the most comprehensive experimental data set, not only providing mineral-chemical analyses of biotite and garnet from the experimental charges, but also of ilmenite, b) least-square calculations showed that with the biotite end-members chosen herein (Table 1), the experimental biotite compositions reported in PD93 could be reproduced with much less residuals than was the case for VM94 or PDB95 biotite data, and c) the bulk composition used in PD93 experiments (HQ-36, supplementary Table 5) best matches that of natural metapelites, as it plots relatively close to the metapelite-average of Forshaw and Pattison (2021 – FP21). On the other hand and shown in Fig. 2a and b, the metagreywacke composition in VM94 experiments (sample CEVP) is much more Si-rich and the synthetic biotite gneiss in PDB95 experiments (sample SBG, crushed from a quartz-diorite, supplementary Table 5) lies off all bulk compositions reported for natural metapelites and greywackes with a markedly larger bulk-TiO2 content (Fig. 2).

Fig. 2
figure 2

a) Bulk-SiO2 vs. bulk-TiO2 b) bulk-Al2O3 vs. bulk-TiO2 (in wt.%) from the metapelite database of Forshaw and Pattison (2021 – magenta dots, n = 217, mean plotted as open square), compared to the bulks used in the experiments of Patino Douce et al. (1993, filled square), Vielzeuf and Montel (1994, filled diamond) and of Patino-Douce and Bird (1995, filled triangle). Black dots represent compositions of the five test samples (supplementary Table 5)

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Ferric-biotite end-member (fbio)

Phonon-calculations with Castep on the ferric-biotite end-member and the processing of these data following Benisek and Dachs (2020) and fitting to a Berman/Brown-type Cp-polynomial yielded:

$$C_{{\text{p}}}^{{{\text{fbio}}}} ({\text{J}}/\left( {{\text{mol}}\cdot{\text{K}}} \right))\, = \,{642}.0 - {3}0{34} \cdot T^{{ - 0.{5}}} - {1}.{5387} \cdot {1}0^{{7}} \cdot T^{{ - {2}}} \, + \,{1}.{839}0 \cdot {1}0^{{9}} \cdot T^{{ - {3}}}$$
(6)

The standard entropy of fbio was determined as Sofbio = 301.69 J/(mol·K) from the Castep phonon-calculations (Table 3) and its \(\Delta {H}_{f}^{o}\) ranging between -5888.6.0 kJ/mol and -5897.3 kJ/mol, for U = 2.5 (default value in Castep) and U = 4.0 eV, respectively. For comparison, the natural mineral-chemical data of biotite and garnet provided by Williams and Grambling (1990), who studied metapelitic rocks from Proterozoic rocks of northern New Mexico, were also used to extract \(\Delta {H}_{f,fbio}^{o}\). These authors also determined the ferric iron content in these minerals by Mössbauer spectroscopy, wet-chemical analysis or by the composition of coexisting oxides in their samples.

From the reaction (abbreviations not yet defined: sil = sillimanite, ky = kyanite, mt = magnetite):

$${\text{fbio}}\, + \,{\text{1 alm}}\, + \,{\text{2 sil}}/{\text{ky}}\, = \,{\text{6 eas}}\, + \,{\text{5 qtz}}\, + \,{\text{3 mt}}$$
(7)

\(\Delta {H}_{f,fbio}^{o}\) was then computed in a similar manner as described above for tbio (mt, qtz, sil and ky were treated as pure phases, and fbio interactions with the other biotite end-members assumed to be ideal). This yielded \(\Delta {H}_{f,fbio}^{o}\)= -5933.6 ± 6.6 kJ/mol (Table 3), around 40 kJ/mol different from the Castep-derived values, which is a deviation of ~ 0.7%. We decided to adopt the more negative \(\Delta {H}_{f}^{o}\) value in concurrence with ideal fbio interactions.

Pyrophyllite end-member (pyp)

It is well known that natural biotites from metapelitic rocks have octahedral Al-contents in excess of that balanced by the Tschmak substitution, i.e. by tetrahedral-Al minus 1 (see e.g. literature review in Guidotti 1984, or Fig. 12 in Dachs and Benisek 2021b). As noted by Guidotti (1984) and displayed in Fig. 3a, octahedral cations in biotite show an universal deficiency, in the order of 0.05 to 0.2 apfu (atoms per formula unit), to the ideal value of 3.0 apfu (for a formula based on 11 oxygens). The interlayer site also typically contains less cations than expected (1.0), with values around 0.9 apfu being most common (Fig. 3b).

Fig. 3
figure 3

a) Octahedral vacancies, VI□, in metapelite biotites from natural sources and from experimental studies on Ti-bearing natural biotite (points > 800 °C). Following Labotka (1983) VI□ was computed as: VI□ = 3 – (Mg + Fe + Fe3+  + VIAl + Ti). Natural sources: Green: Williams and Grambling (1990 – WG90), n = 25; red: Holdaway et al. (1997 – H97), n = 98; dark-blue: Henry and Guidotti (2002 – HG02), Main biotite data set (n = 562). Experimental sources: Magenta: Patino Douce et al. (1993 – PD93); blue: Vielzeuf and Montel (1994 – VM94); brown: Patino Douce and Beard (1995 – PDB95). b) Interlayer-vacancies, A□, in metapelitic biotites from natural sources and from experimental studies using Ti-bearing natural biotite (points > 800 °C). Following Labotka (1983) A□ was computed as: A□ = 1 + VIAl + Fe3+  + 2Ti – IVAl – 2VI□). Natural and experimental sources as in a)

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We thus decided to introduce a di-octahedral pyrophyllite component (pyp), [A]□[M1]□[M2]Al2[(OH)2[T1]Si2[T2]Si2] to account for this. The pyp component generates vacancies in the interlayer position, as well as vacancies on M1 and excess-Al on M2 based on the exchange:

$$^{{\left[ {\text{A}} \right]}} {\text{K}}^{ + } \, + \,^{{[{\text{M1}}]}} {\text{Mg}}^{{{2} + }} \, + \,{2}^{{[{\text{M2}}]}} {\text{Mg}}^{{{2} + }} \, + \,^{{[{\text{T1}}]}} {\text{Al}}^{{{3} + }} \, = \,^{{[{\text{A}}]}} \square \, + \,^{{[{\text{M1}}]}} \square \, + \,{2}^{{[{\text{M2}}]}} {\text{Al}}^{{{3} + }} \, + \,^{{[{\text{T1}}]}} {\text{Si}}^{{{4} + }}$$
(8)

which is obtained by subtracting the phl- from the pyp-composition. This exchange represents a solution of tri-octahedral biotite towards the di-octahedral sheet silicate pyp and is preferred over e.g. using a muscovite-component to model excess octahedral-Al in biotite.

Our Castep calculations using the single-point defect method indicate strongly positive interaction parameters over 100 kJ/mol for mixing of the main biotite components phl and ann towards pyp (Wphlpyp = 116.8 kJ/mol, Wannpyp = 108.2 kJ/mol). Based on these results all remaining W’s involving pyp were set to 120 kJ/mol (Table 2).

Microscopic biotite interaction parameters from Castep calculations

In the context of activity coefficient expressions for multi-site solid solution, it was shown by Powell and Holland (1993 – PH93) that so-called ‘macroscopic’ interaction parameters Wij, appearing in these expressions (Eq. (20) in HP93), describing symmetric mixing between two end-members ij in a solid solution (e.g., Wphlann), can be written as specific linear combinations of ‘microscopic’ same-site and cross-site interaction parameters w (eqs. (7) and (16) in PH93) that relate to mixing of elements on and between sites. This is the basis of the ‘micro-Φ’-approach of Powell et al. (2014), where micro-w’s are approximated from systems where good data exist. Using heuristic arguments, these are then transferred to mineral groups with little or no data and reassembled in order to allow parameterization of macroscopic W’s for these systems. As an example, wFeMg,oct, i.e., same-site mixing of Fe and Mg on an octahedral site, is taken to be 4 kJ per exchange based on experimental data for Fe–Mg olivine (Hackler and Wood 1989; Wiser and Wood 1991). Following this procedure and assuming cross-site terms to be negligible, macroscopic Wannphl would then be given by Wannphl = 3wFeMg(oct) = 12 kJ/mol in the case of biotite (e.g. Holland and Powell 2006), a value used in the biotite activity models Bi(W) and Bio(TCC).

Recently, Benisek and Dachs (2024 – BD24) applied DFT methods to compute enthalpic microscopic interaction parameters for petrologically important substitutions (Mg–Al, Si-Al, Mg-Ti, Mg-Ca and Mg-Fe) for a variety of mineral groups. As shown in this work, micro-w’s correlate with the ‘oxygen-packing fraction’ (OPF) of solid-solutions, whereby this parameter may be envisaged as a kind of density of oxygen in the unit-cell. OPF in turn correlates with the bulk-modulus (stiffness) of minerals. Micro-w’s are thus in principle not interchangeable between mineral groups and BD24 demonstrated that e.g. the micro-w for Mg–Al mixing in biotite is less than half that in diopside-cats pyroxene.

In the following, we present Castep-derived micro-w’s for KFMASH-biotite in more detail.

Phlogopite-annite join

Castep computations in this join, due to the presence of iron, were done in spin-polarized mode for Fe and are complicated by the fact that the so-calculated energies are dependent on the choice of the parameter U (2.5 eV by default). We thus performed several calculations for each required biotite composition using U values of 1.0, 2.0, 2.5 and 3.0 and the energies given in Table 4 are means of these values. The decomposition of macroscopic Wphlann into microscopic w’s is (supplementary Table 4):

$$W_{phlann} = w_{{MgFe\left( {M1} \right)}} + w_{{MgFe\left( {M2} \right)}} + w_{{MgMgFeFe\left( {M1M2} \right)}}$$
(9)
Table 4 Castep energies for various mineral phases of the ann-phl and the phl-eas joins, used to compute microscopic interaction parameters w’s for these binaries, as given in Tab. 5. Castep was run in the LDA non-conserving mode for the phl–eas join and in spin-polarized mode for the ann-phl binary using ultrasoft pseudopotentials. The values given for each composition are means of 4 calculations with U = 1, 2, 2.5 and 3
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In summary, the micro w’s relevant for Mg-Fe mixing in biotite are all close to ideality, i.e., they have only small negative values that are zero within error. The least dependence on U (and thus smallest uncertainty) shows wMgFe(M1) with a value of -0.1 ± 0.8 kJ/mol. Larger uncertainties are indicated for wMgFe(M2) = -0.4 ± 3.6 kJ/mol and wMgMgFeFe(M1M2) = -1.9 ± 2.7 kJ/mol. The reassembled Wphlann is -2.4 ± 5.0 kJ/mol, in reasonable agreement with the Fe–Mg mixing behaviour derived by Dachs and Benisek (2021a) from experimental data, i.e. weak negative deviation from ideality for Fe-rich and weak positive deviation for Mg-rich biotites, as modelled by the two Margules parameters Wphlann = 14.3 kJ/mol and Wannphl = -8.8 kJ/mol (Table 2). From the decomposition of macro-W’s, it then follows that Wphlobi = wMgFe(M1) and Wannobi = wMgFe(M2).

Phlogopite-eastonite join

For macroscopic Wphleas, the decomposition into corresponding micro-w’s is (e.g., Powell et al. 2014; Benisek and Dachs 2024; see supplementary Table 4):

$$W_{phleas} = \left( \frac{1}{4} \right)\left( {4w_{{MgAl\left( {M1} \right)}} + w_{{SiAl\left( {T1} \right)}} - 2w_{{MgAlAlSi\left( {M1T1} \right)}} } \right)$$
(10)

The cross-site term \({w}_{MgAlAlSi(M1T1)}\), describing interactions of Mg, Al and Si between M1 and T1 sites in biotite represents the energy of the reciprocal reaction (e.g., Powell 1977, 1983; Powell et al. 2014; Wood and Nichols 1978; Chatterjee 1991):

$${\text{Mg}}^{{{\text{M1}}}} {\text{Al}}^{{{\text{T1}}}} \, + \,{\text{Al}}^{{{\text{M1}}}} {\text{Si}}^{{{\text{T1}}}} \, = \,{\text{Mg}}^{{{\text{M1}}}} {\text{Si}}^{{{\text{T1}}}} \, + \,{\text{Al}}^{{{\text{M1}}}} {\text{Al}}^{{{\text{T1}}}}$$
(11)

or written in terms of end-member energies:

$$w_{{MgAlAlSi\left( {M1T1} \right)}} = \varepsilon_{{\left( {phl} \right)}} + \varepsilon_{{\left( {east} \right)}} - \varepsilon_{{Al^{T1} \left( {phl} \right)}} - \varepsilon_{{Al^{M1} \left( {phl} \right)}}$$
(12)

Based on the Castep-computed energies (Table 4), cross-site \({w}_{MgAlAlSi(M1T1)}=\) 175.1 kJ/mol. With same-site wMgAl(M1), = 82.5 kJ/mol and wSiAl(T1) = 95.6 kJ/mol from Benisek and Dachs (2024), the recombination of same-site and cross-site micro-w’s according to Eq. (10) then gives a macroscopic Wphleas = 18.8 kJ/mol (Table 5), in good agreement with that resulting from calorimetry (Circone and Navrotsky 1992) or line-broadening in IR spectra (Dachs and Benisek 2019).

Table 5 Microscopic interaction parameters w of the biotite binaries ann-phl and phl-eas, calculated from Castep energies listed in Table 4 (the values given apply for 1 M1 site, 2 M2 and 2 T1 sites). A much more comprehensive set of micro-w’s for various substitutions in petrologically relevant mineral groups is given in Benisek and Dachs (2024). The linear combinations of micro-w’s to obtain macro-W’s (given in bold face), i.e., Wphleas and Wannphl, are also listed (the full set of micro-w’s for the biotite activity model of this study can be found in supplementary Table 4)
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Discussion

Titanium-biotite and ferric-iron biotite end-members (tbio and fbio)

The Castep-derived Sotbio = 328.06 J/(mol·K) yields a slope of reaction (2) that is consistent with the experimental, as well as natural data, i.e., the experimental data of PD93 measured at high temperatures on a peraluminous pelite bulk composition (Fig. 1, dots in magenta) line up with the natural metapelite data of WG90 and H97 that represent metamorphic temperatures between ~ 500 and ~ 650 °C (Fig. 1, dots in green and red). The reasons for preferring the PD93 over the VM94 and PDB95 experimental data to extract \(\Delta {H}_{f,tbio}^{o}\) have been discussed before. If the latter data were used, \(\Delta {H}_{f,tbio}^{o}\) would amount to -6149.44 ± 5.95 kJ/mol, compared to \(\Delta {H}_{f,tbio}^{o}\)= -6131.11 ± 3.33 kJ/mol kJ/mol, as resulting from the preferred PD93 experiments. Caution is nevertheless indicated when applying Bio(D) to high-Ti systems, where the use of the former value may be more appropriate.

To compute \(\Delta {H}_{f,tbio}^{o}\) with Castep, we used four different structural configurations, two representing local-charge balance (LCB) in the tbio end-member and two where the LCB criterion was released, i.e. allowing Ti also to reside on sites more distant to Mg than in the strict LCB situation – probably a more realistic scenario. The resulting \(\Delta {H}_{f,tbio}^{o}\) = −6129 kJ/mol is in good agreement with the value extracted from PD93 experiments.

Figure 4 shows the above discussed experimental data and a bunch of natural biotite data in a plot of Ti apfu vs. temperature compared to calculated Ti contents for bulk HQ-36 at P = 7kbar (PD93 experiments) using Perple_X. With the biotite activity model Bio(D) and thermodynamic properties of biotite end-members from this study (Tables 2 and 3), biotite is stable in various assemblages between ca. 500 °C and 850 °C. At the low-temperature end Ti-contents in biotite are around 0.1 apfu in accordance with the gros of natural data, increasing steadily to values of ~ 0.2 apfu as present in PD93 experiments performed in the range 800 to 850 °C (Fig. 4, line in magenta). A similar behaviour is obtained using Bi(W) (Fig. 4, green line), whereas Bio(TCC) predicts systematically by 0.05–0.08 apfu larger Ti-contents (Fig. 4, dark-yellow line).

Fig. 4
figure 4

Ti-contents of biotite in metapelites (atoms per formula unit – apfu based on 11 oxygens) as function of temperature from experimental studies (points > 800 °C) and natural sources, compared to Perple_X-computed values. Experimental sources: Magenta: Patino Douce et al. (1993 – PD93); blue: Vielzeuf and Montel (1994 – VM94); brown: Patino Douce and Beard (1995 – PDB95). Natural sources: Green: Williams and Grambling (1990 – WG90), n = 25; red: Holdaway et al. (1997 – H97), n = 98; dark-blue: Henry and Guidotti (2002 – HG02), Main biotite data set (n = 562), black: Forshaw and Pattison (2021 – FP21), n = 452. The following temperatures were arbitrarily assigned to their metamorphic zone classification: grt: 490 °C, crd/st/and/ky: 585 °C, sil: 655 °C, kfs: 720 °C. Therefore their data appear vertically aligned. Perple_X-computed Ti-contents of biotite using the activity model Bio(D) and thermodynamic properties of biotite end-members from this study for peraluminous metapelite bulk-composition HQ-36, as used in the experiments of PD93, is shown as thick line in magenta (P = 7 kbar). For comparison, similar calculations with Bi(W) and Bio(TCC) and thermodynamic dataset values for biotite are also shown (green and brown thick lines). Black convex line is the biotite Ti-saturation surface according to Henry et al. (2005)

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In case of the fbio end-member, a direct test of calculated vs. naturally observed Fe3+ content in biotite is not possible, because no bulk-rock compositions were given by WG90 for their samples, but just mineral-chemical data. Instead, we computed the Fe3+ content in biotite for PD93 experiments with Perple_X, implying redox conditions of the QFM buffer, fO2 conditions that supposedly prevailed during their experiments. The values obtained indicate low Fe3+ contents around 0.02 apfu, compared to the estimated values of PD93 that range between 0 and 0.07 apfu Fe3+ (supplementary Table 2).

Pyrophyllite end-member (pyp): Predicted vs. observed AlIV-, AlVI- and excess-AlVI-contents in biotite

In Fig. 5, the observed AlIV-, AlVI- and excess-AlVI-contents of over 1000 natural biotites are plotted as function of temperature using the Main biotite data set (Henry and Guidotti 2002), and the Forshaw and Pattison (2021) biotite data set. The former were collected from peraluminous metapelites that equilibrated between 4 and 6 kbar with a bulk composition probably similar or close to that as used by PD93 in their experiments (sample HQ-36, supplementary Table 5). The Main biotite data can thus serve as test to what extent Perple_X-computed biotite compositions match the natural record. Such AlIV-, AlVI- and excess-AlVI-contents are shown as function of temperature for P = 7 kbar (experimental pressure in PD93 experiments) as brown, dark-blue and red lines in Fig. 5. Three curves appear for AlIV and AlVI, one computed with the biotite activity model Bio(D) from this study, one with Bi(W) and one with Bio(TCC).

Fig. 5
figure 5

AlIV -, AlVI- and AlVIex-contents of biotite in metapelites (apfu based on 11 oxygens) as function of temperature from natural and experimental sources, compared to Perple_X-computed contents for the peraluminous metapelite bulk-composition HQ-36. Data < 800 °C are from natural sources: AlIV-, AlVI- and AlVIex-contents from Main biotites (Henry and Guidotti 2002, – HG02, n = 562) are shown as brown, dark-blue and red points, corresponding values from the Forshaw and Pattison (2021 – FP21) biotite data set (n = 452) as light-green, light-blue and magenta points. The points > 800 °C are the experimental data of Patino Douce et al. (1993 – PD93). Perple_X-computed contents are drawn as thick lines, in brown for the AlIV-, in dark-blue for the AlVI- and in red for the AlVIex-contents of biotite at P = 7 kbar. Biotite activity models used were Bio(D), in combination with thermodynamic properties of biotite end-members from this study, Bi(W) and Bio(TCC) with thermodynamic dataset values for biotite end-members. Note that for AlVIex only one curve, computed with Bio(D), exists (red)

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With Bio(D), AlVI contents of biotite are predicted that match Main-biotite and experimental PD93 compositions over the whole temperature range of 500 to 900 °C.

Computed AlIV- and AlexVI– contents agree with PD93 values, but tend to yield overestimated values in the case of AlIV, and too low values in the case of AlexVI at low temperatures (Fig. 5). Test calculations show that this could be improved by making some of the W’s related to mixing towards pyp temperature dependent, assigning them decreasing values with falling temperature.

Perple_X computations with Bi(W) yield AlIV-contents of biotite in good agreement with natural observations, whereas AlVI is grossly underestimated at T < 700 °C with this model.

Perple_X calculations with Bio(TCC), on the other hand give a too large AlIV, but a correct AlVI–T behaviour with AlVI values close to that obtained using Bio(D).

Predicted vs. observed XFe in biotite

For bulk compositions as used in the Fe–Mg exchange experiments between garnet and biotite of Ferry and Spear (1978 – FS78) and Gessmann et al. (1997 – G97) (supplementary Table 5), recalculated XFe in biotite, derived from Perple_X calculations using Bio(D) and thermodynamic properties of biotite end-members from this study (Tables 2 and 3), is in good agreement with experimental FS78-XFe and somewhat larger (by ca. 0.05 mol%) for G97-XFe (data are compiled in supplementary Table 3, note that XFe includes both ferrous and ferric iron). This is shown in Fig. 6 (open squares) and was already noted by Dachs and Benisek (2021b).

Fig. 6
figure 6

XFe = Fe/(Fe + Mg) in biotite from Fe–Mg exchange experiments between garnet and biotite of Ferry and Spear (1978 – FS78) and of Gessmann et al. (1997 – G97). Experimentally obtained values (XFeexp) are compared to Perple_X-calculated ones (XFecalc) using the biotite activity model Bio(D) and thermodynamic properties of biotite end-members from this study (open squares), using Bi(W) – filled squares, or using Bio(TCC) – filled triangles. Fe includes Fe2+ and Fe3+. Bulk compositions are given in supplementary Table 5. Error bars represent ± 2 σ

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Similar calculations with Bi(W) and Bio(TCC) yield a mostly ca. 0.1 too large XFe in biotite in equilibrium with garnet. Bulk XFe in these experiments was 0.9 (FS78) and 0.78 (G97), respectively.

For bulk-XFe around 0.5, a value more typical for common metapelites, all three activity models give a rather similar XFe vs. T behaviour for biotite at temperature below ~ 800 °C (Fig. 7). For the PD93 experiments, recalculated XFe is a few mol% lower with Bio(D), to a larger extent lower with Bi(W), and in good agreement with experimental values using Bio(TCC) as activity model (data are compiled in supplementary Table 2).

Fig. 7
figure 7

XFe = Fe/(Fe + Mg) in biotite from the experiments of Patino Douce et al. (1993 – PD93) – data > 800 °C, and from natural sources: dark-blue: Henry and Guidotti (2002 – HG02), Main biotite data set (n = 562), light-blue: Forshaw and Pattison (2021 – FP21), n = 452. Perple_X-computed XFe of biotite using the biotite activity model Bio(D) and thermodynamic properties of biotite end-members from this study is shown as thick line in magenta (P = 7 kbar, bulk-composition HQ-36). For comparison, similar calculations with Bi(W) and Bio(TCC) and thermodynamic dataset values for biotite are also shown (blue and brown thick lines)

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Significance of the Castep-computed micro-w’s

Our results obtained on Fe–Mg same-site and cross-site interactions (Table 5) indicate that these are weak, confirming that Fe–Mg mixing in biotite is ideal or nearly so, as already suggested in the petrological literature several times since Müller (1972). As demonstrated by BD24, this applies not only for biotite, but for brucite, pyroxene, olivine, spinel, garnet and perovskite as well, suggesting that ideal Fe–Mg mixing could possibly be more the rule than the exception in many solid-solutions. This would have the impact that activity models could be considerably simplified. We tested this for biotite by reevaluating, under the assumption of ideal Fe–Mg mixing (leaving other W’s the same), the experimental partitioning data of Fe and Mg between biotite and olivine of Zhou (1994), as used by Dachs and Benisek (2021a) to extract \(\Delta {H}_{f,ann}^{o}\). = -5131.6 ± 2.3 kJ/mol along with weakly asymmetric Fe–Mg non-ideality (Tables 2 and 3). This gives a by 2 kJ different value of \(\Delta {H}_{f,ann}^{o}\). = -5133.6 ± 2.5 kJ/mol. The corresponding model is termed ‘Bio(Did)’ and, in addition to Bio(D), Bi(W) and Bio(TCC), was also applied to the garnet/biotite exchange experimental conditions (supplementary Table 3) and to one test sample in the chapter below. Comparing calculated vs. measured biotite compositions, it turns out that the performance of such an ideal Fe–Mg biotite mixing model is almost as good as that of Bio(D).

As emphasized by BD24, there is good agreement between macroscopic W of the phl-eas join computed from reassembling Castep-derived micro-w’s (18.8 kJ/mol), compared to macroscopic Wphleas computed independently from applying the single-defect-method to that join (19 ± 3 kJ/mol), as well as resulting from calorimetry (Circone and Navrotsky 1992) and from line-broadening in IR spectra (Dachs and Benisek 2019) (19 – 24.5 kJ/mol). This places some confidence that the calculated micro w’s are significant, even when charged cells are used. DFT methods thus seem capable to yield reasonable micro-w’s of solid solutions. This opens the possibility towards creating a new generation of activity models, that use DFT- and thus physically based micro-w’s and the accordingly reassembled macro-W’s for petrological calculations. The biotite activity model of this study – Bio(D) – is a first example in this respect. Due to the large number of required interaction parameters in the model, Castep calculations in the present study were, however, confined to the most important binaries involving phl and ann and some assumptions still were made, e.g. concerning pyp- and tbio-mixing with the other biotite end-members.

Test of the new biotite model

Five natural samples of metapelites and metagreywackes, from low- to high grade metamorphic conditions, were chosen to test the new biotite activity model. Their bulk-compositions are given in supplementary Table 5 and are plotted in Figs. 2a and b, showing that they cover SiO2-, Al2O3- and TiO2 contents of ~ 52 to 76 wt.%, ~ 11 to 28 wt.% and ca. 0.5 to 1.3 wt.%, respectively. Mineral-chemical data of relevant phases and assemblages, along with published P–T estimates, are compiled in supplementary Table 6. Recalculated mineral compositions using Perple_X also appear in this table for each sample, allowing a direct comparison of computed results obtained with Bio(D), Bi(W) and Bio(TCC) with observed biotite compositions (all other activity models were identical). Perple_X-generated raw-pseudosections for all samples are given in the Online Resource 7, only that for X567 appears in the text. Note that a Mn end-member has also been added to the Bio(D) model for general application. Similar to Bio(TCC), ideal mixing of all biotite end-members with this Mn-end-member is assumed.

Low-grade chlorite-biotite zone samples 16 and 18

Mather (1970) studied the biotite isograd in lower greenschist facies metapelites and -greywackes of the Dalradian geological unit in Scotland and provided mineral-chemical, as well as bulk-rock chemical data of these rocks. Metapelite sample 16 is located close to the biotite isograd, still in the chlorite zone, whereas sample 18 is a higher grade SiO2-rich metagreywacke from the biotite zone (Mather 1970, their Fig. 1). Both samples contain biotite + chlorite (in major amounts in sample 16, in minor amounts in sample 18) together with phengite, plagioclase, quartz and calcite. As shown by Mather (1970, Table 3), KD = (Fe/Mg)Chl/(Fe/Mg)Bio decreases with metamorphic grade. Perple_X computed pseudosections using Bio(D) for samples 16 and 18 are shown in Online Resource 7 (Figs. S1a, b) and reveal the observed assemblage biotite-chlorite-phengite-albite-quartz-calcite at low temperatures with the additional phases sphene and epidote in small amounts < 3 vol.%.

KD as function of P and T is plotted in Fig. 8 for metapelite sample 16, allowing comparison of KD-contours calculated with Bio(D), Bi(W) and Bio(TCC). Bio(D)-computed contours indicate decreasing KD with temperature (Fig. 8, blue). This is in agreement with the field observation reported in Mather (1970) and could be taken as an indication that biotite and chlorite in the natural low-grade samples approached equilibrium compositions. The contour matching the observed KD = 0.89, is marked in bold and would indicate a temperature of 350 °C at P = 4 kbar for this sample. On the other hand, KD’s that are significantly lower than the observed one for sample 16 result from using Bi(W) and Bio(TCC), respectively.

Fig. 8
figure 8

Perple_X-generated iso-KD contours for sample 16 (Mather 1970). KD = (Fe/Mg)Chl/(Fe/Mg)Bio describes the Fe–Mg distribution between chlorite and biotite, contours in blue were calculated with Bio(D), in red with Bi(W) and in brown with Bio(TCC). Thick blue iso-KD line represents KD = 0.89, as measured in the natural sample

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Mineral-chemical parameters of biotite, chlorite and phengite at 350 °C/4 kbar are given in supplementary Table 6. Bio(D)-calculated AlVI is with 0.26 apfu close to the observed value of 0.31 apfu, while Bi(W)- and Bio(TCC)-predicted AlVI-contents are far too low (around 0.05 apfu). All three biotite activity models predict a too low Ti-content (around 0.04 vs. 0.12 apfu). This indicates that the tbio end-members in these three models, which only differ in the octahedral site-assignement for Ti, but similarily assume deprotonation of the hydroxyl site to achieve charge balance, a scenario that seems to be more relevant at high metamorphic grades (e.g., Cesare et al. 2003), don’t work so well at low temperatures. Computed XFe2+ in biotite and chlorite amount to 0.58 and 0.55 using Bio(D), compared to 0.62 and 0.60 as in sample 16. Due to the comparable XFe2+-difference (0.02–0.03), KD’s are nevertheless similar. Analogous XFe2+-values calculated with Bi(W) and Bio(TCC) are 0.63 and 0.43, and 0.60 and 0.52, respectively, resulting in predicted KD’s that are lower than observed.

Similar mineral-chemical features as described above for sample 16, apply for metagreywacke sample 18 concerning its AlVI- and Ti content in biotite (supplementary Table 6). The naturally observed KD = (Fe/Mg)Chl/(Fe/Mg)Bio in this sample is 0.88 and chlorite vanishes from the biotite-chlorite-phengite-albite-quartz-calcite assemblage above ca. 410 °C at 4 kbar according to the pseudosection (Fig. S1b in Online Resource 7). At these conditions and using Bio(D), computed KD (0.94) somewhat overestimates observed KD, whilst with values of 0.48 and 0.75 computed KD is grossly and somewhat underestimated when Perple_X calculations are done with Bi(W) and Bio(TCC), respectively.

Medium-grade andalusite-staurolite zone sample 980A

Sample 980A stems from the staurolite-andalusite zone of the Silurian Waterville Formation in south-central Maine and contains the assemblage garnet-phengite-biotite-staurolite-andalusite-plagioclase-quartz (Ferry 1980). Classical thermobarometry performed by Ferry (1980) revealed pressures around 3.5 kbar for this area and temperatures ranging between 474 and 554 °C for 980A. Tinkham et al. (2001, their Fig. 12) constructed a MnNCKFMASH P–T pseudosection for this sample and discussed the sequence of predicted and observed assemblages for 980A (Tinkham et al. 2001, their Table 2b). A staurolite + andalusite bearing paragenesis does not appear in their pseudosection, but staurolite is confined to a garnet-biotite-staurolite-muscovite assemblage at ca. 4–5.5 kbar. As can be seen from our recalculated MnNCKFMASHTO-pseudosections for 980A (Fig. S2a-c in Online Resource 7), the observed sequence of assemblages, irrespective which activity model is used for biotite, is correctly predicted, i.e. there is a biotite + chlorite-, followed by a biotite + chlorite + garnet-field at low temperatures (P = 3.5 kbar) passing into a staurolite- and finally an andalusite-bearing field with a small zone of stable biotite + garnet + staurolite + andalusite (+ phengite + plagioclase + ilmenite + quartz) in between. The reason for this better match between predicted and observed assemblages compared to Tinkham et al. (2001) is probably due to the inclusion of TiO2 in our pseudosection calculation and/or improved thermodynamic data since the work of these authors. One notable difference in the topology of the Bio(D)- compared to the Bi(W)- or Bio(TCC)-calculated pseudosections for sample 980A is that Bio(D) predicts chlorite to react out before staurolite comes in (Fig. S2a in Online Resource 7). It remains to be tested, if this applies for metapelite rock compositions in general.

The mineral chemistry of phases constituting the 980A-assemblage (biotite, staurolite, garnet, plagioclase) is relatively well reproduced by all three biotite activity models (supplementary Table 6). The amount of AlVI in biotite calculated with Bio(D) or Bio(TCC) matches best the observed value, while it’s too low resulting from Bi(W). The Ti-content in biotite from 980A (0.09 apfu) is slightly overestimated with Bio(D) (0.12 apfu) and underestimated with Bi(W) and Bio(TCC) (0.08 and 0.07 apfu). XFe2+ in biotite and staurolite, as well as plagioclase and garnet compositions are passably predicted by all three models, with the exception of the spessartine content in garnet, which is generally overestimated. The deviation between observed and calculated XFe2+ for biotite and staurolite is 0.04 for Bio(D). Bi(W) and Bio(TCC) predict a twice this value lower XFe2+ in staurolite.

High-grade sample X567

Pitra and de Waal (2001) studied high-grade metapelites from the Marble Hall Fragment of the Bushveld Complex. They inferred two intrusion related metamorphic episodes: (A) an early-stage paragenesis of chiastolitic andalusite-cordierite-biotite-quartz ± garnet equilibrated at 550–600 °C and ca. 2kbar. In sample X567, this early paragenesis transformed to (B) the peak-event assemblage cordierite-biotite-kalifeldspar-quartz ± garnet that constitutes the matrix, whereas a symplectitic intergrowth of cordierite + spinel has replaced the andalusite porphyroblasts. From their phase diagram analysis, based on pseudosections constructed for the simplified KFMASH system, Pitra and de Waal (2001) argue for temperatures of 720–760 °C, conditions of reduced water activity for this second high-grade metamorphic event and a prograde nature for the formation of the cordierite + spinel symplectites.

Similar KFMASH pseudosections for X567 have been calculated by Tajčmanova et al. (2009) and mor recently by Dachs et al. (2021b) using a former version of Bio(D) that was confined to this simplified system.

With the extended Bio(D) model of this study the sequence of parageneses documented for X567 can be consistently reproduced and the predicted biotite composition agrees well with the observed one. This is shown in the pseudosections of Fig. 9, calculated for X567 using Bio(D). Figure 9a is a P–T pseudosection for water-saturated conditions, Fig. 9b is a T-log(aH2O) pseudosection for P = 2 kbar. At this pressure and H2O in excess, the early-stage (A) paragenesis andalusite-cordierite-biotite-quartz(-plagioclase-ilmenite) ± garnet is stable up to temperatures around 660 °C, grading into the peak-event (B) assemblage cordierite-biotite-kalifeldspar-quartz ± garnet extending up to ca. 710 °C, where the wet solidus is reached. Computed mineral compositions using Bio(D) in this field closely match observed ones at 690 °C/2 kbar (supplementary Table 6). As discussed by Pitra and de Waal (2001), a more realistic scenario for the metamorphic evolution of X567 are, however, water undersaturated conditions (Fig. 9b). The transition from the early-stage to the peak-stage paragenesis is only slightly shifted to lower T’s with decreasing water activity. The observed AlVI-content of 0.48 apfu in biotite is correctly predicted, the corresponding isopleth runs through the (B) stability field. The Ti-content remains nearly constant throughout this field and agrees with the measured one (0.19 apfu in X567). For a reduced water activity of aH2O = 0.7 (log(aH2O) = -0.16), the peak-paragenesis would indicate ~ 665 °C at P = 2 kbar, based on the AlVI-content of 0.48 apfu in biotite. The calculated XFe2+  = 0.77 is also close to its measured value of 0.78 (for further mineral-chemical parameters at these conditions, compared to measured ones see supplementary Table 6). Comparable conditions, resulting from the KSMASH system, have been discussed by Dachs et al. (2021b) as ‘best match’ conditions, i.e., where measured and computed compositions come nearest. With the extension of the former KFMASH biotite activity model they used to the present Bio(D) model, this match is still improved (supplementary Table 6).

Fig. 9
figure 9

a) P–T pseudosection for water-saturated conditions, b) log(aH2O)-T pseudosection for reduced water-activity conditions at 2 kbar for sample X567 (Pitra and de Waal 2001; bulk-rock composition is given in supplementary Table 5). The observed sequence of parageneses is consistent with the succession of phase-fields at reduced aH2O around 0.7 (i.e., log(aH2O) around -0.16). Calculations were done with Perple_X using Bio(D) of this study as activity model for biotite (Tables 2 and 3) and imposing logfO2-conditions of the QFM buffer. Note that no attempt has been made to compute supra-solidus equilibria in a) for a realistic fixed bulk-H2O content. Coloured/dark-grey isopleths are the AlVI-, respectively the Ti content in biotite, XFe in biotite and in cordierite are shown in separate colour density plots. Mineral abbreviations for sub-solidus phase-fields are: and andalusite, Bio biotite, Crd cordierite, Grt garnet, Ilm ilmenite, Pl plagioclase, qz quartz, Kf kalifeldspar, sil sillimanite, Sp spinel (field-numbers are explained in Fig. S3 of Online Resource 7)

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With rising temperature, cordierite + spinel bearing assemblages, without any melt involvement, become stable (above ca. 750 °C for aH2O = 0.7), confirming Pitra and de Waal (2001)’s opinion that these cordierite + spinel symplectites could have a prograde character.

When pseudosection calculations for X567 are repeated with Bi(W) (Fig. S4 in Online Resource 7), the stability field topology and thus succession of parageneses, as discussed above, remain quite similar, as well as roughly the isopleths slopes. Absolute computed values of the AlVI- and Ti contents in biotite are, however, too small to achieve a match to measured analogous quantities in sample X567 (see supplementary Table 6 for comparison of other mineral-chemical parameters). Interestingly, AlVI, as predicted for the peak-event (B) assemblage by Pitra and de Waal (2001), using older thermodynamic data and activity models, is in better agreement with observation (their Fig. 5c, showing AlVI -isochores between 0.3 and 0.5).

With the use of Bio(TCC), the stability field of the peak-event (B) assemblage shrinks to pressured < 2 kbar at water-saturated conditions (Fig. S5 in Online Resource 7). At a reduced water activity of aH2O = 0.7 (log(aH2O) = −0.16), the AlVI-isochore representing AlVI in the natural sample (0.48) would indicate a temperature of ca. 730 °C, around 60 °C higher than resulting from Bio(D). The Ti content and XFe2+ in biotite are, however, somewhat larger than measured in sample X567. For the formation of the cordierite + spinel symplectites, temperatures close to 800 °C would be required.

(Ultra-)High-pressure sample 16Slo12

Li et al. (2020) derived a polymetamorphic P–T history for metapelite 16Slo12 from the (ultra-)high pressure terrane of the Pohorje Mountains in the Eastern Alps (Slovenia) applying various petrological methods, including pseudosection calculations with Perple_X, combined with monazite age dating. The P–T conditions of the earliest Permian metamorphic event, succeeded by an Eoalpine and a Tertiary metamorphic episode, were constrained as 7.5 – 10 kbar at 600–650 °C based on the stability field of the inclusion assemblage bio + stau + ru appearing in porphyroblastic garnet of this rock.

We have repeated Li et al. (2020)’s pseudosection calculation using the bulk-rock composition of 16Slo12 (Fig. S6 in Online Resource 7). All three biotite activity models give rather similar stability fields for this inclusion assemblage consistent with that computed by these authors (pink field in their Fig. 10). Calculated mineral compositions are compared in supplementary Table 6 for pressures of 7 and 9 kbar at T = 630 °C. With a value of 0.52 apfu, compared to 0.54 apfu (measured), octahedral Al is best predicted by Bio(D) at 7 kbar, followed by Bio(TCC), whereas Bi(W) yields a low AlVI of 0.27 apfu. Computed XFe2+ is generally larger in the order of 0.1 – 0.2 than measured XFe2+ in bio of this inclusion assemblage. This discrepancy may be caused, at least partly, by adjustment of the Fe/Mg ratios in biotite, staurolite and garnet to changing P–T conditions during the subsequent Alpine metamorphic (high-pressure) overprints.

Conclusions

Due to the inclusion of the new biotite components tbio and fbio to the former KFMASH model of Dachs and Benisek (2021a), the biotite activity model presented in this study (named Bio(D) in Perple_X), is more generally applicable, because the Ti- and Fe3+ contents of natural biotites can be taken into account. Furthermore, it is the first model that allows the prediction of excess octahedral Al, as present in almost all natural biotites, through incorporation of a pyp component.

DFT-based phonon-calculations were applied to derive the heat capacity functions and standard entropies of tbio and fbio (calorimetric measurements could not be used for that purpose, because it is not possible to synthesize these end-members in a pure form). \(\Delta {H}_{f,tbio}^{o}\) was then extracted from experimental data and is in good agreement with its DFT-calculated analogue. The only thermodynamic quantity that was derived from natural data is \(\Delta {H}_{f,fbio}^{o}\). Generally we think that the use of estimated P–T data combined with measured mineral-chemistries from the ‘natural lab’ for extracting thermodynamic (mixing) quantities should be kept to a minimum, because such a procedure represents a classical ‘circulus vitiosus’. The derived parameters cannot be expected to have a physical relevance but may merely represent fitting quantities.

As shown recently by Benisek and Dachs (2024) and again demonstrated herein, DFT methods seem capable of quantifying microscopic interaction parameters (micro-w’s). For Mg–Al mixing in biotite these are: same-site wMgAl(M1) = 82.5 kJ/mol, same-site wSiAl(T1) = 95.6 kJ/mol, cross-site \({w}_{MgAlAlSi(M1T1)}=\) 175.1 kJ/mol. The recombination of these micro-w’s according to Eq. (10) then gives a macroscopic Wphleas = 18.8 kJ/mol, as used in the Perple_X calculations of this study. Three lines of evidence confirm such a Wphleas close to 20 kJ/mol and the underlying micro w’s: a) results from calorimetry (22.8 ± 18.7 kJ/mol, Circone and Navrotsky 1992), (b) line-broadening in IR spectra of members from the phl-eas binary (25.4 kJ/mol, Dachs and Benisek 2019), c) independent DFT-calculations applying the single-defect method to the phl-eas join (19.0 ± 3.0 kJ/mol, Benisek and Dachs 2024). Due to the dependence of Hmix on the oxygen packing fraction, interaction parameters are not interchangeable between mineral groups, as demonstrated by Benisek and Dachs (2024).

Micro-w’s relevant for Mg-Fe mixing in biotite (wMgFe(M1), wMgFe(M2), \({w}_{MgMgFeFe(M1M2)}\)), have only small negative values that are zero within error.

The following general implications can be drawn for the use of Bio(D) in a pseudosection calculation: (i) Bio(D) gives the best match between observed and calculated AlVI compared to Bi(W) or Bio(TCC) (Fig. 5, supplementary Table 6); (ii) XFe(2+) in biotite is most accurately predicted with Bio(D) for Fe-rich bulk-compositions as used in Fe–Mg exchange experiments (Fig. 6), at more intermediate bulk Fe/Mg ratios all three models yield rather similar XFe2+, deviating not more than 0.05 mol fractions from measured values in the test samples (supplementary Table 6); (iii) the Ti-contents in biotite is underestimated by all three models in the low-grade test samples. In the higher grade samples a reasonable match is observed for calculations with Bio(D) (deviation ≤ 0.03 apfu), whereas Bi(W) tends to slightly under- and Bio(TCC) to somewhat overestimate measured Ti contents (supplementary Table 6, Fig. 4); (iv) for high_Ti systems, Bio(D), as well as Bi(W) are likely to underestimate Ti-contents in biotite; (v) the suggested published successions of parageneses as determined from petrographic studies of the test samples can be consistently interpreted based on the phase-field topologies resulting from Bio(D). These phase-relations do not differ significantly from that as would be obtained by using Bi(W). With Bio(TCC), some phase-equilibria are shifted to maximal 20 °C higher temperatures or ~ 0.7 kbar lower pressures.

Further extension of the Bio(D) model should consider the introduction of two more end-members, taking into account the Na- and F-content in natural biotites.

The biotite activity model of this study may considered a first example of next-generation activity models that utilize DFT methods combined with experimental phase-equilibrium data to derive unknown thermodynamic standard state properties of end-members and their mixing behaviour and that do not longer resort on heuristic assumptions in parameterizing their interaction parameters. Benisek and Dachs (2024) presented Castep-derived micro-w’s for the Mg–Al, Si-Al, Mg-Ti, Mg-Ca and Mg-Fe substitutions. By extending their work to a comprehensive set of relevant exchanges and taking the dependence of micro-w’s on the type of mineral-group, i.e., on the mineral-specific oxygen packing fraction into account, it should be possible in the future to formulate activity models for petrological use that are based on a consistent set of micro-w’s and reassembled macro-W’s.