Abstract
A new activity model for Fe–Mg–Al biotites is formulated, which extends that of Mg–Al biotites (Dachs and Benisek, Contrib Mineral Petrol 174:76, 2019) to the K2O–FeO–MgO–Al2O3–SiO2–H2O (KFMASH) system. It has the two composition variables XMg = Mg/(Mg + Fe2+) and octahedral Al, and Fe–Mg and Mg–Al ordering variables resulting in five linearly independent endmembers: annite (Ann, K[Fe]M1[Fe]2M2[Al0.5Si0.5]2T1[Si]2T2O10(OH)2, phlogopite (Phl, K[Mg]M1[Mg]2M2[Al0.5Si0.5]2T1[Si]2T2O10(OH)2, ordered Fe–Mg biotite (Obi, K[Fe]M1[Mg]2M2[Al0.5Si0.5]2T1[Si]2T2O10(OH)2, ordered eastonite (Eas, K[Al]M1[Mg]2M2[Al]2T1[Si]2T2O10(OH)2, and disordered eastonite (Easd, K[Al1/3Mg2/3]M1[Al1/3Mg2/3]2M2[Al]2T1[Si]2T2O10(OH)2. The methods applied to parameterize the mixing properties of the model were: calorimetry, analysis of existing phase-equilibrium data, line-broadening in powder absorption infrared (IR) spectra, and density functional theory (DFT) calculations. For the calorimetric study, various biotite compositions along the annite–phlogopite, annite–siderophyllite (Sid, K[Al]M1[Fe]2M2[Al]2T1[Si]2T2O10(OH)2), and annite–eastonite joins were synthesized hydrothermally at 700 °C, 4 kbar and logfO2 of around − 20.2, close to the redox conditions of the wüstite–magnetite oxygen buffer at that P–T conditions. The samples were characterised by X-ray powder diffraction (XRPD), energy-dispersive scanning electron microprobe analysis, powder absorption IR spectroscopy, and optical microscopy. The samples were studied further using relaxation calorimetry to measure their heat capacities (Cp) at temperatures from 2 to 300 K. The measured Cp/T was then integrated to get the calorimetric (vibrational) entropies of the samples at 298.15 K. These show linear behaviour when plotted as a function of composition for all three binaries. Excess entropies of mixing are thus zero for the important biotite joins. Excess volumes of mixing are also zero within error for the three binaries Phl-Ann, Ann-Sid, and Ann-Eas. KFMASH biotite, therefore, has excess enthalpies which are independent of pressure and temperature (WGij = WHij). A least-squares procedure was applied in the thermodynamic analysis of published experimental data on the Fe–Mg exchange between biotite and olivine, combined with phase-equilibrium data for phlogopite + quartz stability and experimental data for the Al-saturation level of biotite in the assemblage biotite–sillimanite–sanidine–quartz–H2O to constrain enthalpic mixing parameters and to derive enthalpy of formation values for biotite endmembers. For Fe–Mg mixing in biotite, the most important binary, this gave best-fit asymmetric Margules enthalpy parameters of WHAnnPhl = 14.3 ± 3.4 kJ/mol and WHPhlAnn = −8.8 ± 8.0 kJ/mol (3-cation basis). The resulting asymmetric molar excess Gibbs free energy (Gex) departs only slightly from ideality and is negative at Fe-rich and positive at Mg-rich compositions. Near-ideal activity–composition relationships are thus indicated for the Ann–Phl binary. The presently used low value of − 2 kJ/mol for the enthalpy change of the reaction 2/3 Phl + 1/3 Ann = Obi is generally confirmed by DFT calculations that gave − 2 ± 3 kJ/mol for this ∆HFe–Mg order, indicating that Fe–Mg ordering in biotite is weak. The large enthalpy change of ∆HMg-Al disorder = 34.5 kJ/mol for the disordering of Mg and Al on the M sites in Eas (Dachs and Benisek 2019) is reconfirmed by additional DFT calculations. In combination with WHPhlEas = 10 kJ/mol, which is the preferred value of this study describing mixing along the Phl–Eas join, Mg–Al disordering over the M sites of biotite is predicted to be only significant at high temperatures > 1000 °C. In contrast, it plays no role in metamorphic P–T settings.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Introduction
Numerous papers appeared in the literature dealing with the mixing properties of biotite. For the sake of brevity, all this work is not discussed here in detail, but a chronological list of relevant papers is given in supplementary Table S1. In general, either natural Fe–Mg distribution coefficients (KD) between biotite and another Fe–Mg silicate (mostly garnet), or experimentally determined KDs were used to constrain biotite’s Fe–Mg mixing properties. To extract these properties from empirical and/or experimental calibrations, the mixing properties of the exchange partner of biotite were required. This is a major problem in all these attempts, as the results become dependent on the choice of solution model and the adopted size of mixing parameters of the exchange partner. As a consequence, all types of Fe–Mg mixing behaviour have been proposed, from ideal (Müller 1972; Wones 1972; Ferry and Spear 1978; Schulien 1980; Perchuk and Lavrent’eva 1983; Indares and Martignole 1985; McMullin et al. 1991; Hoisch 1991) to strongly negative (Wones and Eugster 1965) to strongly positive deviation from ideality (Holdaway et al. 1997). Several efforts have also been made to take the effect of AlVI, Fe3+, and Ti in biotite into account (Indares and Martignole 1985; Kleemann and Reinhardt 1994; White et al. 2000, 2014a, 2007; Tajčmanová et al. 2009). The symmetric Margules parameter describing Fe–Mg mixing in biotite in the most recent works was approximated as WGAnnPhl = 12 kJ/mol (3-cation basis) giving a moderate positive deviation from ideality (Holland and Powell 2006; White et al. 2014a). The reasoning behind this is the assumption that Fe–Mg mixing in biotite resembles that in olivine, where it is relatively well established from experimental phase-equilibrium, as well as calorimetric work, that the Margules parameter for symmetrical mixing along the fayalite (Fa)–forsterite (Fo) join, WGFaFo, is around 4 kJ/mol per octahedral site (Davidson and Mukhopadhyay 1984; Wiser and Wood 1991; Kojitani and Akaogi 1994; Berman and Aranovich 1996). On the microscopic level, the underlying hypothesis (wFeMg,one-oct-site)Fe–Mg–silicate = (wFeMg,one-oct-site)olivine = 4 kJ/mol results from the heuristic approach of Powell et al. (2014), used to describe Fe–Mg mixing, including Fe–Mg ordering (see below), in various classes of silicates in cases where experimental data are insufficient or missing, e.g., (White et al. 2014a, b).
With regard to mixing towards the AlVI-bearing endmembers Eas and Sid (Table 1), the only safe ground to stand upon concerning enthalpic mixing behaviour is the solution calorimetric data of Circone and Navrotsky (1992). These indicate a positive ∆Hex for the Phl–Eas join and values of 22.8 ± 18.7 kJ/mol for the enthalpic Margules parameter WHPhlEas. Dachs and Benisek (2019) studied synthetic members of this join with relaxation calorimetry and XRPD. They found ideal entropic and volumetric mixing behaviour, determined revised thermodynamic standard state data for Phl and Eas, and derived an enthalpy change ΔHMg-Al disorder = 34.5 ± 3 kJ/mol for the disordering of Al on the M sites in ordered Eas from DFT calculations.
In the KFMASH system, Berman et al. (2007) provided reversed-phase-equilibrium constraints on the stability of Mg–Fe–Al biotite by experimentally determining the Al saturation level of biotite in the assemblage biotite–sillimanite–sanidine–quartz–H2O. The ∆Hof,Ann value, derived from that experiments, is around 15 kJ/mol more negative than the one extracted by Dachs and Benisek (2015) from (redox-)equilibria in the FASH system (∆Hof,Ann = − 5132.5 ± 2.0 kJ/mol), which is based on the calorimetrically measured entropy of annite of So = 422.9 ± 2.9 J/(mol·K). The thermodynamic analysis of Berman et al. (2007) resulted in symmetric biotite mixing properties with moderate positive deviation from ideality for Fe–Mg and negative deviation for Fe-Al mixing (WGAnnSid = − 8.2 kJ/mol). An even more negative WGAnnSid was proposed earlier by Benisek et al. (1999) for the Ann–Sid join (WGAnnSid = − 29 kJ/mol). These assertions of negative WG’s for the Ann–Sid join conflict with positive ones in the order of 7 kJ/mol as used in recent biotite activity models (Holland and Powell 2006; White et al. 2014a).
Noting that equipartitioning of Fe and Mg on octahedral sites of biotite, as assumed in all earlier work prior to ca. the year 2000, leads to inconsistencies with regard to the formulation of ideal activities, Fe–Mg ordering was introduced based on the ordering reaction 2/3 Phl + 1/3 Ann = Obi (Powell and Holland 1999). By assuming various values for the order parameter Q ≡ XFeM1 – XFeM2 (where XFeM1, XFeM2 are the site fractions of Fe on the M1 or M2 site of biotite, respectively), the enthalpy change, ∆HFe–Mg ordering, of this Fe–Mg ordering reaction was initially constrained to a value of − 10.75 kJ/mol (Powell and Holland 1999) producing a rather pronounced Fe–Mg ordering. It was later changed to − 2 kJ/mol (Holland and Powell 2006), thereby reducing the degree of Fe–Mg ordering, and approximated as − 6.8 kJ/mol by Tajčmanova et al. (2009) to improve the match of model-predicted octahedral Al contents in biotite (AlVI) with natural observations.
The aim of the present contribution is:
-
1.
To determine, using well-characterised synthetic samples, entropic and volumetric mixing properties of the important biotite binaries Ann–Phl, Ann–Sid, and Ann–Eas via relaxation calorimetry and XRPD,
-
2.
to derive enthalpic mixing properties of biotite binaries from the experimental data of Zhou (1994) and Berman et al. (2007), in combination with the evaluation of line broadening in powder absorption IR spectra,
-
3.
to constrain values for the enthalpy change of the ordering reaction 2/3 Phl + 1/3 Ann = Obi from DFT calculations.
The impact of this biotite activity model on phase relations in the KFMASH system is presented in the companion paper.
Methods
Sample synthesis and characterisation
Samples along the Ann–Phl and Ann–Eas binaries were synthesized from gels in a conventional cold-seal hydrothermal apparatus at a temperature of 700 °C and a pressure of 4 kbar. Run durations were about 3 weeks. Details of the hydrothermal apparatus, the applied gel method, and chemicals used in the gel preparation can be found in Dachs (1994) and will not be repeated here.
Redox conditions during synthesis were at a logfO2 of around − 20.2 as measured by hydrogen sensors, close to that of the wüstite–magnetite oxygen buffer at these P–T conditions. Using Mössbauer spectroscopy, Dachs and Benisek (2015) demonstrated that such conditions are sufficiently reducing to keep the Fe3+ content of annite at its well-known minimum of 10 ± 2% (Hazen and Wones 1972; Redhammer et al. 1993). For the synthetic Ann–Phl biotites, we thus corrected our mineral chemistries by taking 10% of Fetot as Fe3+ at the Fe-rich side of the binary and decreasing it linearly to 0% towards Phl (Redhammer et al. 1995). For simplicity, we assume that this amount of Fe3+ is incorporated by a vacancy (□) mechanism, i.e., 3Fe2+(oct) = 2 Fe3+(oct) + □(oct), e.g., (Redhammer et al. 1993). This requires a ferric iron-rich vacancy (□) endmember, K[□]M1[Fe3+]2M2[Al0.5Si0.5]2T1[Si]2T2O10(OH)2 besides Ann and Phl as the third component to compositionally describe our synthetic Fe–Mg biotites. The samples calorimetrically studied will be subsequently referred to as, e.g., Ann50Phl50 for K(Fe1.5Mg1.5)[AlSi3O10(OH)2].
Synthesis products were examined by optical microscopy and by XRPD using a Bruker D8 Advance X-ray diffractometer to check phase purity. Their chemical compositions were determined using an energy-dispersive scanning electron microscope (type: Zeiss Ultra Plus 55 equipped with an Oxford Instrument 50 mm2 SDD EDX detector). Powders were pressed into pellets for this purpose.
Lattice constants were calculated from XRPD patterns collected on a Siemend D-500 diffractometer between 5° and 110° 2θ (Cu-Kα radiation) using the software UnitCell (Holland and Redfern 1997). The lattice constants determined in this way were validated by checking them against a Rietveld refinement, which was done using the software Fullprof (Rodriguez-Carvajal 2001) of the sample Phl60Eas40.
Powder absorption IR spectra were recorded on a Bruker IFS66v/S spectrometer in the wave number region 175–1300 cm−1 using polyethylene and KBr pellets for the low and high wavenumber regions, respectively, to investigate the line broadening as a result of forming the biotite solid solutions (Boffa Ballaran et al. 1999; Salje et al. 2000; Boffa Ballaran 2003). For that purpose, selected absorption peaks were subject to an autocorrelation analysis, using a self-written Mathematica program, from which ∆corr and δ∆corr values were obtained, which are the parameters describing the broadening of the powder absorption IR spectra of the solid solutions relative to the powder absorption IR spectra of the endmembers. The latter were then transformed into excess enthalpies of mixing, ∆Hex, based on the empirical mixing enthalpy—IR line-broadening correlation derived by Etzel and Benisek (2008): ΔHex∆corr = (d + k·normVexcint)·δΔcorr·n, where n = number of atoms per formula unit, d = 13.0/10.5 J·cm·mol−1 for the middle/high-wave number regions and k = 308 J·cm−2 and normVexcint is the integrated excess volume of mixing normalised to one atom per formula unit. Etzel and Benisek (2008) established their correlation between δΔcorr and ΔHmix using data from six binaries. In the meantime, their approach has been applied to other binaries, as well, i.e., ternary feldspars (Benisek et al. 2010), low structural state plagioclases (Benisek et al. 2013), grossular–spessartine binary (Dachs et al. 2014a), almandine–spessartine binary (Dachs et al. 2014b), and the Mg–Al biotite solid solution (Dachs and Benisek, 2019). As shown by Tarantino et al. (2003), the overlap of δΔcorr and calorimetrically measured ΔHmix for the olivine solid solution is remarkable. This correlation was thought to give wrong results for the glaucophane–tremolite solid solution (Jenkins et al. 2014). However, as shown by Benisek and Dachs (2020), if this method is correctly applied, the results agree with the derived ΔHmix values of Jenkins et al (2014). The method using δΔcorr to extract ΔHmix has, however, problems for some binaries, as different IR wave number regions yield different ΔHmix values. However, if the appropriate wave number region is selected, which is indicative of local heterogeneities caused by the substitution and where peak shifts are not a dominant feature (Tarantino et al. 2003), it seems to be a promising tool for determining the heat of mixing or ordering.
Calorimetric methods and data evaluation
Low-temperature heat capacities were measured using a commercially available relaxation calorimeter (the heat capacity option of the Quantum Design® Physical properties measurement system—PPMS). The data were collected in triplicate at 60 different temperatures between 2 and 300 K, using a logarithmic spacing resulting in a higher data density at a lower temperature. The samples consisted of 12–15 mg of crystallites wrapped in thin Al foil and compressed to a ~ 0.5 mm-thick pellet that was then attached to the sample platform of the calorimeter with Apiezon N-grease, to facilitate the required thermal contact.
Further details on the calorimetric technique and measuring procedures have already been described several times and will not be repeated here (Lashley et al. 2003; Dachs and Bertoldi 2005; Kennedy et al. 2007; Dachs and Benisek 2011). All calorimetric data are given in Supplementary Table S2.
The calorimetric (vibrational) molar entropy (Scal) of each compound at 298.15 K was calculated by solving the integral:
Scal corresponds to the standard state (third-law) entropy, So, in the case of an ordered endmember (which assumes ST=0 K = 0). Errors in Scal were estimated according to Dachs and Benisek (2011). The entropy increment 0–2 K, not covered by measured Cp data, is negligible for Mg-rich samples, because absolute Cp values are so small, so that this increment affects Scal only at the second decimal place. The lowermost Cp values of Fe-rich samples from the range 2 – 5 K, on the other hand, were fitted to the equation Cp = a·T3 + b·T5, from which the entropy increment 0–2 K was computed and added to Scal. It amounted to 0.2 J/(mol·K) at maximum.
Computational methods
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, 3s23pPhl, 3s23p2, 3s23p64s1, and 3d64s2 electrons explicitly treated as valence electrons for H, O, Mg, Al, Si, K, and Fe, respectively. The calculations on Fe-containing minerals used the LDA + U approach (Zhou et al. 2004), with U = 2.0–3.0 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 endmember composition having only a single substitutional defect. The energy calculations of the endmembers 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 simulations concerning the enthalpies of ordering/disordering used the same computational methods as described in Dachs and Benisek (2019). The transformation of CASTEP energies (in eV) into enthalpies (in kJ/mol) was done as outlined in Benisek and Dachs (2018, 2020).
Results
A new activity model for Fe–Mg–Al biotites
In the following, we present a new activity model for biotite that is an extension of the model of Dachs and Benisek (2019) for Mg–Al biotite by introducing the Fe-bearing endmembers Ann and ordered Fe–Mg biotite (Obi). This model for the KFMASH system considers 5 linearly independent endmembers, as shown in Table 1, and includes Fe–Mg and Mg–Al order–disorder.
Using a Margules-type expression for its molar excess Gibbs free energy, Gex, as originally formulated by Wohl (1946, 1953) and expanded to multicomponent solutions by Jackson (1989), the resulting Gex expression for the new biotite solution model is:
where each interaction parameter, \(W_{ij}^{G}\), is given by:
The pi’s are the molar proportions of the endmembers and the Qijk ‘s are defined as:
(e.g., Jackson 1989, his Eq. 3). The Cijks are the ternary constants (set to zero due to the absence of appropriate data to calibrate them). Note that each biotite binary is treated as a symmetric solution (i.e., Wij = Wji), except the Phl–Ann join with the two interaction parameters WGPhlAnn and WGAnnPhl (the reason for that is discussed below).
Based on Eq. (2), the activity coefficient, γi, of, e.g., the Phl component is then given by (Jackson 1989, his Eq. 7):
The RTlnγ expressions for the other biotite endmembers can be found by appropriate rotations of subscripts, so for RTlnγEasd: Phl → Easd, Easd → Eas, Eas → Ann, Ann → Obi, Obi → Phl; for RTlnγEas: Phl → Eas, Easd → Ann, Eas → Obi, Ann → Phl, Obi → Easd; for RTlnγAnn: Phl → Ann, Easd → Obi, Eas → Phl, Ann → Easd, Obi → Eas; for RTlnγObi: Phl → Obi, Easd → Phl, Eas → Easd, Ann → Eas, Obi → Ann.
Ideal mixing on site activities, aiid, of the biotite endmembers are:
The site fractions, Xij, expressed in terms of proportions of the five linearly independent endmembers are given by:
The disordering of Mg and Al on the M sites of Eas is modelled via the equilibrium (Dachs and Benisek 2019):
These authors derived an enthalpy of disordering of ΔH(eq. 8) = 34.5 ± 3 kJ/mol from DFT calculations.
To describe Fe–Mg ordering in biotite, we follow Powell and Holland (1999), Holland and Powell (2006) using the internal reaction:
Later, modifications and extensions of this KFMASH biotite activity model (White et al. 2000, 2007, 2014a; Tajčmanová et al. 2009; Powell et al. 2014) are all formulated on the basis of reaction (9) to model Fe–Mg order–disorder. To get a value for the corresponding enthalpy, ΔH(eq. 9), we computed CASTEP energies for Ann, Phl, and Obi. From these energies (Table 2), ΔH(eq. 9)was then calculated by simply subtracting the energy of a mechanical mixture of 2/3 Phl + 1/3 Ann from that of Obi. The CASTEP results for the two Fe-bearing endmembers depend on the value of U in LDA + U calculations. For its default value of 2.5 eV in CASTEP, we obtain ΔH(eq. 9) = − 2.1 kJ/mol. To see the effect of changing U on the resulting ΔH(eq. 9), we have repeated the above computations with values of 2.0 and 3.0 eV for U (Table 2). The dependence of ΔH(eq. 9) on predefined U is moderate, giving a range of reasonable ΔH(eq. 9) values of + 2.5 to -5.2 kJ/mol for U’s varying between 2 and 3 eV. In all further computations, we use a value in the middle of this range, i.e., ΔH(eq. 9) = − 2.0 kJ/mol.
Following Tajčmanova et al. (2009), (Appendix), the pis, representing the stable speciation, can be written as function of the bulk composition parameters by defining proportions pi0 for the fully disordered state and taking into account the stoichiometric constraints imposed by ordering reactions (8) and (9):
The unknown quantities δpEas and δpObi are fractional amounts of the ordered endmembers Eas and Obi, by which the proportions pi0 for the fully disordered state are lowered as result of ordering. The pi0s are related to the bulk parameters XMg = Mg/(Mg + Fe2+) and AlVI = octahedral Al via:
The equilibrium degree of ordering at given P and T is found by solving ΔG = 0 of Eqs. (8) and (9) simultaneously for the two unknowns δpEas and δpObi:
ΔS(eq. 8) equals the configurational entropy of Easd = Rln(27/4) J/(mol·K) and ΔS(eq. 9) = 0 J/(mol·K).
Activities of biotite endmembers are then given by: ai = aiidγi.
Calibrating the mixing parameters and extracting the standard enthalpy of formation values of biotite endmembers
Excess entropies and volumes of mixing (∆S ex, ∆V ex) of biotite binaries
Annite–phlogopite
Atoms per formula unit (apfu) of the synthesized Ann–Phl solid solutions are given in Table 3. They were computed from the average of at least 5, but mostly 10–15 separate microprobe analyses. The synthetic Fe–Mg biotites appeared as fine-grained aggregates of thin (< 1 μ), often pseudohexagonal platelets not exceeding 10 μ in diameter, with a light-greenish colour that gets more intensive towards Fe-rich compositions. The optical examination and XRPD patterns showed impurities of fayalite and sanidine of < 5% in only the Fe-rich biotites. Formula units of Si in all Fe–Mg biotites are 3.0 apfu within error, except in Fe-rich ones, where some Tschermak-substituted Al (and possibly also Fe3+) are present (see footnotes to Table 3). This is reflected by the number of endmembers needed to describe the composition of our synthetic Fe–Mg biotites: Ann, Phl, and small amounts of the Vac end member (decreasing from 10% in Ann90Phl10 to 0% in Phl100) are sufficient for biotites with 0.3 < XMg < 1. The additional Tschermak-type endmembers Sid and Eas in the order of total 6%, 7%, and 14 mol % are required for describing the nominal compositions of Ann80Phl20, Ann90Phl10, and Ann100.
The heat capacities of all Fe–Mg biotites synthesized in this study, as well as those of Ann (Dachs and Benisek 2015) and Phl (Dachs and Benisek 2019), are shown in Fig. 1 in the temperature range 0–320 K. A Cp anomaly, similar to that in annite and interpreted to result from a magnetic phase transition (Dachs and Benisek 2015), can be observed in Ann90Phl10 and less pronounced in Ann80Phl20 as a kink in the slope of the Cp vs. T data at around 50 K. For biotites richer in Mg, this kink vanishes. Integrating the heat capacity data according to Eq. (1), the calorimetric entropy at 298.15 K, Scal, is plotted in Fig. 2 for the Ann–Phl binary as function of XMg (data from Tables 3 and 4). This figure also includes Scal values that have been determined from PPMS Cp measurements made on Ann–Phl samples synthesized and characterised by Redhammer et al. (2005) (Table 4). Combined with Scal = 411.4 ± 2.9 J/(mol·K) for Ann (Dachs and Benisek 2015) and Scal = 319.4 ± 2.2 J/(mol·K) for Phl (Dachs and Benisek 2019), the Scal vs. XMg data all fall on a line connecting the endmember entropies within error, which indicates ideal entropic mixing behaviour for the join. There are thus no significant excess entropies of mixing discernable and WSAnnPhl = 0 [the term excess quantity (vibrational entropy, enthalpy, or volume) for a given binary is the difference between the measured quantity for a specific solid solution composition minus the linear combination of the endmembers representing ideal mixing, and would be given by Xi·Xj·Wij for the simplest case of binary symmetrical mixing between the two endmembers i-j]. The magnetic entropy contribution to Scal for the Ann endmember amounts to 1.5 J/(mol·K) (Dachs and Benisek 2015). For Ann90Phl10 and Ann80Phl20, these contributions are even smaller and cease to exist in more Mg-rich biotites, but no attempt was made to quantify them.
Volumes along the Ann–Phl binary were determined by Hewitt and Wones (1975) and by Redhammer et al. (1995) on synthetic biotites, and Berman et al. (2007) contributed a volume of an intermediate synthetic Ann50Phl50 biotite. Plotting these literature data and that obtained in this study (Tables 3, 4) as a function of composition shows a linear decrease of volumes with XMg, with only minor scatter in the range XMg > 0.4 (Fig. 3). Redhammer et al. (1995) have also demonstrated that the molar volume of a specific solid solution member and of Ann100 shrink as a function of the Fe3+ content, as measured by Mössbauer spectroscopy. Extrapolating these data to Fe3+ = 0 yielded the values V = 15.403(7) J/(bar·mol) for Ann80Phl20, 15.288(18) J/(bar·mol) for Ann60Phl40, 15.181(4) J/(bar·mol) for Ann40Phl60, and 15.081(3) J/(bar·mol) for Ann20Phl80 (Fig. 3, filled squares). Dachs and Benisek (2015) derived a molar volume of 15.53 J/(bar·mol) for ‘ideal’ annite in a similar manner by extrapolating volume data from annites with Fe3+ contents between the minimum values of ~ 10 up to 30%. As these data have a considerable scatter, the extrapolated value of VoAnn is not well established. We prefer in this work to use a linear extrapolation of the volume data of the Ann-Phl join taking into account only volumes of Fe–Mg biotites richer in Mg than XMg 0.4. These biotites have only minor amounts of Fe3+ and the extrapolation to the Fe endmember yields 15.48 J/(bar·mol) for Vo of ‘ideal’ annite. Combined with a Vo = 14.958(3) J/(bar·mol) for Phl (Dachs and Benisek 2019), the existing volume data along the Ann-Phl join can be consistently described as an ideal solution, i.e., excess volumes of mixing are zero for this binary.
Tschermak-substitution related binaries: Phlogopite–eastonite, annite–siderophyllite, and annite – eastonite.
Dachs and Benisek (2019) studied the Phl–Eas join calorimetrically and found that there are no vibrational excess entropies of mixing (i.e., WSPhlEas = 0).
The samples calorimetrically studied herein along the Ann–Sid join were those synthesized by Benisek et al. (1996) and Redhammer et al. (2000). The characterisation of these samples via electron microprobe, XRPD, and optical microscopy can be found in these papers. Scal behaviour shows a linear decrease of Scal with increasing AlVI content (Fig. 4). Therefore, no significant excess entropies of mixing occur on the Ann-Sid binary. Siderophyllite does not exist as an endmember physically, the most AlVI-rich solid solution member has AlVI = 0.75 (Table 5). Linear extrapolation of the Scal-AlVI data then yields a standard entropy of SoSid = 356.0 ± 3.1 J/(mol·K). CASTEP-computed SoSid, on the other hand, is somewhat lower, namely SoSid = 348.6 J/(mol·K) for ordered Sid (Al on M1) and 350.6 J/mol·K for a siderophyllite with Al on M2. From the DSC-measured Cp polynomials of members of the Ann–Sid join (Benisek et al. 1999), we derived the following polynomial for endmember Sid (Cp in J/(mol·K)):
This was done by extrapolating heat capacities calculated from the Cp polynomials of Ann (Dachs and Benisek 2015) and of Sid21 and Sid76 (Benisek et al. 1999, Table 2) as a function of AlVI to endmember Sid composition and refitting these data to Eq. (14). Cp polynomials of intermediate Ann–Sid biotites were not used, because these are affected by excess heat capacities (Benisek et al. 1999).
The volume–composition relationship for the Ann–Sid binary was discussed by Benisek et al. (1996), and it was shown that there is no indication of excess volumes of mixing on this join (i.e., WVAnnSid = 0). The molar volume of Sid was determined as VoSid = 15.06 ± 0.02 J/bar.
The Ann–Eas join is a combination of Fe–Mg and Tschermak exchanges and may be written as:
or in vector notation Mg2VIAlVIAlIVFe-3VISi-1IV. We have synthesized four members of this join (compositional data are given in Table 6) and measured their Scal. Plotting these data vs. 2 Mg + Altot (= AlVI + AlIV), rescaled to range from 0 to 1 in Fig. 4, shows again a linear Scal–composition relationship, i.e., all data plot on a line drawn between So of Ann (411.4 ± 2.9 J/(mol·K), Dachs and Benisek 2015) and So of Eas (294.5 ± 3.0 J/(mol·K) (Dachs and Benisek 2019). Excess entropies of mixing are thus zero and WSAnnEas = 0.
The molar volume–composition behaviour for the synthesized Ann–Eas biotites is shown in Fig. 5 (see Table 7 for structural data). Taking Vo of Ann from above and of Eas from Dachs and Benisek (2019), a picture mirroring that of ∆Hex of the Ann–Phl join emerges, i.e., ∆Vex is slightly positive at Mg (and AlVI)-rich and slightly negative at Fe-rich (and AlVI-poor) compositions. If the ∆Vex data were fitted to an asymmetric Margules model, which can be written for a binary solution 1–2 as (e.g., Cemič 2005):
WVAnnEas = 0.44 ± 0.07 J/bar and WVEasAnn = − 0.36 ± 0.07 J/bar (ρ = − 0.95) would result. Within a 2σ error, however, the modelled volume–composition behaviour deviates only a little bit from ideality (Fig. 5). For now, we prefer to treat the Ann–Eas binary as volumetrically ideal and a more detailed study including more solid solution members of this join would be required to assess if indeed the anticipated slight deviation from ideality really exists.
Excess enthalpies of mixing (∆H ex) of biotite binaries and ∆H o f values of biotite endmembers
Published phase-equilibrium data for the reactions:
were used, in combination with some approximations discussed below, to constrain the WHij parameters and to derive ∆Hof values of biotite endmembers in a single-step least-squares procedure. In this treatment, all required thermodynamic data were taken from Holland and Powell 2011; updated data set tc-ds62.txt) except for biotite endmembers: SoPhl = 330.9 ± 2.2 J/(mol·K), Cp,Phl, SoEas = 294.5 ± 3.0 J/(mol·K) and Cp,Eas from Dachs and Benisek (2019), and SoAnn = 422.9 ± 2.9 J/(mol·K) and Cp,Ann from Dachs and Benisek (2015). A weighting of the experimental data was applied, based on experimental uncertainties and bracket widths (see supplementary materials for more details).
The experimental data processed were (1) the brackets obtained by Bohlen et al. (1983) Clemens (1995), Aranovich and Newton (1998) and Berman et al. (2007) on reaction (17), either in a pure H2O fluid or in fluids with reduced H2O activity (H2O–CO2 or H2O–KCl fluids), (2) the reversals of equilibria (18) and (19) determined by Berman et al. (2007), defining the Al-saturation level of biotite in the assemblage (Fe–Mg-Al)biotite–sillimanite–sanidine–quartz under the presence of water and (3) the experimental Fe–Mg partitioning data between biotite and olivine of Zhou (1994). The reasons for not including experimental garnet–biotite and orthopyroxene–biotite Fe–Mg exchange data are discussed below.
The set of experimental data for reactions (17) and (18) is the same as used by Dachs and Benisek (2019) in their study of Mg–Al biotites.
With regard to reaction (20), the thermodynamic evaluation of Zhou’s partitioning data requires a value for the interaction parameter for symmetrical Fe–Mg mixing in olivine (WGFaFo). As outlined in the discussion, this parameter is relatively well known and we use a value of WGFaFo = 5 kJ/mol (one-site mixing). Information on the AlVI content of Zhou’s biotites comes from the fact that he stated that the two endmembers Phl and a Fe3+-bearing ‘annite’ K(Fe2+2.7Fe3+0.3)[AlSi3O10.3(OH)1.7] were sufficient to describe the compositional variation of his biotites. The measured values of the Al content in biotite were, therefore, not tabulated for his exchange experiments with Ti-free biotites. For biotite bulk compositions with a Si:Al ratio of 3:1 and 0.3 Ti apfu, however, microprobe data exist, and reactant biotites did not show a significant deviation from the Si:Al = 3:1 ratio in experiments conducted at 700 and 860 °C, indicating AlVI ~ 0 (Zhou (1994), his Table 2–14). The same is thus very likely for his Ti-free exchange data. No or only minor AlVI in Zhou’s Ti-free biotites is also confirmed by lattice constants given for synthesized ‘annite’. They yield a molar volume Vo = 15.362 J/bar, which is consistent with Vo of an ‘annite’ with ca. 10% Fe3+ (Dachs and Benisek 2015), whereas more aluminous Fe biotites would have a lower Vo (Benisek et al. 1996). Octahedral Al of biotite in Zhou’s (1994) experiments was thus taken to be negligible and set to zero in our calibration, allowing a parameterization of Fe–Mg mixing-related interaction parameters. The attainment of equilibrium in Zhou’s (1994) study was demonstrated using pairs of experiments that were conducted at similar P–T and logfO2 conditions, but started on opposite sides of the Roozeboom distribution curve (pressure was 1.1 kbar, temperatures were 650, 700, 800, and 860 °C and logfO2 was controlled by the graphite–methane buffer). In one of these experiments, olivines with a high- and biotites with a low Mg content were used and vice versa in the second experiment. XRPD and electron microprobe techniques were used to determine the chemical composition of the fine-grained (5–10 μm) homogeneous run products and simultaneously checked for the occurrence of additional phases. The starting biotites used in Zhou’s (1994) exchange experiments were synthesized from mixes of presynthesized kalsilite and appropriate amounts of oxides representing desired bulk compositions. The Fe-rich product biotites in his exchange experiments with olivine had 10 ± 2% of Fetot as Fe3+ and we used this value applying a linear decrease to 0% for Mg-rich biotites to compute XFeBt = Fe2+/(Fe2+ + Mg).
As can be seen from Eq. (2), our biotite model involves 11 macroscopic WG parameters. From these, WGPhlEas, WGPhlEasd, and WGEasEasd are already known from and discussed in the study of Dachs and Benisek (2019) on the thermodynamic properties of Mg–Al biotite. As noted there, ∆Hof,Eas and WHPhlEas could not be fixed unequivocally, and several combinations of ∆Hof,Eas and WHPhlEas equally well reproduced the experimentally determined Al-saturation level of Mg–Al biotite in the assemblage biotite–sillimanite–sanidine–quartz as determined by Berman et al. (2007). For that reason, Dachs and Benisek (2019) used the experimental data for reaction (18) to establish a correlation between ∆Hof,Eas and WHPhlEas giving:
Based on line-broadening in powder absorption IR spectra and resulting δ∆corr from the mid-wave number IR region, they pinned WHPhlEas to a value of 25.4 kJ/mol which is consistent with the solution calorimetric data of Circone and Navrotsky (1992) (disregarding one of their measurements at XEas = 0.56, which is possibly an outlier), indicating pronounced positive deviation from ideality for the Phl – Eas join. This gave ∆Hof,Eas = − 6358.5 ± 1.4 kJ/mol (Eq. 21). As noted by Dachs and Benisek (2019), if, alternatively, WHPhlEas∆corr = 10 kJ/mol from the high-wave number IR region would be used, still in accordance with the solution calorimetric data except at intermediate XEas, ∆Hof,Eas would change to a value of − 6352.0 ± 1.4 kJ/mol. Because initial attempts to include ∆Hof,Eas and WHPhlEas as free parameters in the least-squares treatment failed, giving unrealistic results, we used WHPhlEas = 10 kJ/mol and Eq. (21) as a constraint in the extraction procedure (see supplementary materials for more details). The reason for preferring 10 kJ/mol over 25.4 kJ/mol for WHPhlEas in our KFMASH biotite solution model is given below.
To further reduce the number of fit parameters, the following approximations were introduced: a) in the case of symmetrical non-ideal mixing, WGPhlObi and WGAnnObi are related to WGAnnPhl via the relations WGPhlObi = 1/3 WGAnnPhl, and WGAnnObi = 2/3 WGPhlAnn (Powell and Holland 1999; Holland and Powell 2006). For our biotite solution model, which is subregular along the Phl – Ann join, we adopted these relations (i.e., assuming WGPhlObi = 1/3(WGAnnPhl + WGPhlAnn)/2 and WGAnnObi = 2/3(WGAnnPhl + WGPhlAnn)/2), implying the assumption that there are no cross-site interactions for Fe–Mg mixing.
Furthermore, due to the equivalence WGEasObi = WGAnnSid, WGEasObi can be determined from experimental constraints pertinent to the Ann-Sid join. For this and the Ann–Eas binary, negative ∆Hexs of a few (< 3) kJ/mol at maximum are indicated from evaluating the line broadening in powder absorption IR spectra (Tables 5 and 6), implying negative values for WHAnnSid (= WHEasObi) and WHAnnEas. We chose to set these parameters to WHAnnSid = − 5 kJ/mol and to WHAnnEas = − 5 kJ/mol (3-cation-basis), representing small negative deviation from ideality for the Ann–Sid and Ann–Eas joins. The available experimental data did not allow to include these interaction parameters in the least-squares procedure, as attempted calculations gave meaningless results. As uncertainty, we assume a value of ± 4 kJ/mol (1 σ).
The two interaction parameters involving disordered eastonite Easd, WGEasdAnn and WGEasdObi were set to zero. Due to the large disordering enthalpy ΔH(eq. 8) = 34.5 ± 3 kJ, the proportion of the Eastd endmember is very low, so that this choice does not affect the least-squares results significantly, as checked by test calculations varying these parameters between − 20 and + 20 kJ/mol.
With the above discussed approximations, the least-squares derived values are:
WHPhlAnn = − 8.8 ± 8.0 kJ/mol, WHAnnPhl = 14.3 ± 3.4 kJ/mol (3-cation basis),
WHPhlEas = 10.0 ± 8.8 kJ/mol (assigned value),
∆Hof,Phl = − 6209.87 ± 0.50 kJ/mol,
∆Hof,Ann = − 5131.55 ± 2.34 kJ/mol,
∆Hof,Eas = − 6352.00 ± 3.72 kJ/mol (using Eq. (21) as a fitting constraint),
∆Hof,Sid = − 5635.83 ± 1.60 kJ/mol.
A matrix containing the correlation coefficients from the least-squares fit is given in Table S3, Table 8 gives a summary of all mixing parameters of the Fe–Mg–Al biotite solution model from this study, and in Table 9, the endmember thermodynamic properties are compiled.
Figure 6 is a plot of the predicted ∆Hex vs. XMg for the Ann–Phl binary. It is negatively deviating from ideality with ca. − 0.5 kJ/mol at maximum at Fe-rich biotite compositions and a positive deviation of around 1.5 kJ/mol occurs at XMg = 0.75. The meaningfulness of such a ∆Hex behaviour is given in the discussion. δ∆corr values derived from line-broadening in powder absorption IR spectra from the high-wave number region of 800–1300 cm−1 (Table 3, Fig. S1) indicate a positive ∆Hex of at maximum ~ 2 kJ/mol. Using the low-wave number region of the spectra (175–640 cm−1), on the other hand, results in a somewhat negative ∆Hex of − 3 kJ/mol at maximum. ∆Hex resulting from δ∆corr values is thus expected to be rather small in magnitude, in agreement with that derived from the exchange experiments of Zhou (1994), but it is not clear from the line broadening if the sign of ∆Hex is positive or negative.
In the initial stage of data fitting, a symmetric mixing model was alternatively tested, which gave ∆Hof,Ann = − 5136.7 ± 2.1 kJ/mol, and WAnnPhl = 9.3 ± 1.8 kJ/mol. This resulted in a positive ∆Hex of 2.3 kJ/mol at maximum. Such a model, however, was discarded, because it has (1) a larger reduced χ2 and is (2) not capable of representing the structural/physical peculiarities of the Ann–Phl join (see below, where DFT calculations are discussed comparing annite’s octahedral site with that in fayalite and ferrosilite). Furthermore, as outlined in part-II, using this ∆Hof,Ann and WAnnPhl from the symmetric model would (3) imply a rather unrealistically large nonideality for Fe–Mg mixing in garnet (WGAlmPy around 12 kJ/mol, 3-site cation basis), to achieve consistency with the experimental data of Ferry and Spear (1978) on the Fe–Mg distribution between biotite and garnet.
Discussion
Standard enthalpy of formation values of biotite endmembers
The extracted ∆Hof,Ann = − 5131.55 ± 2.34 kJ/mol is 13–15 kJ less negative than values of ∆Hof,Ann from thermodynamic databases and other determinations (∆Hof,Ann = − 5144.23 kJ/mol according to Holland and Powell (2011); ∆Hof,Ann = − 5146.1 kJ/mol according to Berman et al. (2007)). One reason for that is the use of the revised standard entropy of Phl from Dachs and Benisek (2019) in our study. So of Phl was determined in that work through relaxation calorimetry on a synthetic pure Phl (Scal,Phl = 319.4 ± 2.2 J/mol·K) and found to be significantly larger compared to previous calorimetric determinations on a natural near endmember Phl, which formed the base for later databases (Scal,Phl = 315.9 ± 1.0 J/mol·K according to Robie and Heminway (1984)). Another reason is that the set of experimental data used in the extraction procedures is not identical.
Our determination of ∆Hof,Ann is in good agreement to that extracted by Dachs and Benisek (2015) from (redox-)equilibria in the FASH system (∆Hof,Ann = − 5132.5 ± 2.0 kJ/mol). If ∆Hof,Ann = − 5131.55 kJ/mol is used to compute the hydrogen fugacity fH2 of the equilibrium:
(all other data from Holland and Powell (2011) at 2 kbar and 700 °C, one gets fH2 = 52 bar, well within 46.3–61.4 bar as determined experimentally for annite (aAnn = 0.64) in equilibrium with sanidine + magnetite + H2 (Dachs 1994; Benisek et al. 1996). The experimental reversals of Dachs and Benisek (1995) on the reaction:
determined experimentally in cold-seal pressure vessels between pressures of 2 and 5 kbar, could also be reproduced, if a small correction of + 1 kJ, well within the uncertainty limits of the least-squares result, is applied to ∆Hof,Ann. The annite standard state properties determined herein (Table 9) thus enable an internal consistency between the Fe–Mg exchange experiments of Zhou (1994) and the experimental data on reactions (22) and (23) in the FASH subsystem (Eugster and Wones 1962; Dachs 1994; Dachs and Benisek 1995).
The calorimetrically determined SoSid = 356.0 ± 3.1 J/(mol·K) of our study is some 20 J/(mol·K) smaller than SoSid = 378.09 J/(mol·K) given by Berman et al. (2007) for this endmember. Enthalpy of formation values from our study compared to theirs differs by ~ 14 kJ/mol (∆Hof,Sid = − 5635.83 ± 1.6 kJ/mol compared to ∆Hof,Sid = − 5621.14 kJ/mol).
The enthalpy of formation values of Ann and Phl and the Fe–Mg mixing properties of biotite (Table 8) derived in our study are not independent, but are correlated to the Fe–Mg mixing properties adopted for olivine that appears as Fe–Mg exchange partner of biotite in Eq. (20). The forsterite (Fo)–fayalite (Fa) system is well studied and most thermodynamic analyses of various phase-equilibrium data point to a moderate positive deviation from ideality that can be modelled with a symmetric macroscopic one-site WGFoFa in the order of 3–6 kJ/mol mol (Davidson and Mukhopadhyay 1984; Wiser and Wood 1991; Berman and Aranovich 1996). The calorimetric study of Kojitani and Akaogi (1994) is with a slight temperature-dependent one-site WGFoFa around 5 kJ/mol in accordance with these assertions. The earlier solution calorimetric measurements of Wood and Kleppa (1981) yielded a similar small positive deviation from ideality that is, however, asymmetric towards the Fe- endmember, characterised by the one-site interaction parameters WGFaFo = 4.2 kJ/mol and WGFoFa = 2.1 kJ/mol.
We have measured the line broadening in powder absorption IR spectra of Fo–Fa olivines studied calorimetrically by Dachs et al. (2007). Our results agree with the above calorimetric and phase-equilibrium determinations and give a similar mean value of WGFoFa = 5 kJ/mol per site (more details are given in supplementary Figs. S2 and S3). Because the majority of studies provide evidence for such a small positive deviation from ideality for Fe–Mg mixing in olivine, we prefer to use WGFoFa = 5 kJ/mol (one-site mixing) in analysing the exchange experiments of Zhou (1994).
It should, however, be noted that the studies of Sack (1980) and Sack and Ghiorso (1989) point to a larger Fe–Mg nonideality in olivine than represented by WGFoFa = 5 kJ/mol (one-site mixing), namely WGFoFa around 10 kJ/mol. If this value would be adopted and symmetrical Fe–Mg mixing in biotite would be assumed in the thermodynamic analysis, the least-squares derived enthalpy of Ann and the interaction parameter for symmetrical Fe–Mg mixing would change to (other enthalpies change only minor):
The consequence of accepting this larger WGFoFa in the thermodynamic treatment would be that Hof,Ann decreases by ~ 12 kJ/mol and that symmetrical Fe–Mg mixing in biotite becomes strongly positive over the whole compositional range implying a rather large ∆Hex in the order of 6.5 kJ/mol at maximum.
Treating, on the other hand, the Phl–Ann join as asymmetric, as preferred in this study, using a WGFoFa = 10 kJ/mol (one-site cation basis) in the least-squares procedure would lead to:
∆Hof,Ann = − 5136.2 ± 1.7 kJ/mol and
WHPhlAnn = 0.9 ± 6.1 kJ/mol, WHAnnPhl = 36.4 ± 2.4 kJ/mol.
Mixing along the Phl–Ann binary would then be predicted to be strongly asymmetric towards the Ann endmember with a large maximal ∆Hex of around 5 kJ/mol (see companion paper for implications on phase relations).
Fe–Mg mixing in biotite: Does it resemble that in olivine?
From the existing Fe–Mg exchange experiments involving biotite, we have only used that of Zhou (1994), because the Fe–Mg mixing properties of its exchange partner olivine seem to be the relative best known. Excess entropies along the Ann–Phl join are ideal (Fig. 2), so that there is no temperature dependence of the mixing properties (as WVijs are zero for all biotite joins, WGij ≡ WHij). We refrained from using experimental garnet–biotite Fe–Mg exchange data (Ferry and Spear 1978; Perchuk and Lavrent’eva 1983; Gessmann et al. 1997), because, first, conflicting data exist, and, second, the Fe–Mg mixing properties of garnet are not as well established as those for olivine and a range of values for the Fe–Mg parameter(s) has been proposed (e.g., Ganguly et al. (1996): WHAlmPy = 2.1 kJ/mol, WHPyAlm = 6.4 kJ/mol; Berman (1990): WHAlmPy = 0.2 kJ/mol, WHPyAlm = 3.7 kJ/mol; Berman and Aranovich (1996): WHAlmPy = 5.1 kJ/mol, WHPyAlm = 6.2 kJ/mol; White et al. (2014a): symmetric WGAlmPy = 2.5 kJ/mol; all values on a 3-cation basis, Alm = almandine, Py = pyrope). Furthermore, existing solution calorimetric work on Alm–Py garnets (Geiger et al. 1987) shows relatively large errors on ∆Hmix prohibiting to extract well-defined WHs for that join.
The experimental orthopyroxene–biotite Fe–Mg exchange data of Fonarev and Konilov (1986) have not been used, because orthopyroxene was notably inhomogeneous in the experimental runs (see part-II for more details). In the companion paper, the existing garnet–biotite and orthopyroxene–biotite Fe–Mg exchange data are, however, discussed and used for validation purposes of our new activity model of biotite.
Our thermodynamic analysis yields a lower reduced χ2 for an asymmetric biotite solution model with WH,AnnPhl = 14.3 ± 3.4 kJ/mol and WH,PhlAnn = − 8.8 ± 8.0 kJ/mol. Fe–Mg mixing behaviour of biotite according to this treatment is thus characterised by a slightly negative Gex at Fe-rich compositions switching to positive Gex in the Mg-rich part of the Phl–Ann binary (Fig. 6). Such a Gex vs. composition behaviour seems reasonable, because Ann does not exist as an endmember due to the misfit between the larger octahedral and the smaller tetrahedral sheet (e.g., Hazen and Wones (1973); Redhammer et al. (1993)). A negative Gex takes this fact into account by destabilising endmember Ann relative to solid solution biotites where some of the Fe is replaced by either Al or Mg. Absolute values of Gex are, however, rather small (< 2 kJ/mol) and not far from ideality. Benisek et al. (1999) have shown that natural biotites, when their amount of AlVI, as balanced by the Tschermak-substitution (AlIV – 1), is plotted vs. Mg/(Mg + Fe), accumulate in a band limited by a tetrahedral rotation angle of α = 8 ± 1° (Hazen and Burnham 1973) (see also Fig. 12 in part-II). Biotites with that structural feature seem to be more stable than others, possibly because the misfit between the tetrahedral sheet and an octahedral layer having this α is minimal and smaller compared to cases where α is either smaller or larger. Along the Ann–Sid join, this situation is realised for biotites around the Ann50Sid50 composition, whereas on the Phl-Eas binary only a small degree of Tschermak substitution is required, i.e., Phl90Eas10 (Benisek et al. 1999, Fig. 10). The thermodynamic mixing behaviour of KFMASH biotites, as proposed in this study, is consistent with that picture: Fe-rich and AlVI-poor biotites around the Ann endmember have smaller α’s (the octahedral sheet is larger, i.e., more stretched compared to the energetically favoured ca. 8° structural situation) and can reach that value by incorporation of either Al (or Fe3+) or Mg or both. This behaviour can be modelled with negative deviation from ideality starting at the Ann endmember towards Sid and Phl. Mg-rich and AlVI-poor biotites, on the other hand, are already in their ‘optimal α-state’ and a further increase of the Tschermak substitution causes α’s larger than the preferred 8° value. This constitutes an energetically more unfavourable situation and is thermodynamically mimicked by positive deviation from ideality along the Phl–Eas binary, as well established from solution calorimetry (Circone and Navrotsky 1992) stabilising AlVI-poor Mg-rich biotites. As a consequence of these opposite mixing behaviours (negative deviation from ideality along the Ann–Sid join, positive deviation for the Phl-Eas binary), Fe–Mg mixing in biotite is then predicted to deviate negatively from ideality for Fe-rich compositions switching to positive deviation at intermediate XMg (Fig. 6).
In the more recent literature and in relation to the development of internally consistent thermodynamic databases and corresponding software, which allow the computation of various types of phase diagrams, Fe–Mg mixing in biotite itself was linked to that in Fe–Mg olivine. This was done by assuming that the value of the microscopic interaction parameter, which describes the energy between a Fe2+ and Mg atom mixing on one octahedral site (wFeMg,oct), is the same as that for olivine, namely around 4 kJ·mol−1. For three octahedral sites as in biotite, WPhlAnn then becomes 12 kJ/mol (Holland and Powell 2006). This idea of defining microscopic interaction w’s from (well) known mineral systems and transferring them to experimentally less or unknown systems is the basis of the heuristic approach of Powell et al. (2014), called ‘micro-Φ’. In this process, the microscopic w’s get reassembled into corresponding macroscopic W’s by multiplying with the minerals site multiplicities and these can then be used in activity model calculations. The asymmetric type of Fe–Mg mixing behaviour which we extracted for biotite from experimental data is not in accordance with that resulting from the micro-Φ’ approach, where the w’s are transferred between different mineral systems. On the microscopic level, the underlying hypothesis (wFeMg,one-oct-site)mineral = (wFeMg,one-oct-site)olivine = 4 kJ/mol can be expected to be a good approximation, if the octahedral sites in ‘mineral’ (e.g., biotite, pyroxene, etc.) have comparable geometries and thus energies to that in olivine. In Table 10, we have listed the range of octahedral Fe–O distances and their mean values for Ann, ferrosilite (Fs) and Fa resulting from additional CASTEP calculations that we have undertaken. Whereas the mean Fe–O distance in ferrosilite is with a value of 2.15 Å quite similar to that in Fayalite (2.14 Å), the corresponding distance in Ann is significantly smaller (2.09 Å). The range of Fe–O distances within an octahedral site in Ann is also smaller compared to that in Fs and Fa (Table 10). The M sites in Ann are thus more ‘compressed’ relative to that in Fa und Fs. Octahedral flattening has been described as a major distortion mechanism besides tetrahedral rotation to accommodate the misfit between octahedral and tetrahedral sheets in biotite (Mercier 2006). In conclusion, our CASTEP calculations show that the transferability of microscopic w’s from olivine to Fe–Mg pyroxenes seems reasonable but not in the case of biotite, where the octahedra are too different compared to the ones in olivine. However, one has to note that the mean site geometry for ferrosilite in this comparison is the mean of two octahedral sites in pyroxene, each having quite different geometries. This gives rise to strong long-range nonconvergent ordering of Fe and Mg between M1 and M2 sites in pyroxene and not in olivine. This last argument shows that micro-Φ approach between different crystal systems should be used with caution.
Fe–Al mixing in biotite
All studies that have extracted biotite mixing properties empirically from data on natural assemblages indicate values not more than a few kJ up to at maximum ≈12 kJ different from zero for the difference WAnnSid – WPhlEas, (≡ WFeAl – WMgAl in older notation, Table S1), i.e., both macroscopic W’s describing the interaction of Fe or Mg with Al should give roughly W’s of the same order (both can still be either positive or negative).
The line broadening in powder absorption IR spectra obtained from members of the Ann–Sid binary provides experimental evidence for a negative deviation from ideality for the ∆Hmix-composition behaviour (WAnnSid in the order of − 12 kJ/mol). Similar to Phl–Ann, excess entropies along this join are zero. A negative WAnnSid = − 29 ± 4 kJ/mol was proposed by Benisek et al. (1996) derived from reaction displacement experiments on the equilibrium Ann = San + Mag + H2. A less negative value (− 8.2 kJ/mol) was proposed by Berman et al. (2007). We have set WAnnSid and WAnnEas to negative values of − 5 kJ/mol. This choice reproduces the Al level of a KFMASH biotite with XMg = 0.57 in the assemblage biotite–sillimanite–sanidine–quartz–H2O, which was experimentally determined by Berman et al. (2007) to lie between 1.78 and 1.83 (AlVI = 0.39–0.42) at 709 °C/2.06 kbar (computed AlVI = 0.4). Our result of negative WAnnSid implies also a negative WEasObi of the same order due to the equality WEasObi = WAnnSid (Powell and Holland 1999; Holland and Powell 2006).
Mg–Al mixing and ordering in biotite
As discussed in Dachs and Benisek (2019) (their Fig. 10), Al is highly concentrated on the M1 site of biotite relative to the M2 site, due to the large enthalpy of disordering of ΔH(eq. 8) = 34.5 ± 3 kJ/mol. Structural data on the ordering of Al among the two M sites in natural biotites are highly controversial, for a discussion on the literature data, see Dachs and Benisek (2019). The evaluation of line broadening in powder absorption IR spectra gave two values for the WPhlEas mixing parameter, namely 25.4 kJ/mol (mid-wave number region) and 10 kJ/mol (high-wave number region). The larger value was preferred by Dachs and Benisek (2019), because it lead to a good agreement with the solution calorimetric data of Circone and Navrotsky (1992) available for Phl–Eas biotites (Dachs and Benisek (2019), their Fig. 7). Based on our calculations in the KFMASH system, we prefer now the smaller value in combination with ∆Hof,Eas = − 6352.0 ± 1.4 kJ/mol. As outlined in Dachs and Benisek (2019) such a WPhlEas around 10 kJ/mol can be considered as minimum value describing nonideality along the Phl–Eas join and reflects the structural situation of short-range order. This choice of setting WPhlEas = 10 kJ/mol is still in accordance with most of the calorimetric data of Circone and Navrotsky (1992) and produces a moderate temperature dependence of AlVI in reasonable agreement with natural observations (Holdaway 1980), showing that the AlVI content in biotite from pelitic assemblages should decrease slightly with increasing T (see part-II for more discussion on this). As a consequence, Al ordering on M1 will be even more pronounced than modelled in Dachs and Benisek (2019) and significant amounts of Al on M2 are not to be expected for metamorphic P–T conditions. At temperatures above 1000 °C, however, Al content on M2 and therefore the proportion of the disordered endmember Easd start rising slightly from values close to zero which could eventually affect the computation of, e.g., mantle equilibria involving Phl-rich biotite.
The enthalpy of Al ordering on the octahedral sites, ΔH(eq. 8), was derived by Dachs and Benisek (2019) from DFT calculations, in which the Al–Si distribution in the tetrahedral sheet of ordered and disordered eastonite (Eas and Easd, respectively) was that of Al avoidance, i.e., Al and Si were alternatively placed in the tetrahedra-ring spanning that sheet. We now did additional DFT calculations for Easd where this constraint was released, allowing a more disordered Si–Al distribution. Such configurations lead to still larger ∆H(eq. 8) values than 34.5 kJ/mol. DFT calculations thus indicate that such a ∆H(eq. 8) is likely to represent the maximum Mg–Al disorder to be expected in natural biotites. This confirms the solution space chosen by White et al. (2014a, b), where octahedral Al was assumed to reside on M1 only.
Fe–Mg ordering in biotite and predicted site preferences
Our KFMASH biotite activity model predicts Fe2+ and Mg site preferences, as shown in Fig. 7 for 600 °C, where [Fe/(Fe + Mg)]M1 and [Fe/(Fe + Mg)]M2 are plotted as a function of bulk XMg in biotite for various AlVI levels. From our DFT calculations, we constrained ΔH(eq. 9)≈ − 2 ± 3 kJ/mol, showing that Fe–Mg ordering in biotite should be weak. This confirms assertions of Holland and Powell (2006) and White et al. (2014a) for this value (ΔH(eq. 9) = − 2 kJ/mol).
Similar to existing ordering models (Holland and Powell 2006; White et al. 2014a), Fe2+ somewhat prefers the M1 site in biotites with XMg > 0.25. For Fe-rich biotites, a switch in this behaviour occurs. In this compositional range, Fe2+ is slightly enriched in the M2 position. This crossover is a direct consequence of the change in sign in ∆Gex from positive to negative in our activity model when going from Phl towards Ann.
Activity–composition relationships
The Phl and Ann activities at 600 °C vs. bulk XMg from this study, for the Ann–Phl binary (AlVI = 0) and for Fe-Mg-Al biotites with AlVI = 0.3, are shown in Fig. 8a and b, respectively. The activities are closer to ideality than would be computed from existing biotite models (Holland and Powell 2006; White et al. 2014a). For these biotite chemistries and at this temperature of 600 °C, our model differs by up to an activity of ca. 0.1 from existing models for biotite. The activities from our modelling are lower by up to this amount, except for the activity of Phl (aPhl) in Mg-rich solid solutions of biotite. The effect of our more ideal biotite activity model on phase relations in the KFMASH system is illustrated in several examples in the companion paper.
Conclusions
-
An important outcome of our study is that excess entropies of mixing are zero for all important biotite binaries (Ann–Phl, Ann–Sid, and Ann–Eas as determined in this work and Phl-Eas as measured by Dachs and Benisek (2019). Excess volumes of mixing are also zero within error for all these joins, so that excess enthalpies are independent of pressure and temperature (WGij = WHij).
-
The thermodynamic analysis of phase-equilibrium data on reactions (17) to (20) gave best-fit results for an asymmetric Gex along the Phl–Ann join, that is negative at Fe-rich and positive at Mg-rich biotite compositions (WHAnnPhl = 14.3 ± 3.4 kJ/mol, WHPhlAnn = − 8.8 ± 8.0 kJ/mol). A value of WGFaFo = 5 kJ/mol for olivine was used in this derivation, which results from published calorimetry, from the majority of phase-equilibrium studies and from our ∆corr analysis of the line broadening in powder absorption IR spectra of olivine. On the structural level, such a Gex behaviour for Fe–Mg mixing in biotite is reasonable, because Fe-rich and AlVI-poor biotites around the Ann endmember have smaller α’s than 8° and can reach that value by incorporation of either Mg or Al (or Fe3+) or both, whereas Mg-rich and AlVI-poor biotites are already in their ‘optimal α-state’ and a further increase of the Tschermak substitution causes α’s larger than the preferred 8° value. This behaviour can be modelled with negative deviation from ideality starting at the Ann endmember towards Phl, positive deviation from ideality along the Phl–Eas binary, and a switch from negative to positive deviation for the Ann–Phl binary at intermediate XMg.
-
DFT calculations show that mean octahedral Fe–O distances are significantly smaller in pure Ann compared to fayalite and ferrosilite. Annites’ octahedral site is thus energetically not comparable to that in fayalite or ferrosilite. Therefore, the micro-Φ approach between different mineral systems should be used with caution.
-
The extracted ∆Hof,Ann = − 5131.55 ± 2.34 kJ/mol is 13–15 kJ less negative than published values, which is largely the result of using the revised standard state data for Phl from Dachs and Benisek (2019) in the extraction procedure. Future updates of internally consistent databases, taking this new Phl standard state data and the experimental data of Zhou (1994) on the Fe–Mg partitioning between biotite and olivine into account, are thus expected to yield a similar less negative ∆Hof,Ann, which is compatible with the experimental record from (redox-) equilibria in the FASH system.
-
The presently used value of ∆H(9) = − 2 kJ/mol for the reaction 2/3 Phl + 1/3 Ann = Obi, describing Fe–Mg ordering in biotite is confirmed by our DFT calculations that gave values for ΔH(eq. 9)≈ − 2 ± 3 kJ/mol. Generally, Fe–Mg ordering in biotite is thus indicated to be weak.
-
The preferred value for WHPhlEas in this study is 10 kJ/mol (as used in present biotite activity models), representing strict local charge balance in the structure. It is compatible with an enthalpy of formation of Eas of ∆Hof,Eas = − 6352.0 ± 3.7 kJ/mol. Fixing WHAnnSid (≡ WHEasObi) and WHAnnEas to − 5 kJ/mol gives a moderate temperature dependence of AlVI in biotite (i.e., decreasing with T). Negative excess enthalpies of mixing for the Ann–Sid and Ann–Eas joins are also indicated by evaluating the line broadening in powder absorption IR spectra of members of these binaries.
-
Additional DFT calculations on disordered Eas reconfirm the relatively large value of ∆H(eq. 8) = 34.5 kJ/mol for the disordering of Eas, as given by Dachs and Benisek (2019). In combination with the preferred smaller value for WPhlEas = 10 kJ/mol of this study (compared to 25.4 kJ/mol as used in Dachs and Benisek (2019), Mg–Al disordering turns out to be only significant at high temperatures > 1000 °C in igneous or mantle equilibria. At metamorphic P–T conditions, Mg–Al disordering is too small to be of importance and Al can be taken to reside on M1 only. For that case, our activity model for biotite could be simplified to one where only Mg and Fe2+ can occupy the M2 site, which is the solution space used by White et al. (2014a)
-
In summary, Fe–Mg mixing in biotite departs only slightly and asymmetrically from ideality. In the companion paper, we study the effects of the new biotite activity model and thermodynamic data to phase relations in the KFMASH system, and we analyse the existing, but partly conflicting, experimental Fe–Mg exchange data between garnet and biotite.
In a forthcoming paper, we will extend our activity model for KFMASH biotite to include endmembers that account for the presence of excess-AlVI, Ti, and Fe3+ in natural biotites.
References
Aranovich LY, Newton RC (1998) Reversed determination of the reaction; phlogopite+quartz = enstatite+potassium feldspar+H2O in the ranges 750–875 degrees C and 2–12 kbar at low H2O activity with concentrated KCl solutions. Am Mineral 83:193–204. https://doi.org/10.2138/am-1998-3-401
Benisek A, Dachs E (2018) The accuracy of standard enthalpies and entropies for phases of petrological interest derived from density-functional calculations. Contrib Mineral Petrol 173:90. https://doi.org/10.1007/s00410-018-1514-x
Benisek A, Dachs E (2020) Excess enthalpy of mixing of mineral solid solutions derived from density-functional calculations. Phys Chem Miner 47:15. https://doi.org/10.1007/s00269-020-01085-8
Benisek A, Dachs E, Redhammer G et al (1996) Activity-composition relationship in Tschermak’s substituted Fe biotites at 700°C, 2 kbar. Contrib Mineral Petrol 125:85–99. https://doi.org/10.1007/s004100050208
Benisek A, Dachs E, Cemic L (1999) Heat capacities of Tschermak substituted Fe-biotite. Contrib Mineral Petrol 135:53–61. https://doi.org/10.1007/s004100050497
Benisek A, Dachs E, Kroll H (2010) A ternary feldspar-mixing model based on calorimetric data: development and application. Contrib Mineral Petrol 160:327–337. https://doi.org/10.1007/s00410-009-0480-8
Benisek A, Dachs E, Carpenter MA (2013) Heat capacity and entropy of low structural state plagioclases. Phys Chem Minerals 40:167–173. https://doi.org/10.1007/s00269-012-0556-2
Berman RG (1990) Mixing properties of Ca-Mg-Fe-Mn garnets. Am Miner 75:328–344
Berman RG, Aranovich LY (1996) Optimized standard state and solution properties of minerals. Contrib Mineral Petrol 126:1–24. https://doi.org/10.1007/s004100050232
Berman RG, Aranovich LY, Rancourt DG, Mercier PHJ (2007) Reversed phase equilibrium constraints on the stability of Mg-Fe-Al biotite. Am Mineral 92:139–150. https://doi.org/10.2138/am.2007.2051
Boffa Ballaran T (2003) Line broadening and enthalpy: some empirical calibrations of solid solution behaviour from IR spectra. Phase Transit 76:137–154. https://doi.org/10.1080/0141159031000076101
Boffa Ballaran T, Carpenter MA, Geiger CA, Koziol AM (1999) Local structural heterogeneity in garnet solid solutions. Phys Chem Miner 26:554–569. https://doi.org/10.1007/s002690050219
Bohlen SR, Boettcher AL, Wall VJ, Clemens JD (1983) Stability of phlogopite-quartz and sanidine-quartz: a model for melting in the lower crust. Contrib Mineral Petrol 83:270–277. https://doi.org/10.1007/BF00371195
Cemič L (2005) The first law of thermodynamics thermodynamics in mineral sciences: an introduction. Springer, Berlin, Heidelberg, pp 78–128
Ceperley DM, Alder BJ (1980) Ground state of the electron gas by a stochastic method. Phys Rev Lett 45:566–569. https://doi.org/10.1103/PhysRevLett.45.566
Circone S, Navrotsky A (1992) Substitution of [6, 4]Al in phlogopite: high-temperature solution calorimetry, heat capacities, and thermodynamic properties of the phlogopite-eastonite join. Am Mineral 77:1191–1205
Clark SJ, Segall MD, Pickard CJ, Hasnip PJ, Probert MIJ, Refson K, Payne MC (2005) First principles methods using CASTEP. Z Kristallogr 220:567–570
Clemens JD (1995) Phlogopite stability in the silica-saturated portion of the system KAlO2-MgO-SiO2-H2O: New data and a reappraisal of phase relations to 1.5 GPa. Am Mineral 80: 982-997
Dachs E (1994) Annite stability revised. 1. Hydrogen-sensor data for the reaction annite = sanidine + magnetite + H2. Contrib Mineral Petrol 117:229–240. https://doi.org/10.1007/BF00310865
Dachs E, Benisek A (1995) The stability of annite+quartz: reversed experimental data for the reaction 2 annite+3 quartz=2 sanidine+3 fayalite +2 H2O. Contrib Mineral Petrol 121:380–387. https://doi.org/10.1007/s004100050103
Dachs E, Benisek A (2011) A sample-saving method for heat capacity measurements on powders using relaxation calorimetry. Cryogenics 51:460–464. https://doi.org/10.1016/j.cryogenics.2011.04.011
Dachs E, Benisek A (2015) Standard-state thermodynamic properties of annite, KFe3[(OH)2AlSi3O10], based on new calorimetric measurements. Eur J Mineral 27:603–616. https://doi.org/10.1127/ejm/2015/0027-2462
Dachs E, Benisek A (2019) A new activity model for Mg–Al biotites determined through an integrated approach. Contrib Mineral Petrol 174:76. https://doi.org/10.1007/s00410-019-1606-2
Dachs E, Bertoldi C (2005) Precision and accuracy of the heat-pulse calorimetric technique low-temperature heat capacities of milligram-sized synthetic mineral samples. Eur J Mineral 17:251–261. https://doi.org/10.1127/0935-1221/2005/0017-0251
Dachs E, Geiger CA, von Seckendorff V, Grodzicki M (2007) A low-temperature calorimetric study of synthetic (forsterite+fayalite) (Mg2SiO4+Fe2SiO4) solid solutions: an analysis of vibrational, magnetic, and electronic contributions to the molar heat capacity and entropy of mixing. J Chem Thermodyn 39:906–933. https://doi.org/10.1016/j.jct.2006.11.009
Dachs E, Geiger CA, Benisek A (2014a) Thermodynamic mixing properties and behavior of grossular–spessartine, (CaxMn1−x)3Al2Si3O12, solid solutions. Geochim Cosmochim Acta 141:294–302. https://doi.org/10.1016/j.gca.2014.06.034
Dachs E, Geiger CA, Benisek A, Grodzicki M (2014b) Thermodynamic mixing properties and behavior of almandine–spessartine solid solutions. Geochim Cosmochim Acta 125:210–224. https://doi.org/10.1016/j.gca.2013.10.005
Davidson PM, Mukhopadhyay DK (1984) Ca-Fe–Mg olivines: phase relations and a solution model. Contrib Mineral Petrol 86:256–263. https://doi.org/10.1007/BF00373671
Etzel K, Benisek A (2008) Thermodynamic mixing behavior of synthetic Ca-Tschermak–diopside pyroxene solid solutions: III. An analysis of IR line broadening and heat of mixing behavior. Phys Chem Miner 35:399–407. https://doi.org/10.1007/s00269-008-0234-6
Eugster HP, Wones DR (1962) Stability relations of the ferruginous biotite, annite. J Petrol 3:82–125. https://doi.org/10.1093/petrology/3.1.82
Ferry JM, Spear FS (1978) Experimental calibration of the partitioning of Fe and Mg between biotite and garnet. Contrib Mineral Petrol 66:113–117. https://doi.org/10.1007/BF00372150
Fonarev VI, Konilov AN (1986) Experimental study of Fe–Mg distribution between biotite and orthopyroxene at P=490 MPa. Contrib Mineral Petrol 93:227–235. https://doi.org/10.1007/BF00371325
Ganguly J, Cheng W, Tirone M (1996) Thermodynamics of aluminosilicate garnet solid solution: new experimental data, an optimized model, and thermometric applications. Contrib Mineral Petrol 126:137–151. https://doi.org/10.1007/s004100050240
Geiger CA, Newton RC, Kleppa OJ (1987) Enthalpy of mixing of synthetic almandine-grossular and almandine-pyrope garnets from high-temperature solution calorimetry. Geochim Cosmochim Acta 51:1755–1763. https://doi.org/10.1016/0016-7037(87)90353-X
Gessmann CK, Spiering B, Raith M (1997) Experimental study of the Fe–Mg exchange between garnet and biotite; constraints on the mixing behavior and analysis of the cation-exchange mechanisms. Am Mineral 82:1225–1240. https://doi.org/10.2138/am-1997-11-1218
Hazen RM, Burnham CW (1973) The crystal structures of one-layer phlogopite and annite. Am Miner 58:889–900
Hazen RM, Wones DR (1972) The effect of cation substitutions on the physical properties of trioctahedral micas. Am Miner 57:103–129
Hewitt DA, Wones DR (1975) Physical properties of some synthetic Fe-Mg-Al trioctahedral biotites. Am Mineral 60:854–862
Hoisch TD (1991) Equilibria within the mineral assemblage quartz + muscovite + biotite + garnet + plagioclase, and implications for the mixing properties of octahedrally-coordinated cations in muscovite and biotite. Contrib Mineral Petrol 108:43–54. https://doi.org/10.1007/BF00307325
Holdaway MJ (1980) Chemical formulae and activity models for biotite, muscovite, and chlorite applicable to pelitic metamorphic rocks. Am Miner 65:711–719
Holdaway MJ, Mukhopadhyay B, Dyar MD et al (1997) Garnet-biotite geothermometry revised; new Margules parameters and a natural specimen data set from Maine. Am Mineral 82:582–595. https://doi.org/10.2138/am-1997-5-618
Holland TJB, Powell R (2006) Mineral activity-composition relations and petrological calculations involving cation equipartition in multisite minerals: a logical inconsistency. J Metamorph Geol 24:851–861. https://doi.org/10.1111/j.1525-1314.2006.00672.x
Holland TJB, Powell R (2011) An improved and extended internally consistent thermodynamic dataset for phases of petrological interest, involving a new equation of state for solids. J Metamorph Geol 29:333–383. https://doi.org/10.1111/j.1525-1314.2010.00923.x
Holland TJB, Redfern S, AT (1997) Unit cell refinement from powder diffraction data: the use of regression diagnostics. Mineral Mag 61:65–77. https://doi.org/10.1180/minmag.1997.061.404.07
Indares A, Martignole J (1985) Biotite–garnet geothermometry in the granulite facies: the influence of Ti and Al in biotite. Am Mineral 70:272–278
Jackson SL (1989) Extension of Wohl’s ternary asymmetric solution model to four and n components. Am Mineral 74:14–17
Jenkins DM, Carpenter MA, Zhang M (2014) Experimental and infrared characterization of the miscibility gap along the tremolite-glaucophane join. Am Miner 99:730–741. https://doi.org/10.2138/am.2014.4590
Kennedy CA, Stancescu M, Marriott RA, White MA (2007) Recommendations for accurate heat capacity measurements using a Quantum Design physical property measurement system. Cryogenics 47:107–112. https://doi.org/10.1016/j.cryogenics.2006.10.001
Kleemann U, Reinhardt J (1994) Garnet-biotite thermometry revisited: The effect of AlVI and Ti in biotite. Eur J Mineral 6:925–942
Kojitani H, Akaogi M (1994) Calorimetric study of olivine solid solutions in the system Mg2SiO4-Fe2SiO4. Phys Chem Miner 20:536–540. https://doi.org/10.1007/BF00211849
Lashley JC, Hundley MF, Migliori A et al (2003) Critical examination of heat capacity measurements made on a Quantum Design physical property measurement system. Cryogenics 43:369–378. https://doi.org/10.1016/S0011-2275(03)00092-4
Li Y, Kowalski PM, Blanca-Romero A et al (2014) Ab initio calculation of excess properties of La1−x(Ln, An)xPO4 solid solutions. J Solid State Chem 220:137–141. https://doi.org/10.1016/j.jssc.2014.08.005
McMullin DW, Berman RG, Greenwood HJ (1991) Calibration of the SGAM thermobarometer for pelitic rocks using data from phase-equilibrium experiments and natural assemblages. Can Mineral 29:889–908
Mercier PHJ (2006) Upper limit of the tetrahedral rotation angle and factors affecting octahedral flattening in synthetic and natural 1M polytype C2/m space group micas. Am Mineral 91:831–849. https://doi.org/10.2138/am.2006.1815
Monkhorst HJ, Pack JD (1976) Special points for Brillouin-zone integrations. Phys Rev B 13:5188–5192. https://doi.org/10.1103/PhysRevB.13.5188
Müller RF (1972) Stability of biotite: a discussion. Am Mineral 57:300–316
Perchuk LL, Lavrent’eva IV (1983) Experimental investigation of exchange equilibria in the system cordierite-garnet-biotite. In: Saxena SK (ed) Kinetics and equilibrium in mineral reactions. Springer, New York, pp 199–239
Pfrommer BG, Côté M, Louie SG, Cohen ML (1997) Relaxation of crystals with the quasi-newton method. J Comput Phys 131:233–240. https://doi.org/10.1006/jcph.1996.5612
Powell R, Holland T (1999) Relating formulations of the thermodynamics of mineral solid solutions; activity modeling of pyroxenes, amphiboles, and micas. Am Mineral 84:1–14. https://doi.org/10.2138/am-1999-1-201
Powell R, White RW, Green ECR et al (2014) On parameterizing thermodynamic descriptions of minerals for petrological calculations. J Metamorph Geol 32:245–260. https://doi.org/10.1111/jmg.12070
Redhammer GJ, Beran A, Dachs E, Amthauer G (1993) A Mössbauer and X-ray diffraction study of annites synthesized at different oxygen fugacities and crystal chemical implications. Phys Chem Miner 20:382–394. https://doi.org/10.1007/BF00203107
Redhammer GJ, Dachs E, Amthauer G (1995) Mössbauer spectroscopic and x-ray powder diffraction studies of synthetic micas on the join annite KFe3AlSi3O10(OH)2-phlogopite KMg3AlSi3O10(OH)2. Phys Chem Miner 22:282–294. https://doi.org/10.1007/BF00202768
Redhammer GJ, Beran A, Schneider J et al (2000) Spectroscopic and structural properties of synthetic micas on the annite-siderophyllite binary: synthesis, crystal structure refinement, Mössbauer, and infrared spectroscopy. Am Mineral 85:449–465. https://doi.org/10.2138/am-2000-0406
Redhammer GJ, Amthauer G, Lottermoser W et al (2005) X-ray powder diffraction and 57Fe – Mössbauer spectroscopy of synthetic trioctahedral micas {K}[Me3]⟨TSi3⟩O10(OH)2, Me = Ni2+, Mg2+, Co2+, Fe2+; T = Al3+, Fe3+. Mineral Petrol 85:89–115. https://doi.org/10.1007/s00710-005-0096-2
Robie RA, Hemingway BS (1984) Heat capacities and entropies of phlogopite (KMg3[AlSi3O10](OH)2) and paragonite (NaAl2[AlSi3O10](OH)2) between 5 and 900 K and estimates of the enthalpies and Gibbs free energies of formation. Am Mineral 69:858–868
Rodriguez-Carvajal J (2001) Recent developments of the program FULLPROF, commission on powder diffraction. IUCr Newsl 26:334
Sack RO (1980) Some constraints on the thermodynamic mixing properties of Fe–Mg orthopyroxenes and olivines. Contrib Mineral Petrol 71:257–269. https://doi.org/10.1007/BF00371667
Sack RO, Ghiorso MS (1989) Importance of considerations of mixing properties in establishing an internally consistent thermodynamic database: thermochemistry of minerals in the system Mg2SiO4-Fe2SiO4-SiO2. Contrib Mineral Petrol 102:41–68. https://doi.org/10.1007/BF01160190
Salje EKH, Carpenter MA, Malcherek T, Boffa Ballaran T (2000) Autocorrelation analysis of infrared spectra from minerals. Eur J Mineral 12:503–519. https://doi.org/10.1127/0935-1221/2000/0012-0503
Schulien S (1980) Mg-Fe partitioning between biotite and a supercritical chloride solution. Contrib Mineral Petrol 74:85–93. https://doi.org/10.1007/BF00375492
Sluiter MHF, Kawazoe Y (2002) Prediction of the mixing enthalpy of alloys. Europhys Lett 57:526. https://doi.org/10.1209/epl/i2002-00493-3
Tajčmanová L, Connolly JAD, Cesare B (2009) A thermodynamic model for titanium and ferric iron solution in biotite. J Metamorph Geol 27:153–165
Tarantino SC, Carpenter MA, Domeneghetti MC (2003) Strain and local heterogeneity in the forsterite–fayalite solid solution. Phys Chem Minerals 30:495–502. https://doi.org/10.1007/s00269-003-0357-8
White P, Holland W (2000) The effect of TiO2 and Fe2O3 on metapelitic assemblages at greenschist and amphibolite facies conditions: mineral equilibria calculations in the system K2O–FeO–MgO–Al2O3–SiO2–H2O–TiO2–Fe2O3. J Metamorph Geol 18:497–511. https://doi.org/10.1046/j.1525-1314.2000.00269.x
White RW, Powell R, Holland TJB (2007) Progress relating to calculation of partial melting equilibria for metapelites. J Metamorph Geol 25:511–527
White RW, Powell R, Holland TJB et al (2014a) New mineral activity-composition relations for thermodynamic calculations in metapelitic systems. J Metamorph Geol 32:261–286. https://doi.org/10.1111/jmg.12071
White RW, Powell R, Johnson TE (2014b) The effect of Mn on mineral stability in metapelites revisited: new a - x relations for manganese-bearing minerals. J Metamorph Geol 32:809–828. https://doi.org/10.1111/jmg.12095
Wiser NM, Wood BJ (1991) Experimental determination of activities in Fe−Mg olivine at 1400 K. Contrib Mineral Petrol 108:146–153. https://doi.org/10.1007/BF00307333
Wohl K (1946) Thermodynamic evaluation of binary and ternary liquid systems. Trans Am Inst Chem Eng 42:215–249
Wohl K (1953) Thermodynamic evaluation of binary and ternary liquid systems. Chem Eng Prog 49:218–221
Wones DR (1972) Stability of biotite: a reply. Am Mineral 57:316–317
Wones DR, Eugster HP (1965) Stability of biotite: experiment, theory and application. Am Mineral 50:1228–1272
Wood BJ, Kleppa OJ (1981) Thermochemistry of forsterite-fayalite olivine solutions. Geochim Cosmochim Acta 45:529–534. https://doi.org/10.1016/0016-7037(81)90185-X
Zhou F (1994) 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
Zhou F, Cococcioni M, Marianetti CA, Morgan D, Ceder G (2004) First-principles prediction of redox potentials in transition-metal compounds with LDA+U. Phys Rev B 70:235121
Acknowledgements
This work was financed by the Austrian Science Fund (FWF) project P28724, which is gratefully acknowledged. We thank E. Forsthofer and colleagues for their professional work in implementing and maintaining the Materials Studio software at the Department of Computer Sciences, Salzburg University, and G. Tippelt for collecting the XRPD patterns. D. Henry (Louisiana State University, USA) is thanked for providing the mineral-chemical data of biotites from Maine. The paper benefited from constructive reviews by R. Sack, B. Myhill, and an anonymous reviewer.
Funding
Open Access funding provided by Paris Lodron University of Salzburg.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Mark S Ghiorso.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary Information
Below is the link to the electronic supplementary material.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Dachs, E., Benisek, A. A new activity model for Fe–Mg–Al biotites: I—Derivation and calibration of mixing parameters. Contrib Mineral Petrol 176, 22 (2021). https://doi.org/10.1007/s00410-020-01770-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00410-020-01770-5