A new activity model for Fe–Mg–Al biotites: I—Derivation and calibration of mixing parameters.

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 X_Mg = 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]₂^M2[Al_0.5Si_0.5]₂^T1[Si]₂^T2O₁₀(OH)₂, phlogopite (Phl, K[Mg]^M1[Mg]₂^M2[Al_0.5Si_0.5]₂^T1[Si]₂^T2O₁₀(OH)₂, ordered Fe–Mg biotite (Obi, K[Fe]^M1[Mg]₂^M2[Al_0.5Si_0.5]₂^T1[Si]₂^T2O₁₀(OH)₂, ordered eastonite (Eas, K[Al]^M1[Mg]₂^M2[Al]₂^T1[Si]₂^T2O₁₀(OH)₂, and disordered eastonite (Easd, K[Al_1/3Mg_2/3]^M1[Al_1/3Mg_2/3]₂^M2[Al]₂^T1[Si]₂^T2O₁₀(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]₂^M2[Al]₂^T1[Si]₂^T2O₁₀(OH)₂), and annite–eastonite joins were synthesized hydrothermally at 700 °C, 4 kbar and logf_O2 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 (C_p) at temperatures from 2 to 300 K. The measured C_p/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 (W^G_ij = W^H_ij). 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 W^H_AnnPhl = 14.3 ± 3.4 kJ/mol and W^H_PhlAnn = −8.8 ± 8.0 kJ/mol (3-cation basis). The resulting asymmetric molar excess Gibbs free energy (G_ex) 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 ∆H_{Fe–Mg order}, indicating that Fe–Mg ordering in biotite is weak. The large enthalpy change of ∆H_{Mg-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 W^H_PhlEas = 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. [ABSTRACT FROM AUTHOR]