A new activity model for Mg–Al biotites determined through an integrated approach

A new activity model for Mg–Al biotites was formulated through an integrated approach combining various experimental results (calorimetry, line-broadening in infrared (IR) spectra, analysis of existing phase-equilibrium data) with density functional theory (DFT) calculations. The resulting model has a sound physical-experimental basis. It considers the three end-members phlogopite (Phl, KMg(3)[(OH)(2)AlSi(3)O(10)]), ordered eastonite (Eas, KMg(2)Al[(OH)(2)Al(2)Si(2)O(10)]), and disordered eastonite (dEas) and, thus, includes Mg–Al order–disorder. The DFT-derived disordering enthalpy, ΔH(dis), associated with the disordering of Mg and Al on the M sites of Eas amounts to 34.5 ± 3 kJ/mol. Various biotite compositions along the Phl–Eas join were synthesised hydrothermally at 700 °C and 4 kbar. The most Al-rich biotite synthesized had the composition X(Eas) = 0.77. The samples were characterised by X-ray diffraction (XRD), microprobe analysis and IR spectroscopy. The samples were studied further using relaxation calorimetry to measure their heat capacities (C(p)) at temperatures from 2 to 300 K and by differential scanning calorimetry between 282 and 760 K. The calorimetric (vibrational) entropy of Phl at 298.15 K, determined from the low-T C(p) measurements on a pure synthetic sample, is S(cal) = 319.4 ± 2.2 J/(mol K). The standard entropy, S(o), for Phl is 330.9 ± 2.2 J/(mol K), which is obtained by adding a configurational entropy term, S(cfg), of 11.53 J/(mol K) due to tetrahedral Al-Si disorder. This value is ~1% larger than those in different data bases, which rely on older calorimetrical data measured on a natural near-Phl mica. Re-analysing phase-equilibrium data on Phl + quartz (Qz) stability with this new S(o), gives a standard enthalpy of formation of Phl, [Formula: see text] = − 6209.83 ± 1.10 kJ/mol, which is 7–8 kJ/mol less negative than published values. The superambient C(p) of Phl is given by the polynomial [J/(mol K)] as follows: [Formula: see text] . Calorimetric entropies at 298.15 K vary linearly with composition along the Phl–Eas join, indicating ideal vibrational entropies of mixing in this binary. The linear extrapolation of these results to Eas composition gives S(o) = 294.5 ± 3.0 J/(mol K) for this end-member. This value is in excellent agreement with its DFT-derived S(o), but ~ 8% smaller than values as appearing in thermodynamic data bases. The DFT-computed superambient C(p) of Eas is given by the polynomial [in J/(mol K)] as follows: [Formula: see text] . A maximum excess enthalpy of mixing, ΔH(ex), of ~6 kJ/mol was derived for the Phl–Eas binary using line-broadening from IR spectra (wavenumber region 400–600 cm(−1)), which is in accordance with ΔH(ex) determined from published solution-calorimetry data. The mixing behaviour can be described by a symmetric interaction parameter [Formula: see text] = 25.4 kJ/mol. Applying this value to published phase-equilibrium data that were undertaken to experimentally determine the Al-saturation level of biotite in the assemblage (Mg–Al)-biotite-sillimanite-sanidine-Qz, gives a [Formula: see text] = − 6358.5 ± 1.4 kJ/mol in good agreement with the independently DFT-derived value of [Formula: see text] = − 6360.5 kJ/mol. Application examples demonstrate the effect of the new activity model and thermodynamic standard state data, among others, on the stability of Mg–Al biotite + Qz. ELECTRONIC SUPPLEMENTARY MATERIAL: The online version of this article (10.1007/s00410-019-1606-2) contains supplementary material, which is available to authorized users.