A new activity model for biotite and its application

A new activity model for biotite is formulated in the system K₂O-FeO-MgO-Al₂O₃-SiO₂-H₂O-TiO₂-O₂ (KFMASHTO), which extends that for the KFMASH system by introducing a titanium-biotite and a ferric-biotite end-member (tbio: K(TiMg₂)[(O)₂(AlSi₃)O₁₀] and fbio: K(Fe³⁺Mg₂)[(OH)₂(Al₂Si₂)O₁₀]), as well as a pyrophyllite end-member (pyp: Al₂[(OH)₂Si₄O₁₀]) that accounts for the presence of octahedral excess-Al in natural biotites. Phonon calculations applying density functional theory (DFT) using the software Castep yielded the standard entropies of tbio and fbio as S^o_tbio = 328.06 J/(mol·K) and S^o_fbio = 301.69 J/(mol·K), and their heat capacity functions. From experimental phase-equilibrium data, the enthalpy of formation value of tbio was constrained as \(\Delta {H}_{f,tbio}^{o}\) = −6124.68 ± 3.33 kJ/mol. Natural data were used to derive \(\Delta {H}_{f,fbio}^{o}\)= −5935.3 ± 6.6 kJ/mol. The single-defect DFT method was applied to parameterize important macroscopic mixing properties (macro-W’s) involving tbio and pyp end-members in the model (fbio was treated ideal). Castep-derived microscopic interaction energies (micro-w’s) are presented herein for KFMASH-biotite. The octahedral same-site (M1) Mg–Al mixing micro-w (w_MgAl(M1)), the same-site tetrahedral Si-Al mixing parameter (w_SiAl(T1)) and the related cross-site term are: w_MgAl(M1) = 82.5 kJ/mol, w_SiAl(T1) = 95.6 kJ/mol (two T1-sites) and \({w}_{MgAlAlSi(M1T1)}=\) 175.1 kJ/mol. The linear combination of these micro-w’s gives a macroscopic W_phleas = 18.8 kJ/mol, that is not transferable to other mineral groups. Micro w’s for Mg-Fe mixing in biotite (w_MgFe(M1), w_MgFe(M2), \({w}_{MgMgFeFe(M1M2)}\)), are all close to ideality. The biotite activity model of this study is thus a first example of next-generation activity models that use DFT- and thus physically based micro-w’s and reassembled macro-W’s for petrological calculations. Test calculations on 5 samples from low- to high-grade metamorphic environments covering metapelite to greywacke bulk-compositions using Perple_X suite of programs illustrate the performance of the new biotite activity model. Computed mineral-chemistries are in all cases in better agreement with measured compositions than resulting from published activity models of biotite.