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.

制定了一种新的Fe-Mg-Al黑云母活动模型，将Mg-Al黑云母的活动模型（Dachs和Benisek，Contrib Mineral Petrol 174：76，2019）扩展到了K ₂ O-FeO-MgO-Al ₂ O _3。 -SiO ₂ -H ₂ O（KFMASH）系统。它具有两个组成变量X _Mg  = Mg /（Mg + Fe ²⁺）和八面体Al，以及Fe–Mg和Mg–Al有序变量，产生五个线性独立的端基：：铁矿（Ann，K [Fe] ^M1 [Fe ] ₂ ^M2 [Al _0.5 Si _0.5 ] ₂ ^T1 [Si] ₂ ^T2 O ₁₀（OH）₂，金云母（Phl，K [Mg] ^M1 [Mg] ₂ ^M2 [Al _0.5 Si _0.5 ] ₂ ^T1 [Si] ₂ ^T2 O ₁₀（OH）₂，有序的Fe–Mg黑云母（Obi，K [Fe] ^M1 [ Mg] ₂ ^M2 [Al _0.5 Si _0.5 ] ₂ ^T1 [Si] ₂ ^T2 O ₁₀（OH）₂，有序膨润土（Eas，K [Al] ^M1 [Mg] ₂ ^M2 [Al] ₂ ^T1 [Si] ₂ ^T2 O₁₀（OH）₂和无序的膨润土（Easd，K [Al _1/3 Mg _2/3 ] ^M1 [Al _1/3 Mg _2/3 ] ₂ ^M2 [Al] ₂ ^T1 [Si] ₂ ^T2 O ₁₀（OH ）₂。用于参数化模型混合特性的方法有：量热法，分析现有的相平衡数据，粉末吸收红外（IR）光谱中的谱线扩展以及密度泛函理论（DFT）计算。研究，沿黑铁矿-金云母，铁矿-铁橄榄石（Sid，K [Al] ^M1 [Fe] ₂^M2 [Al] ₂ ^T1 [Si] ₂ ^T2 O ₁₀（OH）₂），以及在700°C，4 kbar和logf _O2约为− 20.2的水热条件下水合合成了堇青石-锡石连接，接近于白钨矿的氧化还原条件–在该P – T条件下的磁铁矿氧气缓冲液。通过X射线粉末衍射（XRPD），能量色散扫描电子显微探针分析，粉末吸收红外光谱和光学显微镜对样品进行表征。样品使用松弛量热法以测量它们的热容（进一步研究Ç _p在温度为2到300 K的）测量Ç _p然后对/ T进行积分，得到298.15 K时样品的量热（振动）熵。当绘制这三个二进制文件的成分函数时，它们表现出线性行为。因此，对于重要的黑云母连接，混合的过量熵为零。对于三个二进制Phl-Ann，Ann-Sid和Ann-Eas，混合量在误差范围内也为零。因此，KFMASH黑云母具有过多的焓，它们与压力和温度无关（W ^G _ij  =  W ^H _ij）。用最小二乘法对已发表的有关黑云母与橄榄石之间Fe-Mg交换的实验数据进行热力学分析，并结合金云母+石英稳定性的相平衡数据以及黑云母在黑云母中Al饱和度的实验数据。组合黑云母-硅线石-山梨糖-石英-H ₂ O以约束焓混合参数并导出黑云母端成员的形成焓。对于最重要的二元黑云母中的Fe-Mg混合，这给出了最合适的非对称Margules焓参数，W ^H _AnnPhl  = 14.3±3.4 kJ / mol，W ^H _PhlAnn  = -8.8±8.0 kJ / mol（3-阳离子） ）。产生的不对称摩尔过量吉布斯自由能（G_ex）仅偏离理想状态，富铁成分为负，富镁成分为正。因此，对于Ann-Phl二元表示了接近理想的活动-组成关系。DFT计算通常确定了反应中焓变目前使用的− 2 kJ / mol的低值2/3 Phl + 1/3 Ann = Obi，对于∆H _Fe–，该值给出了− 2±3 kJ / mol。_{镁的有序性}，表明黑云母中铁镁的有序性较弱。Δ的大焓变ħ_{的Mg-Al病症} 为Mg和Al的对的Eas（达克斯和Benisek 2019）的M个地点的无序= 34.5千焦/摩尔，通过附加的DFT计算重新确认。与W ^H _PhlEas组合 = 10 kJ / mol，这是该研究描述沿Phl-Eas连接的混合的首选值，预计仅在> 1000°C的高温下，黑云母M位置上的Mg-Al无序化才会显着。相反，它在没有变质作用P -牛逼的设置。