Mantle Convection and Surface Expressions. Группа авторов

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      Based on the volume fractions v_i = x_iV_i/V, the elastic properties of end members can then be combined according to one of the averaging schemes introduced in Section 3.2 to approximate the elastic behavior of the solid solution. Mineral‐physical databases compile elastic and thermodynamic properties for many end members of mantle minerals (Holland et al., 2013; Stixrude & Lithgow‐Bertelloni, 2011). However, the physical properties of some critical end members remain unknown, either because they have not been determined at relevant pressures and temperatures or because the end members are not stable as pure compounds.

      Bridgmanite is believed to be the most abundant mineral in the lower mantle and adopts a perovskite crystal structure. Most bridgmanite compositions can be expressed as solid solutions of the end members MgSiO₃, FeSiO₃, Al₂O₃, and FeAlO₃. Of these end members, only MgSiO₃ is known to have a stable perovskite‐structured polymorph at pressures and temperatures of the lower mantle. FeSiO₃ perovskite is unstable with respect to the post‐perovskite form of FeSiO₃ or a mixture of the oxides FeO and SiO₂ (Caracas and Cohen, 2005; Fujino et al., 2009). Al₂O₃ transforms from corundum to a Rh₂O₃ (II) structured polymorph instead of adopting a perovskite structure (Funamori and Jeanloz, 1997; Kato et al., 2013; Lin et al., 2004). For FeAlO₃, both perovskite and Rh₂O₃ (II) structures have been proposed (Caracas, 2010; Nagai et al., 2005). Although first‐principle calculations can access the elastic properties of compounds in thermodynamically unstable structural configurations, for example for FeSiO₃ and Al₂O₃ in perovskite structures (Caracas & Cohen, 2005; Muir & Brodholt, 2015a; Stackhouse et al., 2005a, 2006), it is unclear to which extent the results are useful for modeling the elastic properties of complex solid solutions that do not extend towards those end member compositions and might be affected by deviations from ideal mixing behavior at intermediate compositions.

      When the physical properties of end members are unknown, it might still be possible to approximate the elastic properties of a solid solution as long as the effects of different chemical substitutions on elastic properties are captured by available experiments or computations on intermediate compositions of the solid solution. Let x_im be the molar fractions of end member i for a set of intermediate compositions for which volumes and elastic properties are known. For each composition m of this set, the molar fractions x_im of all end members form a composition vector x_m of dimension n equal to the total number of end members. If the set of compositions x_m forms a vector basis of ℝⁿ, it is possible to construct a matrix M <span class="dbond"></span> {x_im} that contains the vectors x_m as columns. Any composition vector x of the solid solution can then be transformed into a vector y by using the inverse matrix: y = M⁻¹x. The components of the vector y express the composition x in terms of molar fractions y_m of the intermediate compositions x_m. In this way, the compositions x_m are combined to match the required composition, and their volumes and elastic properties can be combined according to the mixing laws introduced above. It is important to note that this type of mixing is strictly valid only within the compositional limits defined by the compositions x_m that may not cover the full compositional space as spanned by the end members. When volumes can be assumed to mix linearly across the entire solid solution, however, it is possible to extrapolate volumes beyond these compositional limits. The extrapolation of elastic properties requires special caution since negative molar fractions y_m < 0 can lead to unwanted effects when computing bounds on elastic moduli.

Figure 3.3 illustrates the uncertainties that arise from mixing the elastic properties of bridgmanite compositions. P‐ and S‐wave velocities were calculated for bridgmanite solid solutions in the systems MgSiO₃‐Al₂O₃‐FeAlO₃ and MgSiO₃‐Al₂O₃‐FeSiO₃ at 40 GPa and 2000 K using available high‐pressure experimental data on different bridgmanite compositions (Chantel et al., 2012; Fu et al., 2018; Kurnosov et al., 2017; Murakami et al., 2012, 2007) that provided the basis of composition vectors in the approach outlined in the previous paragraph. For all compositions, I adopted the thermoelastic properties given by Zhang et al. (2013). Uncertainties on P‐ and S‐wave velocities due to mixing were estimated as the differences that arise from combining the elastic properties of bridgmanite compositions according to either the Voigt or the Reuss bound relative to velocities of the Voigt‐Reuss‐Hill average, i.e., dlnv = (v^V − v^R)/v^VRH. For bridgmanite compositions that are similar to the compositions studied in experiments and used here to compute P‐ and S‐wave velocities, СКАЧАТЬ