Constant-pressure constant-temperature {it ab initio} molecular dynamics simulations at high temperatures have been used to study MgSiO$_3$ liquid, the major constituent of the Earths lower mantle to conditions of the Earths core-mantle boundary (CMB). We have performed variable-cell {it ab initio} molecular dynamic simulations at relevant thermodynamic conditions across one of the measured melting curves. The calculated equilibrium volumes and densities are compared with the simulations using an orthorhombic perovskite configuration under the same conditions. For molten MgSiO$_3$, we have determined the diffusion coefficients and shear viscosities at different thermodynamic conditions. Our results provide new constraints on the properties of molten MgSiO$_3$ at conditions near the core-mantle boundary. The volume change on fusion is positive throughout the pressure-temperature conditions examined and ranges from 5% at 88 GPa and 3500 K to 2.9% at 120 GPa and 5000 K. Nevertheless, neutral or negatively buoyant melts from (Mg,Fe)SiO$_3$ perovskite compositions at deep lower mantle conditions are consistent with existing experimental constraints on solid-liquid partition coefficients for Fe. Our simulations indicate that MgSiO$_3$ is liquid at 120 GPa and 4500 K, consistent with the lower range of experimental melting curves for this material. Linear extrapolation of our results indicates that the densities of liquid and solid perovskite MgSiO$_3$ will become equal near 180 GPa.
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