Fundamental thermodynamic relations and silicate melting with implications for the constitution of D

TLDR: In this article, the authors describe fundamental thermodynamic relations (Helmholtz free energy as a function of volume and temperature) for solids and liquids, simple physically based expressions which contain all thermodynamic information about a system.

Abstract: We describe fundamental thermodynamic relations (Helmholtz free energy as a function of volume and temperature) for solids and liquids, simple physically based expressions which contain all thermodynamic information about a system. The solid fundamental relation consists of Debye and Birch-Murnaghan finite-strain theory combined in the Mie-Gruneisen framework. The liquid fundamental relation is derived by taking the high-temperature limit of the solid expression. We derive the liquid equation of state, which contains only four parameters, from the liquid fundamental relation and show that it successfully describes measurements of liquid alkali metals, water, and liquid diopside over a wide range of pressure and temperature. We find optimal fundamental relation parameters for diopside, enstatite, ilmenite, and perovskite and find the solid relation to be in excellent agreement with data, including heat capacities, thermal expansion, and MgSiO3 phase equilibria. We then combine the liquid and solid fundamental relations to calculate the melting curves of diopside, enstatite, and perovskite, which are found to be in excellent agreement with experiment. All predicted melting curves have dT/dP slopes which decrease steadily with pressure, eventually becoming negative because of liquid-crystal density inversion. Our predicted melting temperature of perovskite in the D″ region (3750 K) at the base of the mantle is thousands of degrees lower than previous estimates, yet it is consistent with experimental data. The predicted melting curve, although consistent with the lack of widespread melting in the lower mantle, is much lower than recently proposed geotherms in the D″ layer at the base of the mantle. By combining our results with seismic observations of the deep mantle, we propose that the D″ layer consists of magnesiowustite and silica in the form of stishovite or its recently discovered high-pressure modification.