Coordinated Hard Sphere Mixture (CHaSM): A simplified model for oxide and silicate melts at mantle pressures and temperatures

Abstract

We develop a new model to understand and predict the behavior of oxide and silicate melts at extreme temperatures and pressures, including deep mantle conditions like those in the early Earth magma ocean. The Coordinated Hard Sphere Mixture (CHaSM) is based on an extension of the hard sphere mixture model, accounting for the range of coordination states available to each cation in the liquid. By utilizing approximate analytic expressions for the hard sphere model, this method is capable of predicting complex liquid structure and thermodynamics while remaining computationally efficient, requiring only minutes of calculation time on standard desktop computers. This modeling framework is applied to the MgO system, where model parameters are trained on a collection of crystal polymorphs, producing realistic predictions of coordination evolution and the equation of state of MgO melt over a wide range of pressures and temperatures. We find that the typical coordination number of the Mg cation evolves continuously upward from 5.25 at 0 GPa to 8.5 at 250 GPa. The results produced by CHaSM are evaluated by comparison with predictions from published first-principles molecular dynamics calculations, indicating that CHaSM is accurately capturing the dominant physics controlling the behavior of oxide melts at high pressure. Finally, we present a simple quantitative model to explain the universality of the increasing Grü neisen parameter trend for liquids, which directly reflects their progressive evolution toward more compact solid-like structures upon compression. This general behavior is opposite that of solid materials, and produces steep adiabatic thermal profiles for silicate melts, thus playing a crucial role in magma ocean evolution.

Density of magmas at depth

“…Above 35 GPa, Si coordination change and most others are completed, leaving bond compressibility as the main compression mechanism, as in crystals. Solid-like behavior has been proposed for close-packed silicate liquids at very high pressures (Wolf et al, 2015). Magmas density and compressibility are therefore expected to be similar to or not very far from that of crystalline phases, as can be seen from the comparison between molten and crystalline basalt (Fig.3).…”

“…Arguments for a fully molten mantle are: 1) the remnants of the magma ocean atop the core in ULVZs (Williams and Garnero, 1996), 2) the much steeper adiabatic profiles of silicate liquids at extreme pressures than the liquidus (Stixrude et al, 2009;Asimow and Ahrens, 2010;Wolf et al, 2015), 3) the amount of energy released upon impact (Tonks and Melosh, 1993) that is sufficient to melt the whole mantle. Arguments for a partial magma ocean are essentially the depth of equilibrium between molten silicates and molten iron estimated from siderophile elements partitioning at depth, that range between 40 and 60 GPa (Li and Agee, 2001;Bouhifd and Jephcoat, 2011;Siebert et al, 2013), and the incomplete homogeneization of the mantle for noble gas isotopes (Tucker and Mukhopadhyay, 2014).…”

An equation of state for high pressure-temperature liquids (RTpress) with application to MgSiO 3 melt

“…In particular, it must allow for an initially increasing γ value with compression. This general behavior of liquids was discussed in detail in Wolf et al (2015) and a semi-quantitative mechanism was proposed, relating to structural evolution in the liquid, which affects the representative vibrational frequency of the material. Upon compression, the liquid re-orders itself, adopting more compact and solid-like structures.…”

“…The finite strain γ-model, initially developed by Stixrude and Lithgow-Bertelloni (2005) as a more physically-motivated model for solids, however, guarantees the presence of a turnover when applied to liquids, where the Grüneisen parameter always returns to solid-like behavior after sufficient compression. This finite-strain model rests upon the variation of a characteristic vibrational frequency with compression, which was shown by Wolf et al (2015) to naturally capture the entropic properties of MgO melt while automatically incorporating the physically required selflimiting property. To test out the affect of our selection of the finite-strain γ-model, we compare our preferred model to one using the standard powerlaw formulation, which yields a considerably worse fit to the internal energy curves with energy residuals that are ∼3.5 times larger (15 meV/atom).…”

“…The thermophysical properties of these high-pressure molten silicates strongly influence the convection and cooling evolution of the resulting deep magma oceans. Properties such as density, heat capacity, thermal expansion, and compressibility are all more sensitive to pressure and temperature for liquids than their corresponding solid phases, due to their continuous structural evolution (Wolf et al, 2015;Jing and Karato, 2011). The evolutionary path of magma oceans rests on the coupled interactions between thermodynamics and fluid dynamics (e.g.…”