An internal energy-dependent model for the Grüneisen parameter of silicate liquids

We investigated the accuracy of the Mie-Grüneisen approximation, which treats the Grüneisen parameter (γ) as a one-parameter function of volume, for use in describing the thermal equation of state of a silicate liquid. For this study, we focused on a single composition: the diopside-anorthite eutectic, an Fe-free basalt analog that has been extensively studied by shock wave experiments. We tuned an empirical force-field to a small set of ab initio N-V-T molecular dynamics simulations to ensure that it reproduces pressure, heat capacity, and γ at high and low pressures. We then used empirical force-field molecular dynamics simulations in a larger system and for longer run times to ensure accurate extraction of γ at numerous N-V-E state points. To first order, the results show the expected volume-dependence for silicate liquids, with γ increasing as volume decreases. However, there are also significant and systematic variations of γ with internal energy (E) at constant volume. We propose a simple model form that captures the volume and E dependence of γ with only one more free parameter than a typical Mie-Grüneisen formulation. We demonstrate the utility of this new model for well-constrained fitting to sparse shock wave experiment data below 200 GPa, obtaining a marked improvement in the ability to simultaneously fit the pre-heated liquid Hugoniot and points substantially offset from this Hugoniot.