An Internal Energy dependent Grüneisen Parameter Model From Molecular Dynamics Simulations

The Grüneisen parameter (γ) is an essential physical quantity in the study of the thermodynamics of geological materials. According to the widely-used Mie-Grüneisen model,γ depends only on volume (V). However, this approximation lacks theoretical justification in the case of liquids. Experiments do not yet have the precision to test whether γ of silicate liquids has a non-negligible temperature dependence at mantle conditions. If in fact the Mie-Grüneisen approximation is incorrect, then an accurate Grüneisen parameter model that depends on parameters in addition to V is needed.

In this study, we calculated the γ of the eutectic mixture of Anorthite and Diopside liquids in the range of 1500 K to 6500 K and 0 GPa to 150 GPa using both empirical and ab initio molecular dynamics (MD) simulations. Empirical simulation used the Matsui potentials and 1108 atoms in the LAMMPS code; ab initio simulations used PBE-D3 functionals and 277 atoms with an energy cutoff of 520 eV and were run in VASP. We found that the most accurate way to extract γ from simulation output was finite differences between equal volume states while stepping the internal energy (E) in the NVE ensemble. Based on the results, the temperature or E dependence of γ was confirmed and a more refined Grüneisen model with both volume and internal energy dependence was proposed. In our γ(V,E) model, each γ-E isochore has a linear relationship with internal energy and all the γ-E isochores radiate from a single point where the γ is approximately 1. The slopes of γ-E isochores are also approximately linear with respect to V. We then applied the new γ(V,E) model to fit the data from shock experiments on Anorthite-Diopside eutectic mixture. Compared to the old γ(V) model, our new γ(V,E) model captures the trend of the Hugoniot and the offset between hot and cold Hugoniots more accurately.

We also calculated the Wu-Jing parameter (R=P(∂V/∂H)_P) of the same system. R is not, unfortunately, a simple function of pressure only and the enthalpy dependence of R cannot be described by any function as simple as that we proposed for γ. Therefore, it is suggested that we should continue to use thermal pressure in the equation of state rather than switch to the thermal volume description used by the Rice-Walsh or enthalpy equation of state.