The elastic solid solution model for minerals at high pressures and temperatures

Abstract

Non-ideality in mineral solid solutions affects their elastic and thermodynamic properties, their thermobaric stability, and the equilibrium phase relations in multiphase assemblages. At a given composition and state of order, non-ideality in minerals is typically modelled via excesses in Gibbs free energy which are either constant or linear with respect to pressure and temperature. This approach has been extremely successful when modelling near-ideal solutions. However, when the lattice parameters of the solution endmembers differ significantly, extrapolations of thermodynamic properties to high pressures using these models may result in significant errors. In this paper, I investigate the effect of parameterising solution models in terms of the Helmholtz free energy, treating volume (or lattice parameters) rather than pressure as an independent variable. This approach has been previously applied to models of order-disorder, but the implications for the thermodynamics and elasticity of solid solutions have not been fully explored. Solid solution models based on the Helmholtz free energy are intuitive at a microscopic level, as they automatically include the energetic contribution from elastic deformation of the endmember lattices. A chemical contribution must also be included in such models, which arises from atomic exchange within the solution. Derivations are provided for the thermodynamic properties of n-endmember solutions. Examples of the use of the elastic model are presented for the alkali halides, pyroxene, garnet, and bridgmanite solid solutions. Elastic theory provides insights into the microscopic origins of non-ideality in a range of solutions, and can make accurate predictions of excess enthalpies, entropies, and volumes as a function of volume and temperature. In solutions where experimental data are sparse or contradictory, the Helmholtz free energy approach can be used to assess the magnitude of excess properties and their variation as a function of pressure and temperature. The formulation is expected to be useful for geochemical and geophysical studies of the Earth and other planetary bodies.

Elastic Properties of the Pyrite‐Type FeOOH‐AlOOH System From First‐Principles Calculations

“…This allowed for the direct comparison of the static calculations of the velocities of bridgmanite (Wentzcovitch et al, 2004) against those of (Al,Fe)O 2 H, in the absence of additional thermoelastic calculations, enabling first-order estimates of the velocity and density contrasts produced by intermediate compositions of pyrite-type (Al,Fe)O 2 H at lower mantle pressures. Although increasing attention has been paid recently to how non-ideality of solid solutions affect their elastic properties (e.g., Myhill, 2018), the THOMPSON ET AL.…”

“…This allowed for the direct comparison of the static calculations of the velocities of bridgmanite (Wentzcovitch et al., 2004) against those of (Al,Fe)O 2 H, in the absence of additional thermoelastic calculations, enabling first‐order estimates of the velocity and density contrasts produced by intermediate compositions of pyrite‐type (Al,Fe)O 2 H at lower mantle pressures. Although increasing attention has been paid recently to how non‐ideality of solid solutions affect their elastic properties (e.g., Myhill, 2018), the similar chemistry, structure, and unit cell volumes of the Fe‐ and Al‐endmembers suggest that non‐ideality in the high‐pressure (Al,Fe)OOH system is minimal. For example, experimentally determining unit cell volumes of CaCl 2 ‐type (Al,Fe)OOH trend linearly with measured Fe/(Fe + Al) ratios at ambient (i.e., fixed) pressure and temperature.…”

Research on Mechanical Properties of High-Pressure Anhydrite Based on First Principles

“…A three-dimensional model of anhydrite (CaSO 4 ) crystal was first established. Because of the orthogonality of anhydrite crystals, there are nine independent elastic constants, which can be obtained through the following formula [18,19]:…”

“…Taking the wave vector k directions as [100], [001] and [110], and substituting Equation (19) into Equation (18) to obtain the longitudinal wave velocity and transverse wave velocity in each direction yields the following:…”

Beyond the Vegard's law: solid mixing excess volume and thermodynamic potentials prediction, from end-members

“…supercell's size determination, and ii) exploring the related disordered atomic arrangements. In this view, being able to infer some fundamental properties of a solid mixing using only the EMs would result in a significant enhancement to modelling and understanding such systems [21,22].…”