How the HPx-eos are made

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This page gives a nuts-and-bolts description of how the HPx-eos are put together, which should be of interest to the general user. Downloads of technical information are included for those interested in implementing the HPx-eos. See also: details of the individual HPx-eos families, references to the dataset and HPx-eos papers, and practical information for users.

On this page:

Building a mineral HPx-eos

Construction of an HPx-eos for a solid solution phase begins with deciding which mixing sites to consider, and which ions will be allowed to mix on them. On this basis, an independent set of end-members are chosen which can usually be subdivided into compositional and ordered end-members. Compositional end-members are chosen to allow the x-eos to encompass the desired compositional range, while ordered end-members are used to describe how ions are distributed over mixing sites when there is more than one option for doing this.

We then need:

  1. a recipe for making the thermodynamic properties of each end-member;
  2. activity-composition relations, to describe the thermodynamic properties associated with mixing of end-members.

Making end-members

A recipe is needed for the thermodynamic properties of each end-member in the solid solution – specifically the end-member’s Gibbs energy, G, as a function of pressure and temperature. Fundamentally, such end-members are built on the thermal-pressure equation of state used in the Holland & Powell (2011) internally-consistent dataset, and the tabulated and optimised properties within it.

Compositional end-members

Many compositional end-members already appear in the dataset, in which case the dataset end-member is almost always adopted directly. For example:

Recipe for diopside (di; C2/c):   "take 1 diopside from the dataset".

Many other compositional end-members, however, do not appear in the dataset. These are made from linear combinations of dataset end-members, along with an extra energetic term, ∆G, describing the difference between the linear combination of dataset end-members and the true HPx-eos end-member. For example:

Recipe for high-temperature clinoenstatite (cenh; C2/c):
“take 1 ortho-enstatite (Pbca) from the dataset
+ an energetic term describing the Pbca→C2/c transition”.

Recipe for Cr-diopside (crdi):
“take 1 disordered Ca-tschermaks pyroxene
+ 1 kosmochlor,
– 1 jadeite from the dataset,
+ an energetic term as needed to approximate the
true G(P,T) for Cr-diopside”.

The additional energetic term ∆G is always linear in P and T: ∆G = a + b T + c P, providing a minimal number of parameters to calibrate. Clearly, in the recipe for high-temperature clinoenstatite above, the ∆G term has a well-defined physical meaning, while the use of dataset ortho-enstatite in the recipe is likely to provide a reasonable approximation to the true, non-linear P– and T-dependence of G(cenh). Conversely, the choice of end-members in the recipe for Cr-diopside is arbitrary (though they must generate the right composition), and the ∆G term simply represents the difference between G for this arbitrary combination of unrelated end-members, and the true value. End-members created by such a process only ever appear in small proportions, and the ∆G term is never well-known.

Occasionally a ∆G term is added to an end-member that can be simply taken from the dataset (∆G(mod)). This is done, usually for end-members that are never present in substantial proportions, or for an end-member for which the dataset calibration is particularly in doubt. See the appropriate published paper for details of individual cases.

Ordered end-members

Ordered end-members can always be made by combining a subset of the compositional end-members in the same x-eos. A ∆G term is then added, representing the energetics of the ordering on sites at that composition.

The Holland & Powell dataset

The Holland & Powell dataset ultimately supplies the G(P,T) information, minus the ∆G terms, for making the HPx-eos end-members. It also supplies pure, or stoichiometric, end-members such as quartz.

Extant families of HPx-eos are built on version 6 of the dataset (Holland & Powell, 2011). This applies a modified Tait equation of state for solid end-members, as summarised in the pdf below, showing an implementation equivalent to that used in THERMOCALC 3.50 and above. The implementation includes handling of order-disorder in end-members by Landau and the symmetric formalism (see Holland & Powell, 2011, for more details), and contrasting treatment of thermal expansion in mineral and silicate melt end-members.

The dataset grows and evolves all the time, with versions 6.2 and 6.33 currently in widespread use. The relevant versions can be downloaded with the appropriate HPx-eos families. The software THERMOCALC can be used to extract thermodynamic data from the dataset. The history of the dataset and the thinking behind it can be traced through the published papers. Release notes below list changes in the evolution of versions 6.xx:

Activity-composition relations

The formulation of activity-composition (a-x) relations is based on the regular solution: ideal entropy of mixing on sites, with non-ideal enthalpic interaction between each pair of end-members represented by a single, symmetrical, macroscopic interaction parameter, W. To this approach has been added asymmetry of enthalpic interactions, via a van laar formulation, and the possibility of linear P and T dependent of W parameters. Furthermore, the entropic contribution from a given site (typically the tetrahedral site in chain silicates) may in some cases be reduced to 1/2 or 1/4 of the ideal entropy.

Full details are given in the listed papers, while a summary (excerpt from ECRG’s PhD thesis) can be downloaded here:

Details of the a-x calibrations used in extant HPx-eos are given on the pages for the appropriate HPx-eos families.

Calibration

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Find out more

For more information, see the list of HPx-eos resources. If this doesn’t help, feel free to ask the Discussion Group.

References

Holland & Powell (2011) J Metamorph Geol 29 333-383.