The HPx-eos

This section gives an overview of the HPx-eos – the building blocks of our phase equilibrium modelling tool. Elsewhere in this section we try to make the HPx-eos accessible and transparent for those who would use them through THERMOCALC or another program. See our pages on how the HPx-eos are built (for users and implementers), which families of HPx-eos are available for modelling different rock types, and what you should know about using the HPx-eos before you get started.

An introduction to the HPx-eos

Each one of the HPx-eos is an equation of state (eos), representing a geological phase (such as the olivine solid solution, (Mg,Fe,Ca)2SiO4, or a silicate melt) which in general shows compositional (x) variation. They are founded on the Holland & Powell internally-consistent dataset. An equation of state provides the relationships among the various thermodynamic properties of a phase and the physical conditions it experiences. In the well known “G-X loop” representation of phase equilibrium (Figure 1), the HPx-eos for a phase is identical to that phase’s GX loop.

Figure 1: Construction of phase equilibria in G-X (Gibbs free energy – composition) space at constant pressure and temperature (P,T). The figure shows G-X loops for phases A, B and C (phase C exhibits a solvus at intermediate compositions). Blue dashed common tangents indicate equilibria between phases that could be mass-balanced at bulk composition X1; black dots indicate equilibrium compositions. The most stable assemblage that could form in rock X1 is coexisting phase A with composition a and phase B with composition b (dark blue tangent). The next most stable assemblage is pure phase B with composition X1, followed by coexisting A and C (mid-blue tangent), followed by two coexisting phases of C’s structure and compositions c and d (light blue tangent). The HPx-eos of a phase provides its G-X loop.

An HPx-eos is built out of a set of compositional end-members – e.g. the end-members Mg2SiO4, Fe2SiO4 and CaMgSiO4 for olivine – and a set of activity-composition (a-x) relations to describe the thermodynamics of mixing those end-members. The thermodynamic properties of the end-members are taken from the Holland & Powell internally-consistent dataset. The Holland & Powell dataset collates a large body of disparate data on the thermodynamics of mineral end-members, and optimises it, adjusting some values with reference to their uncertainties in order to reconcile all of the data within a thermodynamic framework.* Somewhat less formally, the HPx-eos embody the same goal of internal consistency. They can be thought of as a means of converting the full range of available thermodynamic data on geological phases into a tool for predicting equilibrium phase relations.

The HPx-eos were first developed within the realm of metamorphic petrology, with the aim of modelling mineral assemblages in metapelites (White, Powell & Holland, 2007; White, Powell, Holland et al, 2014). Early incarnations involved ideal mixing of the Holland & Powell dataset end-members (Powell, Holland & Worley, 1998). They have since grown much more complex, acquiring asymmetric mixing properties (Holland & Powell, 2003) and elaborate order-disorder (e.g. Green, Holland & Powell, 2007), and today may routinely occupy a 10-dimensional composition space (Holland, Green & Powell, 2018). They have also spread into the domains of metabasite (Green, White, Diener et al, 2016) and igneous rocks in the crust and upper mantle (Holland, Green & Powell, 2018).

We’re aware that recent rapid changes to the HPx-eos, starting with the upgrade to version 6 of the Holland & Powell dataset, have left users with a bewildering array of HPx-eos to choose from. Whereas each HPx-eos represents a single phase, a single phase may be represented by multiple HPx-eos – three each, for example, for clinopyroxene and for silicate melt. We are working towards a single library of HPx-eos, one per phase. In the meantime we provide guidance on choosing the most appropriate HPx-eos in the section on the HPx-eos families.

*Experimental data and citation

Of course, the data used in calibration does not grow on trees, but represents the expertise, labour and time of some hundreds of experimentalists. As a community we need urgently to grapple with the problem of how to give these authors credit for work that has been subsumed into the Holland & Powell or equivalent dataset. For any given use of the HPx-eos, most or all of the 653 works cited in the Holland & Powell (2011) dataset paper are implicated, along with many others that contributed to the calibration of a-x relations. It is not practical to include a full list of these works in every paper that makes use of the HPx-eos. Perhaps credit could be automatically associated with the ORCiD of all contributing authors, or perhaps a change in attitude to citation metrics is key. Either way, just as this problem arises from the deeply intertwined nature of the experimental observations exploited by the HPx-eos, so it reflects the deep inter-reliance of subdisciplines within the geoscience community.


Green et al (2007) Am Mineral 92 1181-1189. Green et al (2016) J Metamorph Geol 34 845-892. Holland & Powell (2003) Contrib Mineral Petrol 145 492-501. Holland & Powell (2011) J Metamorph Geol 29 333-383. Holland et al (2018) J Petrol 59 881-900. Powell et al (1998) J Metamorph Geol 16: 577-588. White et al (2007) J Metamorph Geol 25 511-527. White et al (2014) J Metamorph Geol 32 261-286.