With a thermal-pressure EoS the total pressure at a given V,T is:
P(V,Tref) is determined only from the isothermal equation of state for the material at the reference temperature, but with the ‘observed’ volume from P and T.
Pth(T) is the pressure that would be created by increasing the T from Tref to T at constant volume, along an isochor. The thermal pressure at Tref is thus zero, so at Tref the EoS reduces to the isothermal EoS, as required. The thermal pressure at other temperatures clearly depends on the bulk modulus.
All of these thermal-pressure EoS use a model for the phonon density of states to calculate the isochoric heat capacity Cv. The thermal pressure is then obtained through the Grüneisen relationship that states αKT = γCV. Thus the change in thermal pressure along an isochor becomes an integral over the isochoric heat capacity:
Along an isochor the volume remains constant, and the quasi-harmonic approximation means that the value of γ is solely dependent on volume. Along an isochor γ does not change and can also be removed from the integral:
So the thermal pressure is solely a function of the heat capacity, which is defined by a model for the phonon density of states. Each of the models used has a characteristic temperature which defines a characteristic energy of the phonon density of states. In the Debye model, this is the Debye temperature θD, in the Einstein oscillator models it is the Einstein temperature θE.
The quasi-harmonic approximation behind these EoS states that the phonon frequencies, and thus the characteristic temperature of the phonon density of states (θD or θE), is only a function of volume. The coefficient relating the phonon energy change to volume change is the thermal Grüneisen parameter γ:
If the variation of γ with volume is expressed as:
then it follows that the characteristic temperature varies as
q-compromise models
In refinements the values of θ0, q and γ0 are often highly-correlated and their values cannot be determined independently. Therefore we have introduced into EosFit 7c q-compromise models for both the MGD and Einstein oscillator thermal pressure models (Kroll et al., 2019, Angel et al., 2020) .
Careful reading of Holland and Powell (2011) reveals that their EoS is also a q-compromise model. In these models:
The parameter q is therefore not a parameter in these q-compromise models, leaving θ0 and γ0 as refineable parameters (but see discussion for the Holland-Powell (2011) which is treated differently).
One of the key properties of q-compromise models is that the thermal pressure is only a function of T. This means that the isochors are parallel to one another, and that the isothermal bulk modulus KT is constant along each isochor while the thermal expansion coefficient is not (Kroll et al., 2012).
EoS files written by EosFit version 7.6 for q-compromise models for MGD EoS, and files for the Einstein oscillator model in either form, will not be read and interpreted by earlier versions of EosFit, because these models are new in v7.6
Warning: volume and pressure units in MGD and Einstein EoS
The big disadvantage of these EoS is the need to use the molar volume V to convert the thermal energy into the thermal pressure (pressure having units of energy per unit volume). This makes the parameters of these EoS specific to a single combination of specific volume and pressure scales, consistent with the units used for the bulk modulus. This limits the transferability and ease of use of the EoS, and results in frequent calculation errors. This warning does not apply to the Holland-Powell model because of the different way it is parameterised.
Make sure that you set the vscale and the pscale correctly in EosFit7 before using the MGD and Einstein EoS!!
And make sure before refinement to data that you also set the vscale and the pscale correctly in the data files.
Note that EosFit does not support MGD EoS for linear data.
Warning: assumptions
For further discussion of details of the MGD EoS see Anderson (1995) and Angel et al. (2017b). Also be aware that all of these thermal-pressure EoS are based on the assumptions of the quasi-harmonic approximation (QHA) which states that the frequencies of the vibrational modes of the solid (i.e. the phonons) do not change along isochors. This is true for isotropic and cubic materials, because the thermal pressure (the ratio of the linear thermal expansion to the compressibility in the same direction) is required to be isotropic by symmetry. In lower-symmetry crystals, this may not be true, and the QHA is violated, and QHA-based EoS may not fit the available data and may not represent the PVT and elasticity of the crystal correctly. See Zaffiro et al. (2019) for a discussion and example.
Warning: failure at high T and low P
For high temperatures and low pressures the volume at P and T is larger than the volume at reference conditions. Therefore the compressional part of the EoS must be used in expansion. A large expansions the compressional EoS may become physically-invalid.This limit is sensitive to the combined effects of the EoS parameters K'0T, q and the Grüneisen parameter γ0 (Angel et al. 2019). Large values of q, which correspond to a rapid decrease in phonon mode frequencies with increasing volume, can also lead to the bulk modulus becoming zero at high pressures and temperatures that are not particularly extreme for planetary geotherms. These EoS therefore have an extremely limited P and T regime over which they are both valid and have physically-meaningful properties. Eosfit issues warnings when these limits are exceeded.