In "isothermal EoS" the properties such as V0T and K0T at P = 0 are calculated at the temperature of interest, and then used in an isothermal EoS at the needed temperature. In EosFit7 any thermal expansion model can be combined with any isothermal equation of state and two types of models for the variation of the bulk modulus with temperature, which can be selected with the cross command in the input utility.
Linear dK/dT
The simplest approach of assuming a linear variation of K0 with temperature, so is constant, is certainly justified at high temperatures by direct measurements of the bulk moduli of many materials by elasticity measurements (e.g. as summarised in Anderson, 1995). In combination with an expression for thermal expansion that allows
, and an isothermal EoS with
, this approach includes all second derivatives of the volume with respect to the intensive variables P and T, and is thus algebraically internally consistent. This is because
provides the cross-derivative
; thus if
then α does not change with pressure.
However, this approach has two problems:
Hellfrich-Connolly model
In order for any ‘isothermal type’ EoS to be thermodynamically correct, it must ensure that all of αV, (dKTR/dT)P and (dK'TR/dT)P tend to zero as T approaches absolute zero. This precludes simple polynomial models for the volume as a function of T at room P, which have no such requirement.
We implement the Kroll et al. (2012) version of Holland-Powell thermal expansion, but we explicitly use the Anderson-Grüneisen (Anderson 1995) parameter δT in place of (1+K'TR). The expression for volume at temperature then becomes:
The use of δT and K'TR as independent parameters allows data with any bulk modulus variation along an isochor to be fit. This approach also completely separates the thermal and baric parts of the EoS while maintaining a reasonable physical basis in the Einstein oscillator model; any description of the thermal expansion and temperature dependence of the bulk moduli is completely separated from the description of the isothermal compression. This also allows this thermal expansion equation to be combined with any isothermal equation of state for compression.
The same Anderson-Grueneisen parameter δT also controls the variation of the bulk modulus with temperature:
and this provides the correct behaviour of KTR at low temperatures, because KTR follows the volume variation and Equation (12) ensures that αV goes to zero as the temperature approaches absolute zero. Because there is no simple thermodynamic expression for the temperature variation of K'TR we introduce a simple expression for its temperature variation:
When the additional parameter δ' is not zero, this ensures in combination with that (dK'TR/dT)P becomes zero at low temperatures in the same way as in the MGD and Holland-Powell thermal-pressure EoS. For more discussion and details, see Angel et al (2018).
Calculation of heat capacities.
For isothermal types of PVT EoS, the heat capacity at constant volume, CV, is calculated as:
And the heat capacity at constant pressure is:
in which the volume V and the thermal expansion α, and the bulk modulus KT are all calculated from the EoS. The γ is the thermal Grüneisen parameter. In EosFit7c the EoS must be expressed in terms of molar volumes (cm3/mol, set by vscale) and either kbar or GPa as pressure units (set by pscale) in order to obtain heat capacities in J/mol/K. If the pscale or vscale of the EoS cannot be interpreted, no heat capacities are calculated.
The variation of the thermal Grüneisen parameter is:
which means when q = 0, the thermal Grüneisen parameter is constant and equal to γ0.
In isothermal EoS the values of γ and q only appear when adiabatic moduli are converted to isothermal moduli. Therefore, the values of γ0 and q can only be refined if adiabatic moduli are present in the dataset. The situation is different for thermal-pressure type EoS