Simplified Landau Model
Within EosFit we have implemented a simplified version of a Landau model for the change in V (or linear dimensions such as cell parameters) through a continuous phase transition. This description in this section is written in terms of volume. Within EosFit linear quantities are handled the same way by cubing the lengths to produce 'volume-like' equations. A full description of the theory and implementation in Eosfit7 is given in Angel et al. (2017).
The basic idea is that the volume of the low-symmetry phase can be expressed as:
in which
is the volume of the high-symmetry phase unaffected by any transition effects and
is the 'spontaneous strain' arising from the phase transition. This means that without the transition, the volume of the phase would be
and the transition induces an additional strain of
. Note that this implicitly assumes that the spontaneous strain is sufficiently small that its variation with P or T can be described in terms of infinitesimal strains.
The value of is defined by a ‘normal’ EoS. The transition is characterised by how the spontaneous strain
evolves with pressure and temperature. In EosFit we implement a simple Landau-type power-law model:
for the variation with temperature at any fixed P
for the variation with pressure at any fixed T
The parameters a are scaling parameters for the strain, and the exponent β provides the necessary power-law term. Note that this expression may not be sufficient to describe the volume variations of phases with complex transitions, for example those involving significant coupling between order parameters. Neither is low-temperature saturation of the spontaneous strain implemented.The choice of using the absolute values of, e.g. , ensures that the equations work for both
or
.
These two equations, for isothermal and isobaric data, are implemented in EosFit as transition models 1 and 2. They have refineable parameters a and β which are assumed constants with P and T. The transition point (or
) is settable by the user but often cannot be refined, as such refinements are usually unstable.
For PVT equations, if the phase boundary is a straight line in P-T space, with a constant slope and
at P = 0, then the transition temperature at any given pressure P is given as:
And it follows that:
It is not clear, in the case of PVT data, whether it is necessary to include a term for the pressure dependency of the a coefficient. In version 7.2, a linear dependency with constant was included in the cfml_eos module, but it was not used in the console implementation. In version 7.3 this was removed. Instead curvature of the phase boundary in P-T space can be described as d2T/dPtr2.
This approach yields a self-consistent set of equations. In addition to the parameters to describe the spontaneous strain variation in the equations listed above, the user also has to specify whether the low-symmetry phase is the high-temperature or the low-temperature (or pressure) phase.
Within EosFit the parameters such as the volume and the bulk modulus are always calculated for the stable phase at the requested T and P; Differentiation of the expression of the volume with respect to pressure yields the compressibility or the bulk modulus of the low-symmetry phase (Angel et al. 2017):
High-symmetry softening
If there is also pre-transition softening in the high-symmetry phase it can be modeled in the same way. Now we write so that
is the volume of the high-symmetry phase unaffected by softening, and
is the volume in the neighbourhood of the transition which is affected by softening. This approach is simplistic but fits the data for quartz very precisely.
The high symmetry softening is then described by additional parameters and
which can be refined to the data. Note that this must often be done with caution as for some transitions, such as quartz, there is significant elastic softening in the high-symmetry phase near to the transition, but no obvious spontaneous strain. In this case other data, such as the measured elastic parameters, must be used to constrain the parameters used in EosFit.
If no softening in the high-symmetry phase is present, then the parameter should be left set at zero in EosFit.