Equilibrium andalusite <--> kyanite

Thermodynamics of kyanite is here evaluated from frequencies and static energies evaluated ab initio, and processed within the framework offered by the Quasi-Harmonic approximation (QHA)

The modified Kieffer model is used to estimate the contribution of the acoustic modes to the Helmholtz free energy

The following command is a directive issued in order to show plots directly in the current notebook, instead of being done in a separate windows

In the following cell, the program is loaded; the input file is shown and the Kieffer model is applied to estimate the free energy contribution from the acoustic modes, in a given temperature range. The $F_{acoustic}(T)$ function, in numerical form, is stored in a stack from which values of $F$, at any temperature in the range, are recovered by numerical interpolation of the stored values by the method get_value of the kieffer class

The info.show command displays various parameters set in the input.txt file

A 3^rd order Birch-Murnaghan EoS (BM3) is computed at 300K; this is done by fitting a volume integrated form of the EoS to the $F(V)$ numerical values.

The same result is obtained by fitting the BM3 EoS to the $P(V)$ set of values at 300K; pressures $P$ at each volume are computed as derivatives of $F$ with respect to $V$, at constant temperature:

A comparison of computed $C_P$ and $S$ with available experimental data is possible; $K_p$ is fixed at the value estimated at 300K

In the following, the equilibrium kyanite <--> andalusite will be analysed. To begin with, the equilibrium curve in the P/T space is calculated by using the Holland & Powell (H&P) thermodynamics data. Such data for kyanite are shown by the method info of the ky instance of the class mineral:

The equilibrium curve is computed by the command equilib in the 500 - 700K temperature range (12 points)

Now, thermodynamics data for kyanite are computed from quantum-mechanical (QM) calculations. These regard:

  1. bulk modulus and its pressure derivative;
  2. thermal expansion as a function of $T$;
  3. specific heat (at constant pressure) as function of $T$;
  4. entropy at the reference state (T=298.15 K, P=1 bar)

The volume at the reference state is the experimental one: as it is well known, quantum-mechanical calculations generally overestimate cell volumes; this fact leads to an overestimation of the $PV$ contribution to the Gibbs free energy ($G=F+PV$) which generally becomes significant at high pressures.

The $G$ free energy in the reference ($G_0$) is also from the H&P database: the $G$ energy computed at the QM level is some absolute energy of the mineral phase, whereas the corresponding quantity from the H&P database is the energy of formation from the elements. In order to properly fix the zero of the energy scale (that is, to put both kyanite and andalusite on the same energy scale), the $G_0$ must be properly set.

All of such calculations, together with the loading of such quantities in the instance ky of the mineral class, are performed by the upload_mineral command:

Now, ky.info will show the new data stored in the database, for kyanite:

The impact of the contribution form acoustic modes for the present case is well shown by switching off the Kieffer model and by repeating the computation:

Indeed, the equilibrium curve is shifted at significantly higher pressures and the Clapeyron slope significantly increases.

The frequencies used in the Kieffer model can be set by the freq method of the kieffer class (or the input file, under the KIEFFER keyword):