Mineral Property Modifiers

Standard thermal equations of state typically consider the effects of strain and quasiharmonic lattice vibrations on the energies of minerals at given temperatures and pressures. There are a number of additional processes, such as isochemical order-disorder and magnetic effects which also contribute to the total energy of a phase, which are not typically included in standard equations of state. Corrections for these additional processes can, to first order, be applied as additive modifiers, ignoring any explicit coupling with the equation of state.

Burnman currently includes implementations of the following:

In all cases, the excess Gibbs energy \(\mathcal{G}\) and first and second partial derivatives with respect to pressure and temperature are calculated. The effective thermodynamic properties of each phase are then modified in a self-consistent manner; specifically:

\[\begin{split}\mathcal{G} = \mathcal{G}_{\text{o}} + \mathcal{G}_{\text{m}}, \\ S = S_{\text{o}} - \left(\frac{\partial \mathcal{G}}{\partial T}\right)_{\text{m}}, \\ V = V_{\text{o}} + \left(\frac{\partial \mathcal{G}}{\partial P}\right)_{\text{m}}, \\ K_T = V / \left( \left(\frac{V_{\text{o}}}{K_{T\text{o}}}\right) - \left(\frac{\partial^2\mathcal{G}}{\partial P^2}\right)_{\text{m}} \right), \\ C_p = C_{p\text{o}} - T \left(\frac{\partial ^2 \mathcal{G}}{\partial T^2}\right)_{\text{m}}, \\ \alpha = \left( \alpha_{\text{o}} V_{\text{o}} + \left(\frac{\partial^2 \mathcal{G}}{\partial P \partial T}\right)_{\text{m}} \right) / V\end{split}\]

where the subscript “o” refers to properties calculated from the underlying equation of state, and the subscript “m” refers to contributions from the property modifier. Other derived properties are then calculated from these modified thermodynamic properties using standard thermodynamic relationships:

\[\begin{split}\mathcal{H} = \mathcal{G} + T S, \\ \mathcal{F} = \mathcal{G} - P V, \\ C_v = C_p - V T \alpha^2 K_T, \\ \gamma = \frac{\alpha K_T V}{C_v}, \\ K_S = K_T \frac{C_p}{C_v}\end{split}\]

Importantly, this allows us to stack modifications such as multiple Landau transitions in a simple and straightforward manner. In the burnman code, we add property modifiers as an attribute to each mineral as a list. For example:

from burnman.minerals import SLB_2011

stv = SLB_2011.stishovite()

stv.property_modifiers = [["landau", {"Tc_0": -4250.0, "S_D": 0.012, "V_D": 1e-09}],
                          ["linear", {"delta_E": 1.e3, "delta_S": 0., "delta_V": 0.}]]

Each modifier is a list with two elements, first the name of the modifier type, and second a dictionary with the required parameters for that model. A list of parameters for each model is given in the following sections.

Landau tricritical model (Putnis, 1992) (available as landau)

burnman.eos.property_modifiers.landau_excesses(pressure, temperature, params)

Applies a tricritical Landau correction to the properties of an endmember which undergoes a displacive phase transition. These transitions are not associated with an activation energy, and therefore they occur rapidly compared with seismic wave propagation.

This correction follows [Putnis92], and is done relative to the completely ordered state (at 0 K). It therefore differs in implementation from both [SLB11] and [HollandPowell11], who compute properties relative to the completely disordered state and standard states respectively. The current implementation is preferred, as the excess entropy (and heat capacity) terms are equal to zero at 0 K.

\[Tc = Tc_0 + \frac{V_D P}{S_D}\]

If the temperature is above the critical temperature, Q (the order parameter) is equal to zero, and the Gibbs free energy is simply that of the disordered phase:

\[\begin{split}\mathcal{G}_{\textrm{dis}} = -S_D \left( \left( T - Tc \right) + \frac{Tc_0}{3} \right), \\ \frac{\partial \mathcal{G}}{\partial P}_{\textrm{dis}} = V_D, \\ \frac{\partial \mathcal{G}}{\partial T}_{\textrm{dis}} = -S_D\end{split}\]

If temperature is below the critical temperature, Q is between 0 and 1. The Gibbs energy can be described thus:

\[\begin{split}Q^2 = \sqrt{\left( 1 - \frac{T}{Tc} \right)}, \\ \mathcal{G} = S_D \left((T - Tc) Q^2 + \frac{Tc_0 Q^6}{3} \right) + \mathcal{G}_{\textrm{dis}}, \\ \frac{\partial \mathcal{G}}{\partial P} = - V_D Q^2 \left(1 + \frac{T}{2 Tc} \left(1. - \frac{Tc_0}{Tc} \right) \right) + \frac{\partial \mathcal{G}}{\partial P}_{\textrm{dis}}, \\ \frac{\partial \mathcal{G}}{\partial T} = S_D Q^2 \left(\frac{3}{2} - \frac{Tc_0}{2 Tc} \right) + \frac{\partial \mathcal{G}}{\partial T}_{\textrm{dis}}, \\ \frac{\partial^2 \mathcal{G}}{\partial P^2} = V_D^2 \frac{T}{S_D Tc^2 Q^2} \left( \frac{T}{4 Tc} \left(1. + \frac{Tc_0}{Tc} \right) + Q^4 \left(1. - \frac{Tc_0}{Tc} \right) - 1 \right), \\ \frac{\partial^2 \mathcal{G}}{\partial T^2} = - \frac{S_D}{Tc Q^2} \left(\frac{3}{4} - \frac{Tc_0}{4 Tc} \right), \\ \frac{\partial^2 \mathcal{G}}{\partial P \partial T} = \frac{V_D}{2 Tc Q^2} \left(1 + \left(\frac{T}{2 Tc} - Q^4 \right) \left(1 - \frac{Tc_0}{Tc} \right) \right)\end{split}\]

Parameters

Parameter	Description
Tc_0	Critical temperature at reference pressure (K)
S_D	Entropy parameter (J/mol/K)
V_D	Volume parameter (m³/mol)

Landau tricritical model (Stixrude and Lithgow-Bertelloni, 2022) (available as landau_slb_2022)

burnman.eos.property_modifiers.landau_slb_2022_excesses(pressure, temperature, params)

Applies a tricritical Landau correction to the properties of an endmember which undergoes a displacive phase transition. This correction follows [SLB22], and is done relative to the state with order parameter \(Q=1\).

The order parameter of this formulation can exceed one, at odds with [Putnis92], but in better agreement with atomic intuition [SLB22]. Nevertheless, this implementation is still not perfect, as the excess entropy (and heat capacity) terms are not equal to zero at 0 K. \(Q\) is limited to values less than or equal to 2 to avoid unrealistic stabilisation at ultrahigh pressure.

Parameters

Parameter	Description
Tc_0	Critical temperature at reference pressure (K)
S_D	Entropy parameter (J/mol/K)
V_D	Volume parameter (m³/mol)

Landau tricritical model (Holland and Powell, 1998) (available as landau_hp)

burnman.eos.property_modifiers.landau_hp_excesses(pressure, temperature, params)

Applies a tricritical Landau correction similar to that described above. However, this implementation follows [HollandPowell11], who compute properties relative to the standard state.

Includes the correction published within landaunote.pdf (Holland, pers. comm), which ‘corrects’ the terms involving the critical temperature Tc / Tc*.

Note that this implementation allows the order parameter Q to be greater than one.

\[Tc = Tc0 + \frac{V_D P}{S_D}\]

If the temperature is above the critical temperature, Q (the order parameter) is equal to zero. Otherwise

\[Q^2 = \sqrt{\left( \frac{Tc - T}{Tc0} \right)}\]

\[\begin{split}\mathcal{G} = Tc_0 S_D \left( Q_0^2 - \frac{Q_0 ^ 6}{3} \right) \\ - S_D \left( Tc Q^2 - Tc_0 \frac{Q ^ 6}{3} \right) \\ - T S_D \left( Q_0^2 - Q^2 \right) + P V_D Q_0^2, \\ \frac{\partial \mathcal{G}}{\partial P} = -V_D \left( Q^2 - Q_0^2 \right), \\ \frac{\partial \mathcal{G}}{\partial T} = S_D \left( Q^2 - Q_0^2 \right), \\\end{split}\]

The second derivatives of the Gibbs free energy are only non-zero if the order parameter exceeds zero. Then

\[\begin{split}\frac{\partial^2 \mathcal{G}}{\partial P^2} = -\frac{V_D^2}{2 S_D Tc_0 Q^2}, \\ \frac{\partial^2 \mathcal{G}}{\partial T^2} = -\frac{S_D}{2 Tc_0 Q^2}, \\ \frac{\partial^2 \mathcal{G}}{\partial P \partial T} = \frac{V_D}{2 Tc_0 Q^2}\end{split}\]

Parameters

Parameter	Description
P_0	Reference pressure (Pa)
T_0	Reference temperature (K)
Tc_0	Critical temperature at reference pressure (K)
S_D	Entropy parameter (J/mol/K)
V_D	Volume parameter (m³/mol)

Linear in P and T (available as linear)

burnman.eos.property_modifiers.linear_excesses(pressure, temperature, params)

A simple linear correction in pressure and temperature.

The excess Gibbs energy and its derivatives are given by:

\[\begin{split}\mathcal{G} = \Delta \mathcal{E} - T \Delta S + P \Delta V, \\ \frac{\partial \mathcal{G}}{\partial T} = - \Delta S, \\ \frac{\partial \mathcal{G}}{\partial P} = \Delta V, \\ \frac{\partial^2 \mathcal{G}}{\partial T^2} = 0, \\ \frac{\partial^2 \mathcal{G}}{\partial P^2} = 0, \\ \frac{\partial^2 \mathcal{G}}{\partial T \partial P} = 0\end{split}\]

This form of excess is extremely useful as a first order tweak to free energies (especially in solid solutions where data on all endmembers may not be available).

Parameters

Parameter	Description
delta_E	Energy correction (J/mol)
delta_S	Entropy correction (J/mol/K)
delta_V	Volume correction (m³/mol)

Bragg-Williams model (available as bragg_williams)

burnman.eos.property_modifiers.bragg_williams_excesses(pressure, temperature, params)

The Bragg-Williams model is a symmetric solution model with an excess configurational entropy term determined by the specifics of order-disorder in the mineral, multiplied by some empirical factor. Expressions for the excess Gibbs free energy can be found in [HP96].

Excess properties assume that order-disorder processes are rapid compared with the timescales of pressure and temperature changes (i.e., equilibrium is maintained).

This may not be reasonable for order-disorder, especially for slow or coupled diffusers (Si-Al, for example). Properties for minerals that order-disorder slowly should be calculated using an explicit solid solution model.

Parameters

Parameter	Description
deltaH	Enthalpy change (J/mol)
deltaV	Volume change (m³/mol)
Wh	Enthalpy interaction parameter (J/mol)
Wv	Volume interaction parameter (m³/mol)
n	Related to the number of available sites for ordering
factor	An empirical factor

Magnetic ordering (Chin et al., 1987) (available as magnetic_chs)

burnman.eos.property_modifiers.magnetic_excesses_chs(pressure, temperature, params)

This model approximates the excess energy due to magnetic ordering. It was originally described in [CHS87]. The expressions used in this implementation can be found in [Sun91].

Parameters

Parameter	Description
structural_parameter	A dimensionless parameter related to the crystal structure
curie_temperature	A list of length 2: zero pressure Curie temperature (K) and pressure dependence (K/Pa)
magnetic_moment	A list of length 2: zero pressure magnetic moment (Bohr magnetons) and pressure dependence (Bohr magnetons/Pa)

Debye model excess (Helmholtz) (available as debye)

burnman.eos.property_modifiers.debye_excesses(pressure, temperature, params)

Applies an excess contribution based on a Debye model. The excess Gibbs energy and its derivatives are given by:

\[\begin{split}\mathcal{G} = F_{\textrm{Debye}}, \\ \frac{\partial \mathcal{G}}{\partial T} = - S_{\textrm{Debye}}, \\ \frac{\partial \mathcal{G}}{\partial P} = 0, \\ \frac{\partial^2 \mathcal{G}}{\partial T^2} = - \frac{C_{V,\textrm{Debye}}}{T}, \\ \frac{\partial^2 \mathcal{G}}{\partial P^2} = 0, \\ \frac{\partial^2 \mathcal{G}}{\partial T \partial P} = 0\end{split}\]

The excess heat capacity tends toward a constant value at high temperature.

Parameters

Parameter	Description
Cv_inf	Heat capacity at infinite temperature (J/mol/K).
Theta_0	Debye temperature (K).

Debye model excess (internal energy) (available as debye_delta)

burnman.eos.property_modifiers.debye_delta_excesses(pressure, temperature, params)

Applies an excess contribution based on a Debye model. The excess Gibbs energy and its derivatives are given by:

\[\begin{split}\mathcal{G} = - U_{\textrm{Debye}}, \\ \frac{\partial \mathcal{G}}{\partial T} = - C_{V,\textrm{Debye}}, \\ \frac{\partial \mathcal{G}}{\partial P} = 0, \\ \frac{\partial^2 \mathcal{G}}{\partial T^2} = - \frac{dC_{V,\textrm{Debye}}}{dT}, \\ \frac{\partial^2 \mathcal{G}}{\partial P^2} = 0, \\ \frac{\partial^2 \mathcal{G}}{\partial T \partial P} = 0\end{split}\]

The excess entropy tends toward a constant value at high temperature and behaves like the heat capacity of a Debye model at finite temperature.

Parameters

Parameter	Description
S_inf	Entropy at infinite temperature (J/mol/K).
Theta_0	Einstein temperature (K).

Einstein model excess (Helmholtz) (available as einstein)

burnman.eos.property_modifiers.einstein_excesses(pressure, temperature, params)

Applies an excess contribution based an Einstein model. The excess Gibbs energy and its derivatives are given by:

\[\begin{split}\mathcal{G} = F_{\textrm{Einstein}}, \\ \frac{\partial \mathcal{G}}{\partial T} = - S_{\textrm{Einstein}}, \\ \frac{\partial \mathcal{G}}{\partial P} = 0, \\ \frac{\partial^2 \mathcal{G}}{\partial T^2} = - \frac{C_{V,\textrm{Einstein}}}{T}, \\ \frac{\partial^2 \mathcal{G}}{\partial P^2} = 0, \\ \frac{\partial^2 \mathcal{G}}{\partial T \partial P} = 0\end{split}\]

The excess heat capacity tends toward a constant value at high temperature.

Parameters

Parameter	Description
Cv_inf	Heat capacity at infinite temperature (J/mol/K).
Theta_0	Einstein temperature (K).

Einstein model excess (internal energy) (available as einstein_delta)

burnman.eos.property_modifiers.einstein_delta_excesses(pressure, temperature, params)

Applies an excess contribution based an Einstein model. The excess Gibbs energy and its derivatives are given by:

\[\begin{split}\mathcal{G} = - U_{\textrm{Einstein}}, \\ \frac{\partial \mathcal{G}}{\partial T} = - C_{V,\textrm{Einstein}}, \\ \frac{\partial \mathcal{G}}{\partial P} = 0, \\ \frac{\partial^2 \mathcal{G}}{\partial T^2} = - \frac{dC_{V,\textrm{Einstein}}}{dT}, \\ \frac{\partial^2 \mathcal{G}}{\partial P^2} = 0, \\ \frac{\partial^2 \mathcal{G}}{\partial T \partial P} = 0\end{split}\]

The excess entropy tends toward a constant value at high temperature and behaves like the heat capacity of an Einstein model at finite temperature.

Parameters

Parameter	Description
S_inf	Entropy at infinite temperature (J/mol/K).
Theta_0	Einstein temperature (K).