# This file is part of BurnMan - a thermoelastic and thermodynamic toolkit
# for the Earth and Planetary Sciences.
# Copyright (C) 2012 - 2024 by the BurnMan team, released under the GNU
# GPL v2 or later.
import scipy.optimize as opt
from . import equation_of_state as eos
import warnings
import numpy as np
# Try to import the jit from numba. If it is
# not available, just go with the standard
# python interpreter
try:
from numba import jit
except ImportError:
def jit(nopython=True):
def decorator(fn):
return fn
return decorator
[docs]
@jit(nopython=True)
def make_params(K0, K0_prime, K_infinity_prime):
a = (
16.0 * np.power(K0_prime, 3.0)
+ 84.0 * np.power(K0_prime, 2.0)
+ 192.0 * K0_prime
- 972.0 * K_infinity_prime
+ 1177.0
)
b = 2.0 * np.power(K0_prime, 2.0) + 7.0 * K0_prime - 27.0 * K_infinity_prime + 38.0
omega = np.power((a + np.sqrt(a * a - 32.0 * b * b * b)), 1.0 / 3.0)
C = (
(11.0 / 6.0)
+ (1.0 / 3.0) * K0_prime
- K_infinity_prime
+ (np.power(2, -1.0 / 3.0) / 6) * omega
+ (np.power(2, 1.0 / 3.0) / 3) * (b / omega)
)
B = K_infinity_prime - 1.0
A = K0 / (B - 0.5 * C + np.power(B + C, 2.0))
return A, B, C
[docs]
class MACAW(eos.IsothermalEquationOfState):
"""
Class for the MACAW equation of state
detailed in :cite:`Lozano2022`.
This equation of state has no temperature dependence.
.. list-table::
:widths: 25 75 20
:header-rows: 1
* - Parameter
- Description
- Units
* - ``F_0``
- Reference Helmholtz free energy.
- :math:`\\text{J/mol}`
* - ``P_0``
- Reference pressure.
- :math:`\\text{Pa}`
* - ``V_0``
- Reference volume.
- :math:`\\text{m}^3`
* - ``K_0``
- Reference bulk modulus.
- :math:`\\text{Pa}`
* - ``Kprime_0``
- Pressure derivative of the bulk modulus at reference pressure.
- Dimensionless
* - ``Kprime_inf``
- Infinite pressure derivative of the bulk modulus.
- Dimensionless
"""
[docs]
def isothermal_bulk_modulus_reuss(self, pressure, temperature, volume, params):
"""
Returns isothermal bulk modulus :math:`K_T` :math:`[Pa]` as a function of pressure :math:`[Pa]`,
temperature :math:`[K]` and volume :math:`[m^3]`.
"""
A, B, C = make_params(params["K_0"], params["Kprime_0"], params["Kprime_inf"])
Vrel = volume / params["V_0"]
term1 = A * np.power(Vrel, -(B + 1))
term2 = np.exp((2.0 / 3.0) * C * (1 - np.power(Vrel, 1.5)))
term3 = np.power(C * np.power(Vrel, 1.5) + B, 2.0) - (
0.5 * C * np.power(Vrel, 1.5) - B
)
return term1 * term2 * term3
[docs]
def volume(self, pressure, temperature, params):
"""
Get the Vinet volume at a reference temperature for a given
pressure :math:`[Pa]`. Returns molar volume in :math:`[m^3]`
"""
def delta_pressure(x):
return self.pressure(0.0, x, params) - pressure
V = opt.brentq(delta_pressure, 0.1 * params["V_0"], 1.5 * params["V_0"])
return V
[docs]
def pressure(self, temperature, volume, params):
"""
Returns pressure :math:`[Pa]` as a function of volume :math:`[m^3]`.
"""
A, B, C = make_params(params["K_0"], params["Kprime_0"], params["Kprime_inf"])
Vrel = volume / params["V_0"]
term1 = A * np.power(Vrel, -(B + 1.0))
term2 = np.exp((2.0 / 3.0) * C * (1.0 - np.power(Vrel, 1.5)))
term3 = C * np.power(Vrel, 1.5) + B
return term1 * term2 * term3 - A * (B + C) + params["P_0"]
def _molar_helmholtz_energy(self, pressure, temperature, volume, params):
"""
Returns the Helmholtz energy :math:`\\mathcal{F}` of the mineral. :math:`[J/mol]`
"""
A, B, C = make_params(params["K_0"], params["Kprime_0"], params["Kprime_inf"])
Vrel = volume / params["V_0"]
I1 = -params["V_0"] * (
np.power(Vrel, -B) * np.exp((2.0 / 3.0) * C * (1.0 - np.power(Vrel, 1.5)))
- 1.0
)
I0 = (-A * (B + C) + params["P_0"]) * params["V_0"] * (Vrel - 1.0)
return params["F_0"] - A * I1 - I0
[docs]
def gibbs_energy(self, pressure, temperature, volume, params):
"""
Returns the Gibbs free energy :math:`\\mathcal{G}` of the mineral. :math:`[J/mol]`
"""
return (
self._molar_helmholtz_energy(pressure, temperature, volume, params)
+ pressure * volume
)
[docs]
def shear_modulus(self, pressure, temperature, volume, params):
"""
Returns shear modulus :math:`G` of the mineral. :math:`[Pa]`
"""
return 1.0e99
[docs]
def validate_parameters(self, params):
"""
Check for existence and validity of the parameters.
The value for :math:`K'_{\\infty}` is thermodynamically bounded
between 5/3 and :math:`K'_0` :cite:`StaceyDavis2004`.
"""
if "F_0" not in params:
params["F_0"] = 0.0
if "P_0" not in params:
params["P_0"] = 1.0e5
if "E_0" in params:
raise KeyError(
"Isothermal equations of state should be "
"defined in terms of Helmholtz free energy "
"F_0, not internal energy E_0."
)
# Check that all the required keys are in the dictionary
expected_keys = ["V_0", "K_0", "Kprime_0", "Kprime_inf"]
for k in expected_keys:
if k not in params:
raise KeyError("params object missing parameter : " + k)
# Finally, check that the values are reasonable.
if params["P_0"] < 0.0:
warnings.warn("Unusual value for P_0", stacklevel=2)
if params["V_0"] < 1.0e-7 or params["V_0"] > 1.0e-3:
warnings.warn("Unusual value for V_0", stacklevel=2)
if params["K_0"] < 1.0e9 or params["K_0"] > 1.0e13:
warnings.warn("Unusual value for K_0", stacklevel=2)
if params["Kprime_0"] < 0.0 or params["Kprime_0"] > 10.0:
warnings.warn("Unusual value for Kprime_0", stacklevel=2)
if params["Kprime_inf"] < 1 + 45.0 / 29.0:
warnings.warn("Value for Kprime_inf below recommended value", stacklevel=2)
if params["Kprime_inf"] > params["Kprime_0"]:
warnings.warn("Kprime_inf should be less than Kprime_0", stacklevel=2)