from __future__ import absolute_import
# This file is part of BurnMan - a thermoelastic and thermodynamic toolkit for the Earth and Planetary Sciences
# Copyright (C) 2012 - 2017 by the BurnMan team, released under the GNU
# GPL v2 or later.
import numpy as np
import scipy.optimize as opt
from . import equation_of_state as eos
from ..utils.math import bracket
import warnings
def bulk_modulus_fourth(volume, params):
"""
compute the bulk modulus as per the fourth order
birch-murnaghan equation of state. Returns bulk
modulus in the same units as the reference bulk
modulus. Pressure must be in :math:`[Pa]`.
"""
x = params["V_0"] / volume
f = 0.5 * (pow(x, 2.0 / 3.0) - 1.0)
Xi = (3.0 / 4.0) * (4.0 - params["Kprime_0"])
Zeta = (3.0 / 8.0) * (
(params["K_0"] * params["Kprime_prime_0"])
+ params["Kprime_0"] * (params["Kprime_0"] - 7.0)
+ 143.0 / 9.0
)
K = (
5.0
* f
* pow((1.0 + 2.0 * f), 5.0 / 2.0)
* params["K_0"]
* (1.0 - (2.0 * Xi * f) + (4.0 * Zeta * pow(f, 2.0)))
) + (
pow(1.0 + (2.0 * f), 7.0 / 2.0)
* params["K_0"]
* (1.0 - (4.0 * Xi * f) + (12.0 * Zeta * pow(f, 2.0)))
)
return K
def volume_fourth_order(pressure, params):
func = lambda x: birch_murnaghan_fourth(params["V_0"] / x, params) - pressure
try:
sol = bracket(func, params["V_0"], 1.0e-2 * params["V_0"])
except:
raise ValueError(
"Cannot find a volume, perhaps you are outside of the range of validity for the equation of state?"
)
return opt.brentq(func, sol[0], sol[1])
def birch_murnaghan_fourth(x, params):
"""
equation for the fourth order birch-murnaghan equation of state, returns
pressure in the same units that are supplied for the reference bulk
modulus (params['K_0'])
"""
f = 0.5 * (pow(x, 2.0 / 3.0) - 1.0)
Xi = (3.0 / 4.0) * (4.0 - params["Kprime_0"])
Zeta = (3.0 / 8.0) * (
(params["K_0"] * params["Kprime_prime_0"])
+ params["Kprime_0"] * (params["Kprime_0"] - 7.0)
+ 143.0 / 9.0
)
return (
3.0
* f
* pow(1.0 + 2.0 * f, 5.0 / 2.0)
* params["K_0"]
* (1.0 - (2.0 * Xi * f) + (4.0 * Zeta * pow(f, 2.0)))
+ params["P_0"]
)
[docs]class BM4(eos.EquationOfState):
"""
Base class for the isothermal Birch Murnaghan equation of state. This is fourth order in strain, and
has no temperature dependence.
"""
[docs] def volume(self, pressure, temperature, params):
"""
Returns volume :math:`[m^3]` as a function of pressure :math:`[Pa]`.
"""
return volume_fourth_order(pressure, params)
[docs] def pressure(self, temperature, volume, params):
return birch_murnaghan_fourth(volume / params["V_0"], params)
[docs] def isothermal_bulk_modulus(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]`.
"""
return bulk_modulus_fourth(volume, params)
[docs] def adiabatic_bulk_modulus(self, pressure, temperature, volume, params):
"""
Returns adiabatic bulk modulus :math:`K_s` of the mineral. :math:`[Pa]`.
"""
return bulk_modulus_fourth(volume, params)
[docs] def shear_modulus(self, pressure, temperature, volume, params):
"""
Returns shear modulus :math:`G` of the mineral. :math:`[Pa]`
"""
return 0.0
[docs] def entropy(self, pressure, temperature, volume, params):
"""
Returns the molar entropy :math:`\mathcal{S}` of the mineral. :math:`[J/K/mol]`
"""
return 0.0
[docs] def molar_internal_energy(self, pressure, temperature, volume, params):
"""
Returns the internal energy :math:`\mathcal{E}` of the mineral. :math:`[J/mol]`
"""
x = np.power(volume / params["V_0"], -1.0 / 3.0)
x2 = x * x
x4 = x2 * x2
x6 = x4 * x2
x8 = x4 * x4
xi1 = 3.0 * (4.0 - params["Kprime_0"]) / 4.0
xi2 = (
3.0
/ 8.0
* (
params["K_0"] * params["Kprime_prime_0"]
+ params["Kprime_0"] * (params["Kprime_0"] - 7.0)
)
+ 143.0 / 24.0
)
intPdV = (
-9.0
/ 2.0
* params["V_0"]
* params["K_0"]
* (
(xi1 + 1.0) * (x4 / 4.0 - x2 / 2.0 + 1.0 / 4.0)
- xi1 * (x6 / 6.0 - x4 / 4.0 + 1.0 / 12.0)
+ xi2 * (x8 / 8 - x6 / 2 + 3.0 * x4 / 4.0 - x2 / 2.0 + 1.0 / 8.0)
)
)
return -intPdV + params["E_0"]
[docs] def gibbs_free_energy(self, pressure, temperature, volume, params):
"""
Returns the Gibbs free energy :math:`\mathcal{G}` of the mineral. :math:`[J/mol]`
"""
# G = int VdP = [PV] - int PdV = E + PV
return (
self.molar_internal_energy(pressure, temperature, volume, params)
+ volume * pressure
)
[docs] def molar_heat_capacity_v(self, pressure, temperature, volume, params):
"""
Since this equation of state does not contain temperature effects, simply return a very large number. :math:`[J/K/mol]`
"""
return 1.0e99
[docs] def molar_heat_capacity_p(self, pressure, temperature, volume, params):
"""
Since this equation of state does not contain temperature effects, simply return a very large number. :math:`[J/K/mol]`
"""
return 1.0e99
[docs] def thermal_expansivity(self, pressure, temperature, volume, params):
"""
Since this equation of state does not contain temperature effects, simply return zero. :math:`[1/K]`
"""
return 0.0
[docs] def grueneisen_parameter(self, pressure, temperature, volume, params):
"""
Since this equation of state does not contain temperature effects, simply return zero. :math:`[unitless]`
"""
return 0.0
[docs] def validate_parameters(self, params):
"""
Check for existence and validity of the parameters
"""
if "E_0" not in params:
params["E_0"] = 0.0
if "P_0" not in params:
params["P_0"] = 0.0
# If G and Gprime are not included this is presumably deliberate,
# as we can model density and bulk modulus just fine without them,
# so just add them to the dictionary as nans
if "G_0" not in params:
params["G_0"] = float("nan")
if "Gprime_0" not in params:
params["Gprime_0"] = float("nan")
# Check that all the required keys are in the dictionary
expected_keys = ["V_0", "K_0", "Kprime_0"]
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_prime_0"] > 0.0 or params["Kprime_prime_0"] < -10.0:
warnings.warn("Unusual value for Kprime_prime_0", stacklevel=2)