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(volume, params):
"""
compute the bulk modulus as per the third 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. / 3.) - 1.0)
K = pow(1. + 2. * f, 5. / 2.) * (params['K_0'] + (3. * params['K_0'] * params['Kprime_0'] -
5 * params['K_0']) * f + 27. / 2. * (params['K_0'] * params['Kprime_0'] - 4. * params['K_0']) * f * f)
return K
def birch_murnaghan(x, params):
"""
equation for the third order birch-murnaghan equation of state, returns
pressure in the same units that are supplied for the reference bulk
modulus (params['K_0'])
"""
return 3. * params['K_0'] / 2. * (pow(x, 7. / 3.) - pow(x, 5. / 3.)) \
* (1. - .75 * (4. - params['Kprime_0']) * (pow(x, 2. / 3.) - 1.)) + params['P_0']
def volume(pressure, params):
"""
Get the birch-murnaghan volume at a reference temperature for a given
pressure :math:`[Pa]`. Returns molar volume in :math:`[m^3]`
"""
func = lambda x: birch_murnaghan(params['V_0'] / x, params) - pressure
try:
sol = bracket(func, params['V_0'], 1.e-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 shear_modulus_second_order(volume, params):
"""
Get the birch murnaghan shear modulus at a reference temperature, for a
given volume. Returns shear modulus in :math:`[Pa]` (the same units as in
params['G_0']). This uses a second order finite strain expansion
"""
x = params['V_0'] / volume
G = params['G_0'] * pow(x, 5. / 3.) * (
1. - 0.5 * (pow(x, 2. / 3.) - 1.) * (5. - 3. * params['Gprime_0'] * params['K_0'] / params['G_0']))
return G
def shear_modulus_third_order(volume, params):
"""
Get the birch murnaghan shear modulus at a reference temperature, for a
given volume. Returns shear modulus in :math:`[Pa]` (the same units as in
params['G_0']). This uses a third order finite strain expansion
"""
x = params['V_0'] / volume
f = 0.5 * (pow(x, 2. / 3.) - 1.0)
G = pow((1. + 2. * f), 5. / 2.) * (params['G_0'] + (3. * params['K_0'] * params['Gprime_0'] - 5. * params['G_0']) * f + (
6. * params['K_0'] * params['Gprime_0'] - 24. * params['K_0'] - 14. * params['G_0'] + 9. / 2. * params['K_0'] * params['Kprime_0']) * f * f)
return G
[docs]class BirchMurnaghanBase(eos.EquationOfState):
"""
Base class for the isothermal Birch Murnaghan equation of state. This is third order in strain, and
has no temperature dependence. However, the shear modulus is sometimes fit to a second order
function, so if this is the case, you should use that. For more see :class:`burnman.birch_murnaghan.BM2` and :class:`burnman.birch_murnaghan.BM3`.
"""
[docs] def volume(self, pressure, temperature, params):
"""
Returns volume :math:`[m^3]` as a function of pressure :math:`[Pa]`.
"""
return volume(pressure, params)
[docs] def pressure(self, temperature, volume, params):
return birch_murnaghan(params['V_0'] / volume, 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(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(volume, params)
[docs] def shear_modulus(self, pressure, temperature, volume, params):
"""
Returns shear modulus :math:`G` of the mineral. :math:`[Pa]`
"""
if(self.order == 2):
return shear_modulus_second_order(volume, params)
elif(self.order == 3):
return shear_modulus_third_order(volume, params)
[docs] def entropy(self, pressure, temperature, volume, params):
"""
Returns the molar entropy :math:`\mathcal{S}` of the mineral. :math:`[J/K/mol]`
"""
return 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./3.)
x2 = x*x
x4 = x2*x2
x6 = x4*x2
x8 = x4*x4
xi1 = 3.*(4. - params['Kprime_0'])/4.
intPdV = (-9./2. * params['V_0'] * params['K_0'] *
((xi1 + 1.)*(x4/4. - x2/2. + 1./4.) -
xi1*(x6/6. - x4/4. + 1./12.)))
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.e99
[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.e99
[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.
[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.
[docs] def validate_parameters(self, params):
"""
Check for existence and validity of the parameters
"""
if 'E_0' not in params:
params['E_0'] = 0.
if 'P_0' not in params:
params['P_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', 'G_0', 'Gprime_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.:
warnings.warn('Unusual value for P_0', stacklevel=2)
if params['V_0'] < 1.e-7 or params['V_0'] > 1.e-3:
warnings.warn('Unusual value for V_0', stacklevel=2)
if params['K_0'] < 1.e9 or params['K_0'] > 1.e13:
warnings.warn('Unusual value for K_0', stacklevel=2)
if params['Kprime_0'] < 0. or params['Kprime_0'] > 10.:
warnings.warn('Unusual value for Kprime_0', stacklevel=2)
if params['G_0'] < 0.0 or params['G_0'] > 1.e13:
warnings.warn('Unusual value for G_0', stacklevel=2)
if params['Gprime_0'] < -5. or params['Gprime_0'] > 10.:
warnings.warn('Unusual value for Gprime_0', stacklevel=2)
[docs]class BM3(BirchMurnaghanBase):
"""
Third order Birch Murnaghan isothermal equation of state.
This uses the third order expansion for shear modulus.
"""
def __init__(self):
self.order = 3
[docs]class BM2(BirchMurnaghanBase):
"""
Third order Birch Murnaghan isothermal equation of state.
This uses the second order expansion for shear modulus.
"""
def __init__(self):
self.order = 2