# This file is part of BurnMan - a thermoelastic and thermodynamic toolkit for the Earth and Planetary Sciences
# Copyright (C) 2012 - 2015 by the BurnMan team, released under the GNU
# GPL v2 or later.
from __future__ import absolute_import
import numpy as np
import scipy.optimize as opt
import scipy.integrate as integrate
import warnings
# 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(fn):
return fn
from . import birch_murnaghan as bm
from . import debye
from . import equation_of_state as eos
from ..utils.math import bracket
@jit
def _grueneisen_parameter_fast(V_0, volume, gruen_0, q_0):
"""global function with plain parameters so jit will work"""
x = V_0 / volume
f = 1. / 2. * (pow(x, 2. / 3.) - 1.)
a1_ii = 6. * gruen_0 # EQ 47
a2_iikk = -12. * gruen_0 + 36. * \
gruen_0 * gruen_0 - 18. * q_0 * gruen_0 # EQ 47
nu_o_nu0_sq = 1. + a1_ii * f + (1. / 2.) * a2_iikk * f * f # EQ 41
return 1. / 6. / nu_o_nu0_sq * (2. * f + 1.) * (a1_ii + a2_iikk * f)
def _intgroverVdV(V_0, volume, gruen_0, q_0):
return integrate.quad(lambda x: _grueneisen_parameter_fast(V_0, x, gruen_0, q_0)/x, V_0, volume)[0]
@jit
def _delta_pressure(x, pressure, temperature, V_0, T_0, Cv, a1_ii, a2_iikk, b_iikk, b_iikkmm):
f = 0.5 * (pow(V_0 / x, 2. / 3.) - 1.)
nu_o_nu0_sq = 1. + a1_ii * f + (1. / 2.) * a2_iikk * f * f # EQ 41
gr = 1. / 6. / nu_o_nu0_sq * (2. * f + 1.) * (a1_ii + a2_iikk * f)
return (1. / 3.) * (pow(1. + 2. * f, 5. / 2.)) * ((b_iikk * f) + (0.5 * b_iikkmm * f * f)) \
+ gr * Cv * (temperature - T_0) / x - pressure # EQ 21
[docs]class DKS_S(eos.EquationOfState):
"""
Base class for the finite strain solid equation of state detailed
in :cite:`deKoker2013` (supplementary materials).
"""
[docs] def volume_dependent_q(self, x, params):
"""
Finite strain approximation for :math:`q`, the isotropic volume strain
derivative of the grueneisen parameter.
"""
f = 1. / 2. * (pow(x, 2. / 3.) - 1.)
a1_ii = 6. * params['grueneisen_0'] # EQ 47
a2_iikk = -12. * params['grueneisen_0'] + 36. * pow(
params['grueneisen_0'], 2.) - 18. * params['q_0'] * params['grueneisen_0'] # EQ 47
nu_o_nu0_sq = 1. + a1_ii * f + (1. / 2.) * a2_iikk * f * f # EQ 41
gr = 1. / 6. / nu_o_nu0_sq * (2. * f + 1.) * (a1_ii + a2_iikk * f)
if np.abs(params['grueneisen_0']) < 1.e-10: # avoids divide by zero if grueneisen_0 = 0.
q = 1. / 9. * (18. * gr - 6.)
else:
q = 1. / 9. * \
(18. * gr - 6. - 1. / 2. / nu_o_nu0_sq *
(2. * f + 1.) * (2. * f + 1.) * a2_iikk / gr)
return q
def _isotropic_eta_s(self, x, params):
"""
Finite strain approximation for :math:`eta_{s0}`, the isotropic shear
strain derivative of the grueneisen parameter.
"""
f = 1. / 2. * (pow(x, 2. / 3.) - 1.)
a2_s = -2. * params['grueneisen_0'] - 2. * params['eta_s_0'] # EQ 47
a1_ii = 6. * params['grueneisen_0'] # EQ 47
a2_iikk = -12. * params['grueneisen_0'] + 36. * pow(
params['grueneisen_0'], 2.) - 18. * params['q_0'] * params['grueneisen_0'] # EQ 47
nu_o_nu0_sq = 1. + a1_ii * f + \
(1. / 2.) * a2_iikk * pow(f, 2.) # EQ 41
gr = 1. / 6. / nu_o_nu0_sq * (2. * f + 1.) * (a1_ii + a2_iikk * f)
# EQ 46 NOTE the typo from Stixrude 2005:
eta_s = - gr - \
(1. / 2. * pow(nu_o_nu0_sq, -1.) * pow((2. * f) + 1., 2.) * a2_s)
return eta_s
# calculate isotropic thermal pressure, see
# Matas et. al. (2007) eq B4
def _thermal_pressure(self, T, V, params):
gr = self.grueneisen_parameter(0., T, V, params) # P not important
return params['Cv'] * (T - params['T_0']) * gr
[docs] def volume(self, pressure, temperature, params):
"""
Returns molar volume. :math:`[m^3]`
"""
T_0 = params['T_0']
V_0 = params['V_0']
Cv = params['Cv']
a1_ii = 6. * params['grueneisen_0'] # EQ 47
a2_iikk = -12. * params['grueneisen_0'] + 36. * pow(
params['grueneisen_0'], 2.) - 18. * params['q_0'] * params['grueneisen_0'] # EQ 47
b_iikk = 9. * params['K_0'] # EQ 28
b_iikkmm = 27. * params['K_0'] * (params['Kprime_0'] - 4.) # EQ 29z
# we need to have a sign change in [a,b] to find a zero. Let us start with a
# conservative guess:
args = (pressure, temperature, V_0, T_0,
Cv, a1_ii, a2_iikk, b_iikk, b_iikkmm)
try:
sol = bracket(_delta_pressure, params[
'V_0'], 1.e-2 * params['V_0'], args)
except ValueError:
raise Exception(
'Cannot find a volume, perhaps you are outside of the range of validity for the equation of state?')
return opt.brentq(_delta_pressure, sol[0], sol[1], args=args)
[docs] def pressure(self, temperature, volume, params):
"""
Returns the pressure of the mineral at a given temperature and volume [Pa]
"""
gr = self.grueneisen_parameter(
0.0, temperature, volume, params) # does not depend on pressure
b_iikk = 9. * params['K_0'] # EQ 28
b_iikkmm = 27. * params['K_0'] * (params['Kprime_0'] - 4.) # EQ 29
f = 0.5 * (pow(params['V_0'] / volume, 2. / 3.) - 1.) # EQ 24
P = (1. / 3.) * (pow(1. + 2. * f, 5. / 2.)) \
* ((b_iikk * f) + (0.5 * b_iikkmm * pow(f, 2.)))\
+ gr * params['Cv'] * (temperature - params['T_0']) / volume # EQ 21
return P
[docs] def grueneisen_parameter(self, pressure, temperature, volume, params):
"""
Returns grueneisen parameter :math:`[unitless]`
"""
return _grueneisen_parameter_fast(params['V_0'], volume, params['grueneisen_0'], params['q_0'])
[docs] def isothermal_bulk_modulus(self, pressure, temperature, volume, params):
"""
Returns isothermal bulk modulus :math:`[Pa]`
"""
E_th_diff = params['Cv'] * (temperature - params['T_0'])
gr = self.grueneisen_parameter(pressure, temperature, volume, params)
q = self.volume_dependent_q(params['V_0'] / volume, params)
K = bm.bulk_modulus(volume, params) \
+ (gr + 1. - q) * (gr / volume) * E_th_diff \
- (pow(gr, 2.) / volume) * E_th_diff
return K
[docs] def adiabatic_bulk_modulus(self, pressure, temperature, volume, params):
"""
Returns adiabatic bulk modulus. :math:`[Pa]`
"""
K_T = self.isothermal_bulk_modulus(
pressure, temperature, volume, params)
alpha = self.thermal_expansivity(pressure, temperature, volume, params)
gr = self.grueneisen_parameter(pressure, temperature, volume, params)
K_S = K_T * (1. + gr * alpha * temperature)
return K_S
[docs] def shear_modulus(self, pressure, temperature, volume, params):
"""
Returns shear modulus. :math:`[Pa]`
"""
T_0 = params['T_0']
eta_s = self._isotropic_eta_s(params['V_0'] / volume, params)
E_th_diff = params['Cv'] * (temperature - params['T_0'])
return bm.shear_modulus_third_order(volume, params) - eta_s * (E_th_diff) / volume
[docs] def molar_heat_capacity_v(self, pressure, temperature, volume, params):
"""
Returns heat capacity at constant volume. :math:`[J/K/mol]`
"""
return params['Cv']
[docs] def molar_heat_capacity_p(self, pressure, temperature, volume, params):
"""
Returns heat capacity at constant pressure. :math:`[J/K/mol]`
"""
alpha = self.thermal_expansivity(pressure, temperature, volume, params)
gr = self.grueneisen_parameter(pressure, temperature, volume, params)
C_v = self.molar_heat_capacity_v(pressure, temperature, volume, params)
C_p = C_v * (1. + gr * alpha * temperature)
return C_p
[docs] def thermal_expansivity(self, pressure, temperature, volume, params):
"""
Returns thermal expansivity. :math:`[1/K]`
"""
C_v = self.molar_heat_capacity_v(pressure, temperature, volume, params)
gr = self.grueneisen_parameter(pressure, temperature, volume, params)
K = self.isothermal_bulk_modulus(pressure, temperature, volume, params)
alpha = gr * C_v / K / volume
return alpha
[docs] def gibbs_free_energy(self, pressure, temperature, volume, params):
"""
Returns the Gibbs free energy at the pressure and temperature of the mineral [J/mol]
"""
G = self.helmholtz_free_energy(
pressure, temperature, volume, params) + pressure * volume
return G
[docs] def molar_internal_energy(self, pressure, temperature, volume, params):
"""
Returns the internal energy at the pressure and temperature of the mineral [J/mol]
"""
return self.helmholtz_free_energy(pressure, temperature, volume, params) + \
temperature * \
self.entropy(pressure, temperature, volume, params)
[docs] def entropy(self, pressure, temperature, volume, params):
"""
Returns the entropy at the pressure and temperature of the mineral [J/K/mol]
"""
S_0 = params['S_0']
gruen_0 = params['grueneisen_0']
q_0 = params['q_0']
S_th = params['Cv'] * (np.log(temperature/params['T_0']) +
_intgroverVdV(params['V_0'], volume, gruen_0, q_0))
return S_0 + S_th
[docs] def enthalpy(self, pressure, temperature, volume, params):
"""
Returns the enthalpy at the pressure and temperature of the mineral [J/mol]
"""
return self.helmholtz_free_energy(pressure, temperature, volume, params) + \
temperature * self.entropy(pressure, temperature, volume, params) + \
pressure * volume
[docs] def helmholtz_free_energy(self, pressure, temperature, volume, params):
"""
Returns the Helmholtz free energy at the pressure and temperature of the mineral [J/mol]
"""
V_0 = params['V_0']
gruen_0 = params['grueneisen_0']
q_0 = params['q_0']
x = V_0 / volume
f = 1. / 2. * (pow(x, 2. / 3.) - 1.)
b_iikk = 9. * params['K_0'] # EQ 28
b_iikkmm = 27. * params['K_0'] * (params['Kprime_0'] - 4.) # EQ 29
T_0 = params['T_0']
T = temperature
S_0 = params['S_0']
Cv = params['Cv']
F_0 = params['E_0'] - T_0 * S_0
F_cmp = 0.5 * b_iikk * f * f * V_0 + (1. / 6.) * V_0 * b_iikkmm * f * f * f
F_th = - S_0 * (T - T_0) - \
Cv * (T * np.log(T/T_0) - (T - T_0)) - \
Cv * (T - T_0) * _intgroverVdV(V_0, volume, gruen_0, q_0)
return F_0 + F_cmp + F_th
[docs] def validate_parameters(self, params):
"""
Check for existence and validity of the parameters
"""
if 'T_0' not in params:
params['T_0'] = 300.
# If eta_s_0 is not included this is presumably deliberate,
# as we can model density and bulk modulus just fine without it,
# so just add it to the dictionary as nan
# The same goes for the standard state Helmholtz free energy
if 'eta_s_0' not in params:
params['eta_s_0'] = float('nan')
if 'E_0' not in params:
params['E_0'] = float('nan')
# First, let's check the EoS parameters for Tref
bm.BirchMurnaghanBase.validate_parameters(
bm.BirchMurnaghanBase(), params)
# Now check all the required keys for the
# thermal part of the EoS are in the dictionary
expected_keys = ['Cv', 'grueneisen_0', 'q_0', 'eta_s_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['T_0'] < 0.:
warnings.warn('Unusual value for T_0', stacklevel=2)
if params['Cv'] < 0. or params['Cv'] > 1000.:
warnings.warn('Unusual value for Cv', stacklevel=2)
if params['grueneisen_0'] < -0.005 or params['grueneisen_0'] > 10.:
warnings.warn('Unusual value for grueneisen_0', stacklevel=2)
if params['q_0'] < -10. or params['q_0'] > 10.:
warnings.warn('Unusual value for q_0', stacklevel=2)
if params['eta_s_0'] < -10. or params['eta_s_0'] > 10.:
warnings.warn('Unusual value for eta_s_0', stacklevel=2)