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 scipy.optimize as opt
from scipy.special import gamma, gammainc
from . import equation_of_state as eos
from ..tools import bracket
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(fn):
return fn
@jit
def _delta_PoverK_from_P(PoverK, pressure, K_0, Kprime_0, Kprime_inf):
return PoverK - (pressure/K_0)*np.power((1. - Kprime_inf*PoverK), Kprime_0/Kprime_inf) # eq. 58
@jit
def _delta_PoverK_from_V(PoverK, V, V_0, K_0, Kprime_0, Kprime_inf):
Kprime_ratio = Kprime_0 / Kprime_inf
return ( np.log( V_0 / V ) +
Kprime_ratio / Kprime_inf * np.log(1. - Kprime_inf * PoverK) +
(Kprime_ratio - 1.) * PoverK ) # eq. 61
def _upper_incomplete_gamma(z, a):
"""
An implementation of the non-regularised upper incomplete gamma
function. Computed using the relationship with the regularised
lower incomplete gamma function (scipy.special.gammainc).
Uses the recurrence relation wherever z<0.
"""
n = int(-np.floor(z))
if n > 0:
z = z + n
u_gamma = (1. - gammainc(z, a))*gamma(z)
for i in range(n):
z = z - 1.
u_gamma = (u_gamma - np.power(a, z)*np.exp(-a))/z
return u_gamma
else:
return (1. - gammainc(z, a))*gamma(z)
def _PoverK_from_P(pressure, params):
"""
Calculates the pressure:bulk modulus ratio
from a given pressure using brentq optimization
"""
args = ((pressure - params['P_0']), params['K_0'],
params['Kprime_0'], params['Kprime_inf'])
return opt.brentq(_delta_PoverK_from_P,
1./(params['Kprime_inf'] - params['Kprime_0']) + np.finfo(float).eps,
1./params['Kprime_inf'] - np.finfo(float).eps,
args=args)
def _PoverK_from_V(volume, params):
"""
Calculates the pressure:bulk modulus ratio
from a given volume using brentq optimization
"""
args = (volume, params['V_0'], params['K_0'],
params['Kprime_0'], params['Kprime_inf'])
return opt.brentq(_delta_PoverK_from_V,
1./(params['Kprime_inf'] - params['Kprime_0']) + np.finfo(float).eps,
1./params['Kprime_inf'] - np.finfo(float).eps,
args=args)
def bulk_modulus(pressure, params):
"""
Returns the bulk modulus at a given pressure
"""
PoverK = _PoverK_from_P(pressure, params)
K = params['K_0']*np.power((1. - params['Kprime_inf']*PoverK), -
params['Kprime_0']/params['Kprime_inf'])
return K
def shear_modulus(pressure, params):
"""
Returns the shear modulus at a given pressure
"""
G = ( params['G_0']/params['K_0'] * bulk_modulus(pressure, params) -
(params['G_0']/params['K_0']*params['Kprime_inf'] - params['Gprime_inf']) * pressure )
return G # eq. 78
class RKprime(eos.EquationOfState):
"""
Class for the isothermal reciprocal K-prime equation of state
detailed in :cite:`StaceyDavis2004`. This equation of state is
a development of work by :cite:`Keane1954` and :cite:`Stacey2000`,
making use of the fact that :math:`K'` typically varies smoothly
as a function of :math:`P/K`, and is thermodynamically required to
exceed 5/3 at infinite pressure.
It is worth noting that this equation of state rapidly becomes
unstable at negative pressures, so should not be trusted to provide
a good *HT-LP* equation of state using a thermal pressure
formulation. The negative root of :math:`dP/dK`
can be found at :math:`K/P = K'_{\infty} - K'_0`,
which corresponds to a bulk modulus of
:math:`K = K_0 ( 1 - K'_{\infty}/K'_0 )^{K'_0/K'_{\infty}}`
and a volume of
:math:`V = V_0 ( K'_0 / (K'_0 - K'_{\infty}) )^{K'_0/{K'}^2_{\infty}} \exp{(-1/K'_{\infty})}`.
This equation of state has no temperature dependence.
"""
def volume(self, pressure, temperature, params):
"""
Returns volume :math:`[m^3]` as a function of pressure :math:`[Pa]`.
"""
Kprime_ratio = params['Kprime_0']/params['Kprime_inf']
PoverK = _PoverK_from_P(pressure, params)
V = params['V_0'] * np.exp( Kprime_ratio/params['Kprime_inf'] *
np.log(1. - params['Kprime_inf'] * PoverK) +
(Kprime_ratio - 1.) * PoverK ) # Eq. 61
return V
def pressure(self, temperature, volume, params):
"""
Returns pressure :math:`[Pa]` as a function of volume :math:`[m^3]`.
"""
PoverK = _PoverK_from_V(volume, params)
return params['P_0'] + ( params['K_0'] * PoverK *
np.power(1. - params['Kprime_inf'] * PoverK,
-params['Kprime_0']/params['Kprime_inf']) )
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(pressure, params)
def adiabatic_bulk_modulus(self, pressure, temperature, volume, params):
"""
Returns adiabatic bulk modulus :math:`K_s` of the mineral. :math:`[Pa]`.
"""
return bulk_modulus(pressure, params)
def shear_modulus(self, pressure, temperature, volume, params):
"""
Returns shear modulus :math:`G` of the mineral. :math:`[Pa]`
"""
return shear_modulus(pressure, params)
def entropy(self, pressure, temperature, volume, params):
"""
Returns the molar entropy :math:`\mathcal{S}` of the mineral. :math:`[J/K/mol]`
"""
return 0.
def _intVdP(self, xi, params):
a = params['Kprime_inf']
b = (params['Kprime_0']/params['Kprime_inf']/params['Kprime_inf'] -
params['Kprime_0']/params['Kprime_inf'] - 1.)
c = params['Kprime_0'] - params['Kprime_inf']
f = (params['Kprime_0']/params['Kprime_inf'] - 1.)
i1 = float( params['V_0'] * params['K_0'] *
np.exp(f / a) * np.power(a, b - 1.) /
np.power(f, b + 2.) *
( f * params['Kprime_0'] * _upper_incomplete_gamma( b + 1. ,
f * (1./a - xi) ) -
a * c * _upper_incomplete_gamma( b + 2., f * (1./a - xi) ) ) )
return i1
def gibbs_free_energy(self, pressure, temperature, volume, params):
"""
Returns the Gibbs free energy :math:`\mathcal{G}` of the mineral. :math:`[J/mol]`
"""
# G = E0 + int VdP (when S = 0)
K = self.isothermal_bulk_modulus(pressure, temperature, volume, params)
return params['E_0'] + params['P_0']*params['V_0'] + self._intVdP((pressure - params['P_0'])/K, params) - self._intVdP(0., params)
def molar_internal_energy(self, pressure, temperature, volume, params):
"""
Returns the internal energy :math:`\mathcal{E}` of the mineral. :math:`[J/mol]`
"""
# E = G - PV (+ TS)
return ( self.gibbs_free_energy(pressure, temperature, volume, params) - pressure*volume)
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
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
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.
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.
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 'E_0' not in params:
params['E_0'] = 0.
if 'P_0' not in params:
params['P_0'] = 0.
# If G and Gprime_inf 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_inf' not in params:
params['Gprime_inf'] = float('nan')
# Check that all the required keys are in the dictionary
expected_keys = ['V_0', 'K_0', 'Kprime_0', 'Kprime_inf', 'G_0', 'Gprime_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.:
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['Kprime_inf'] < 5./3. or params['Kprime_inf'] > params['Kprime_0']:
warnings.warn('Unusual value for Kprime_inf', stacklevel=2) # eq. 17
if params['G_0'] < 0.0 or params['G_0'] > 1.e13:
warnings.warn('Unusual value for G_0', stacklevel=2)
if params['Gprime_inf'] < -5. or params['Gprime_inf'] > 10.:
warnings.warn('Unusual value for Gprime_inf', stacklevel=2)
