Source code for burnman.eos.slb

# 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 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 ..tools 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)


@jit
def _delta_pressure(x, pressure, temperature, V_0, T_0, Debye_0, n, a1_ii, a2_iikk, b_iikk, b_iikkmm):

    f = 0.5 * (pow(V_0 / x, 2. / 3.) - 1.)
    debye_temperature =  Debye_0 * \
        np.sqrt(1. + a1_ii * f + 1. / 2. * a2_iikk * f * f)
    E_th = debye.thermal_energy(
        temperature, debye_temperature, n)  # thermal energy at temperature T
    E_th_ref = debye.thermal_energy(
        T_0, debye_temperature, n)  # thermal energy at reference temperature
    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 * (E_th - E_th_ref) / x - pressure  # EQ 21


[docs]class SLBBase(eos.EquationOfState): """ Base class for the finite strain-Mie-Grueneiesen-Debye equation of state detailed in :cite:`Stixrude2005`. For the most part the equations are all third order in strain, but see further the :class:`burnman.slb.SLB2` and :class:`burnman.slb.SLB3` classes. """ def _debye_temperature(self, x, params): """ Finite strain approximation for Debye Temperature [K] x = ref_vol/vol """ 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 return params['Debye_0'] * np.sqrt(1. + a1_ii * f + 1. / 2. * a2_iikk * f * f)
[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) eta_s = - gr - \ (1. / 2. * pow(nu_o_nu0_sq, -1.) * pow((2. * f) + 1., 2.) * a2_s) # EQ 46 NOTE the typo from Stixrude 2005 return eta_s # calculate isotropic thermal pressure, see # Matas et. al. (2007) eq B4 def _thermal_pressure(self, T, V, params): Debye_T = self._debye_temperature(params['V_0'] / V, params) gr = self.grueneisen_parameter(0., T, V, params) # P not important P_th = gr * debye.thermal_energy(T, Debye_T, params['n']) / V return P_th
[docs] def volume(self, pressure, temperature, params): """ Returns molar volume. :math:`[m^3]` """ T_0 = params['T_0'] Debye_0 = params['Debye_0'] V_0 = params['V_0'] n = params['n'] 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, Debye_0, n, 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] """ debye_T = self._debye_temperature(params['V_0'] / volume, params) gr = self.grueneisen_parameter( 0.0, temperature, volume, params) # does not depend on pressure E_th = debye.thermal_energy(temperature, debye_T, params['n']) E_th_ref = debye.thermal_energy( params['T_0'], debye_T, params['n']) # thermal energy at reference temperature 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 * (E_th - E_th_ref) / 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]` """ T_0 = params['T_0'] debye_T = self._debye_temperature(params['V_0'] / volume, params) gr = self.grueneisen_parameter(pressure, temperature, volume, params) E_th = debye.thermal_energy( temperature, debye_T, params['n']) # thermal energy at temperature T E_th_ref = debye.thermal_energy( T_0, debye_T, params['n']) # thermal energy at reference temperature C_v = debye.heat_capacity_v( temperature, debye_T, params['n']) # heat capacity at temperature T C_v_ref = debye.heat_capacity_v( T_0, debye_T, params['n']) # heat capacity at reference temperature q = self.volume_dependent_q(params['V_0'] / volume, params) K = bm.bulk_modulus(volume, params) \ + (gr + 1. - q) * ( gr / volume ) * (E_th - E_th_ref) \ - (pow(gr, 2.) / volume) * (C_v * temperature - C_v_ref * T_0) 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'] debye_T = self._debye_temperature(params['V_0'] / volume, params) eta_s = self._isotropic_eta_s(params['V_0'] / volume, params) E_th = debye.thermal_energy(temperature, debye_T, params['n']) E_th_ref = debye.thermal_energy(T_0, debye_T, params['n']) if self.order == 2: return bm.shear_modulus_second_order(volume, params) - eta_s * (E_th - E_th_ref) / volume elif self.order == 3: return bm.shear_modulus_third_order(volume, params) - eta_s * (E_th - E_th_ref) / volume else: raise NotImplementedError("")
[docs] def heat_capacity_v(self, pressure, temperature, volume, params): """ Returns heat capacity at constant volume. :math:`[J/K/mol]` """ debye_T = self._debye_temperature(params['V_0'] / volume, params) return debye.heat_capacity_v(temperature, debye_T, params['n'])
[docs] def 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.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.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 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] """ x = params['V_0'] / volume f = 1. / 2. * (pow(x, 2. / 3.) - 1.) Debye_T = self._debye_temperature(params['V_0'] / volume, params) S = debye.entropy(temperature, Debye_T, params['n']) return S
[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] """ x = params['V_0'] / volume f = 1. / 2. * (pow(x, 2. / 3.) - 1.) Debye_T = self._debye_temperature(params['V_0'] / volume, params) F_quasiharmonic = debye.helmholtz_free_energy( temperature, Debye_T, params['n'] ) - \ debye.helmholtz_free_energy( params['T_0'], Debye_T, params['n']) b_iikk = 9. * params['K_0'] # EQ 28 b_iikkmm = 27. * params['K_0'] * (params['Kprime_0'] - 4.) # EQ 29 F = params['F_0'] + \ 0.5 * b_iikk * f * f * params['V_0'] + (1. / 6.) * params['V_0'] * b_iikkmm * f * f * f +\ F_quasiharmonic return F
[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 'F_0' not in params: params['F_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 = [ 'molar_mass', 'n', 'Debye_0', '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['molar_mass'] < 0.001 or params['molar_mass'] > 10.: warnings.warn('Unusual value for molar_mass', stacklevel=2) if params['n'] < 1. or params['n'] > 1000.: warnings.warn('Unusual value for n', stacklevel=2) if params['Debye_0'] < 1. or params['Debye_0'] > 10000.: warnings.warn('Unusual value for Debye_0', 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)
class SLB3(SLBBase): """ SLB equation of state with third order finite strain expansion for the shear modulus (this should be preferred, as it is more thermodynamically consistent.) """ def __init__(self): self.order = 3 class SLB2(SLBBase): """ SLB equation of state with second order finite strain expansion for the shear modulus. In general, this should not be used, but sometimes shear modulus data is fit to a second order equation of state. In that case, you should use this. The moral is, be careful! """ def __init__(self): self.order = 2