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 - 2015 by the BurnMan team, released under the GNU
# GPL v2 or later.
import scipy.optimize as opt
from . import equation_of_state as eos
from ..tools import bracket
import warnings
import numpy as np
def bulk_modulus(volume, params):
"""
Compute the bulk modulus as per the Morse potential
equation of state.
Returns bulk modulus in the same units as
the reference bulk modulus.
Pressure must be in :math:`[Pa]`.
"""
VoverV0 = volume / params['V_0']
x = (params['Kprime_0'] - 1.)*(1. - np.power(VoverV0, 1./3.))
K = params['K_0']*( ( 2./(params['Kprime_0'] - 1.) *
np.power(VoverV0, -2./3.) *
(np.exp(2.*x) - np.exp(x)) ) +
( np.power(VoverV0, -1./3.) *
(2.*np.exp(2.*x) - np.exp(x)) ) )
return K
def shear_modulus(volume, params):
"""
Shear modulus not currently implemented for this equation of state
"""
return 0.
def morse_potential(VoverV0, params):
"""
Equation for the Morse Potential equation of state,
returns pressure in the same units that are supplied
for the reference bulk modulus (params['K_0'])
"""
x = (params['Kprime_0'] - 1.)*(1. - np.power(VoverV0, 1./3.))
return ( 3. * params['K_0'] / (params['Kprime_0'] - 1.) *
np.power(VoverV0, -2./3.) *
(np.exp(2.*x) - np.exp(x)) ) + params['P_0']
def volume(pressure, params):
"""
Get the Morse Potential volume at a
reference temperature for a given pressure :math:`[Pa]`.
Returns molar volume in :math:`[m^3]`
"""
func = lambda V: morse_potential(V / params['V_0'], 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])
[docs]class Morse(eos.EquationOfState):
"""
Class for the isothermal Morse Potential equation of state
detailed in :cite:`Stacey1981`.
This equation of state 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(pressure, params)
[docs] def pressure(self, temperature, volume, params):
return morse_potential(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(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]`
"""
return shear_modulus(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 = (params['Kprime_0'] - 1)*(1 - np.power(volume/params['V_0'], 1./3.))
intPdV = ( 9./2. * params['V_0'] * params['K_0'] /
np.power(params['Kprime_0'] - 1., 2.) *
(2.*np.exp(x) - np.exp(2.*x) - 1.) )
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]`
"""
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)