# 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 . import equation_of_state as eos
import warnings
from math import exp
def bulk_modulus(volume, params):
"""
compute the bulk modulus as per the
Vinet equation of state. Reference bulk
modulus should be in :math:`[Pa]`.
"""
x = volume / params['V_0']
eta = (3. / 2.) * (params['Kprime_0'] - 1.)
K = (params['K_0'] * pow(x, -2. / 3.)) * \
(1 + ((eta * pow(x, 1. / 3.) + 1.) * (1. - pow(x, 1. / 3.)))) * \
exp(eta * (1. - pow(x, 1. / 3.)))
return K
def vinet(x, params):
"""
equation for the Vinet equation of state, returns
pressure in the same units that are supplied for the reference bulk
modulus (params['K_0']), which should be in math:`[Pa]`.
"""
eta = (3. / 2.) * (params['Kprime_0'] - 1.)
return 3. * params['K_0'] * (pow(x, -2. / 3.)) * (1. - (pow(x, 1. / 3.))) \
* exp(eta * (1. - pow(x, 1. / 3.))) + params['P_0']
def volume(pressure, params):
"""
Get the Vinet volume at a reference temperature for a given
pressure :math:`[Pa]`. Returns molar volume in :math:`[m^3]`
"""
func = lambda x: vinet(x / params['V_0'], params) - pressure
V = opt.brentq(func, 0.1 * params['V_0'], 1.5 * params['V_0'])
return V
[docs]class Vinet(eos.EquationOfState):
"""
Base class for the isothermal Vinet equation of state.
References for this equation of state are :cite:`vinet1986`
and :cite:`vinet1987`. This equation of state actually
predates Vinet by 55 years :cite:`Rydberg1932`,
and was investigated further by :cite:`Stacey1981`.
"""
[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 vinet(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]`
Currently not included in the Vinet EOS, so omitted.
"""
return 0.
[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 = pow(volume/params['V_0'], 1./3.)
eta = (3. / 2.) * (params['Kprime_0'] - 1.)
intPdV = (9.* params['V_0'] * params['K_0'] / (eta*eta) *
((1. - eta*(1. - x))*exp(eta*(1. - 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]`
"""
# 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.
# G is not included in the Vinet EOS so we shall set them to NaN's
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']
for k in expected_keys:
if k not in params:
raise KeyError('params object missing parameter : ' + k)
# now check that the values are reasonable. I mostly just
# made up these values from experience, and we are only
# raising a warning. Better way to do this? [IR]
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'] < -5. or params['Kprime_0'] > 10.:
warnings.warn('Unusual value for Kprime_0', stacklevel=2)