# 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.
from __future__ import absolute_import
import numpy as np
from . import processchemistry
from . import constants
# Try to import the jit from numba. If it is
# not available, just go with the standard
# python interpreter
try:
import os
if (('NUMBA_DISABLE_JIT' in os.environ
and int(os.environ['NUMBA_DISABLE_JIT']) == 1)):
raise ImportError("NOOOO!")
from numba import njit
except ImportError:
def njit(fn):
return fn
def _ideal_activities_fct(molar_fractions, endmember_occupancies, n_endmembers,
n_occupancies, site_multiplicities,
endmember_configurational_entropies):
site_occupancies = np.dot(molar_fractions, endmember_occupancies)
a = np.power(site_occupancies,
endmember_occupancies * site_multiplicities).prod(-1)
normalisation_constants = np.exp(endmember_configurational_entropies
/ constants.gas_constant)
return normalisation_constants * a
def _non_ideal_hessian_fct(phi, molar_fractions, n_endmembers, alpha, W):
q = np.eye(n_endmembers) - phi*np.ones((n_endmembers, n_endmembers))
sum_pa = np.dot(molar_fractions, alpha)
hess = np.einsum('m, i, ij, jk, mk->im', -alpha/sum_pa, -alpha, q, W, q)
hess += hess.T
return hess
def _non_ideal_interactions_fct(phi, molar_fractions, n_endmembers, alpha, W):
# -sum(sum(qi.qj.Wij*)
# equation (2) of Holland and Powell 2003
q = np.eye(n_endmembers) - phi*np.ones((n_endmembers, n_endmembers))
# The following are equivalent to
# np.einsum('i, ij, jk, ik->i', -self.alphas, q, self.Wx, q)
Wint = -alpha * (q.dot(W)*q).sum(-1)
return Wint
@njit
def _non_ideal_hessian_subreg(p, n_endmembers, W):
hess = np.zeros((n_endmembers, n_endmembers))
for k in range(n_endmembers):
for m in range(n_endmembers):
for i in range(n_endmembers):
for j in range(n_endmembers):
dik = 1. if i == k else 0.
djk = 1. if j == k else 0.
dim = 1. if i == m else 0.
djm = 1. if j == m else 0.
hess[k, m] += W[i, j] * ((djk*djm*p[i] - dik*dim*p[j])
+ (dik*djm + djk*dim)
* (p[j] - p[i])
+ (djk + djm)
* (p[i]*(p[i] - 2.*p[j]))
- (dik + dim)
* (p[j]*(p[j] - 2.*p[i]))
+ 3.*p[i]*p[j]
* (p[j] - p[i])
+ (((dik + dim) - p[i])
* ((djk + djm) - p[j])
+ p[i]*p[j])
/ 2.)
return hess
@njit
def _non_ideal_interactions_subreg(p, n_endmembers, W):
Wint = np.zeros(n_endmembers)
for k in range(n_endmembers):
for i in range(n_endmembers):
for j in range(n_endmembers):
dik = 1. if i == k else 0.
djk = 1. if j == k else 0.
Wint[k] += W[i, j]/2. * ((p[i] - dik)*(p[j] - djk)
* (2.*(p[i] - p[j]) - 1.)
+ djk*p[i]*p[i] - dik*p[j]*p[j])
return Wint
def logish(x, eps=1.e-5):
"""
2nd order series expansion of log(x) about eps:
log(eps) - sum_k=1^infty (f_eps)^k / k
Prevents infinities at x=0
"""
f_eps = 1. - x/eps
mask = x > eps
ln = np.where(x <= eps, np.log(eps) - f_eps - f_eps*f_eps/2., 0.)
ln[mask] = np.log(x[mask])
return ln
def inverseish(x, eps=1.e-5):
"""
1st order series expansion of 1/x about eps: 2/eps - x/eps/eps
Prevents infinities at x=0
"""
mask = x > eps
oneoverx = np.where(x <= eps, 2./eps - x/eps/eps, 0.)
oneoverx[mask] = 1./x[mask]
return oneoverx
[docs]class SolutionModel(object):
"""
This is the base class for a solution model, intended for use
in defining solid solutions and performing thermodynamic calculations
on them. All minerals of type :class:`burnman.SolidSolution` use
a solution model for defining how the endmembers in the solid solution
interact.
A user wanting a new solution model should define the functions included
in the base class. All of the functions in the base class return zero,
so if the user-defined solution model does not implement them,
they essentially have no effect, and the Gibbs free energy and molar
volume of a solid solution will be equal to the weighted arithmetic
averages of the different endmember values.
"""
def __init__(self):
"""
Does nothing.
"""
pass
[docs] def excess_gibbs_free_energy(self, pressure, temperature, molar_fractions):
"""
Given a list of molar fractions of different phases,
compute the excess Gibbs free energy of the solution.
The base class implementation assumes that the excess gibbs
free energy is zero.
Parameters
----------
pressure : float
Pressure at which to evaluate the solution model. [Pa]
temperature : float
Temperature at which to evaluate the solution. [K]
molar_fractions : list of floats
List of molar fractions of the different endmembers in solution
Returns
-------
G_excess : float
The excess Gibbs free energy
"""
return np.dot(np.array(molar_fractions), self.excess_partial_gibbs_free_energies(pressure, temperature, molar_fractions))
[docs] def excess_volume(self, pressure, temperature, molar_fractions):
"""
Given a list of molar fractions of different phases,
compute the excess volume of the solution.
The base class implementation assumes that the excess volume is zero.
Parameters
----------
pressure : float
Pressure at which to evaluate the solution model. [Pa]
temperature : float
Temperature at which to evaluate the solution. [K]
molar_fractions : list of floats
List of molar fractions of the different endmembers in solution
Returns
-------
V_excess : float
The excess volume of the solution
"""
return np.dot(molar_fractions,
self.excess_partial_volumes(pressure, temperature, molar_fractions))
[docs] def excess_entropy(self, pressure, temperature, molar_fractions):
"""
Given a list of molar fractions of different phases,
compute the excess entropy of the solution.
The base class implementation assumes that the excess entropy is zero.
Parameters
----------
pressure : float
Pressure at which to evaluate the solution model. [Pa]
temperature : float
Temperature at which to evaluate the solution. [K]
molar_fractions : list of floats
List of molar fractions of the different endmembers in solution
Returns
-------
S_excess : float
The excess entropy of the solution
"""
return np.dot(molar_fractions,
self.excess_partial_entropies(pressure, temperature, molar_fractions))
[docs] def excess_enthalpy(self, pressure, temperature, molar_fractions):
"""
Given a list of molar fractions of different phases,
compute the excess enthalpy of the solution.
The base class implementation assumes that the excess enthalpy is zero.
Parameters
----------
pressure : float
Pressure at which to evaluate the solution model. [Pa]
temperature : float
Temperature at which to evaluate the solution. [K]
molar_fractions : list of floats
List of molar fractions of the different endmembers in solution
Returns
-------
H_excess : float
The excess enthalpy of the solution
"""
return (self.excess_gibbs_free_energy(pressure, temperature, molar_fractions)
+ temperature*self.excess_entropy(pressure, temperature, molar_fractions))
[docs] def excess_partial_gibbs_free_energies(self, pressure, temperature, molar_fractions):
"""
Given a list of molar fractions of different phases,
compute the excess Gibbs free energy for each endmember of the solution.
The base class implementation assumes that the excess gibbs
free energy is zero.
Parameters
----------
pressure : float
Pressure at which to evaluate the solution model. [Pa]
temperature : float
Temperature at which to evaluate the solution. [K]
molar_fractions : list of floats
List of molar fractions of the different endmembers in solution
Returns
-------
partial_G_excess : numpy array
The excess Gibbs free energy of each endmember
"""
return np.zeros_like(molar_fractions)
[docs] def excess_partial_entropies(self, pressure, temperature, molar_fractions):
"""
Given a list of molar fractions of different phases,
compute the excess entropy for each endmember of the solution.
The base class implementation assumes that the excess entropy
is zero (true for mechanical solutions).
Parameters
----------
pressure : float
Pressure at which to evaluate the solution model. [Pa]
temperature : float
Temperature at which to evaluate the solution. [K]
molar_fractions : list of floats
List of molar fractions of the different endmembers in solution
Returns
-------
partial_S_excess : numpy array
The excess entropy of each endmember
"""
return np.zeros_like(molar_fractions)
[docs] def excess_partial_volumes(self, pressure, temperature, molar_fractions):
"""
Given a list of molar fractions of different phases,
compute the excess volume for each endmember of the solution.
The base class implementation assumes that the excess volume
is zero.
Parameters
----------
pressure : float
Pressure at which to evaluate the solution model. [Pa]
temperature : float
Temperature at which to evaluate the solution. [K]
molar_fractions : list of floats
List of molar fractions of the different endmembers in solution
Returns
-------
partial_V_excess : numpy array
The excess volume of each endmember
"""
return np.zeros_like(np.array(molar_fractions))
[docs]class MechanicalSolution (SolutionModel):
"""
An extremely simple class representing a mechanical solution model.
A mechanical solution experiences no interaction between endmembers.
Therefore, unlike ideal solutions there is no entropy of mixing;
the total gibbs free energy of the solution is equal to the
dot product of the molar gibbs free energies and molar fractions
of the constituent materials.
"""
def __init__(self, endmembers):
self.n_endmembers = len(endmembers)
self.formulas = [e[1] for e in endmembers]
[docs] def excess_gibbs_free_energy(self, pressure, temperature, molar_fractions):
return 0.
[docs] def excess_volume(self, pressure, temperature, molar_fractions):
return 0.
[docs] def excess_entropy(self, pressure, temperature, molar_fractions):
return 0.
[docs] def excess_enthalpy(self, pressure, temperature, molar_fractions):
return 0.
[docs] def excess_partial_gibbs_free_energies(self, pressure, temperature, molar_fractions):
return np.zeros_like(molar_fractions)
[docs] def excess_partial_volumes(self, pressure, temperature, molar_fractions):
return np.zeros_like(molar_fractions)
[docs] def excess_partial_entropies(self, pressure, temperature, molar_fractions):
return np.zeros_like(molar_fractions)
[docs] def activity_coefficients(self, pressure, temperature, molar_fractions):
return np.ones_like(molar_fractions)
[docs] def activities(self, pressure, temperature, molar_fractions):
return np.ones_like(molar_fractions)
[docs]class IdealSolution (SolutionModel):
"""
A very simple class representing an ideal solution model.
Calculate the excess gibbs free energy and entropy due to configurational
entropy, excess volume is equal to zero.
"""
def __init__(self, endmembers):
self.n_endmembers = len(endmembers)
self.formulas = [e[1] for e in endmembers]
# Process solid solution chemistry
processchemistry.process_solution_chemistry(self)
self._calculate_endmember_configurational_entropies()
def _calculate_endmember_configurational_entropies(self):
S_conf = -constants.gas_constant * (self.site_multiplicities
* self.endmember_occupancies
* logish(self.endmember_occupancies)).sum(-1)
self.endmember_configurational_entropies = S_conf
[docs] def excess_partial_gibbs_free_energies(self, pressure, temperature, molar_fractions):
return self._ideal_excess_partial_gibbs(temperature, molar_fractions)
[docs] def excess_partial_entropies(self, pressure, temperature, molar_fractions):
return self._ideal_excess_partial_entropies(temperature, molar_fractions)
[docs] def excess_partial_volumes(self, pressure, temperature, molar_fractions):
return np.zeros((self.n_endmembers))
[docs] def gibbs_hessian(self, pressure, temperature, molar_fractions):
hess_S = self._ideal_entropy_hessian(temperature, molar_fractions)
return -temperature*hess_S
[docs] def entropy_hessian(self, pressure, temperature, molar_fractions):
hess_S = self._ideal_entropy_hessian(temperature, molar_fractions)
return hess_S
[docs] def volume_hessian(self, pressure, temperature, molar_fractions):
return np.zeros((len(molar_fractions), len(molar_fractions)))
def _configurational_entropy(self, molar_fractions):
site_occupancies = np.dot(molar_fractions, self.endmember_occupancies)
conf_entropy = - constants.gas_constant * (site_occupancies
* self.site_multiplicities
* logish(site_occupancies)).sum(-1)
return conf_entropy
def _ideal_excess_partial_gibbs(self, temperature, molar_fractions):
return -temperature*self._ideal_excess_partial_entropies(temperature, molar_fractions)
def _ideal_excess_partial_entropies(self, temperature, molar_fractions):
return -constants.gas_constant * self._log_ideal_activities(molar_fractions)
def _ideal_entropy_hessian(self, temperature, molar_fractions):
hessian = -constants.gas_constant * self._log_ideal_activity_derivatives(molar_fractions)
return hessian
def _log_ideal_activities(self, molar_fractions):
site_occupancies = np.dot(molar_fractions, self.endmember_occupancies)
lna = (self.endmember_occupancies * self.site_multiplicities
* logish(site_occupancies)).sum(-1)
normalisation_constants = (self.endmember_configurational_entropies
/ constants.gas_constant)
return lna + normalisation_constants
def _log_ideal_activity_derivatives(self, molar_fractions):
site_occupancies = np.dot(molar_fractions, self.endmember_occupancies)
n_site_atoms = self.endmember_occupancies[0].dot(self.site_multiplicities)
dlnadp = np.einsum('ij, j, mj',
self.endmember_occupancies,
self.site_multiplicities*inverseish(site_occupancies),
self.endmember_occupancies) - n_site_atoms
return dlnadp
def _ideal_activities(self, molar_fractions):
return _ideal_activities_fct(molar_fractions,
self.endmember_occupancies,
self.n_endmembers,
self.n_occupancies,
self.site_multiplicities,
self.endmember_configurational_entropies)
[docs] def activity_coefficients(self, pressure, temperature, molar_fractions):
return np.ones_like(molar_fractions)
[docs] def activities(self, pressure, temperature, molar_fractions):
return self._ideal_activities(molar_fractions)
[docs]class AsymmetricRegularSolution (IdealSolution):
"""
Solution model implementing the asymmetric regular solution model formulation as described in :cite:`HP2003`.
"""
def __init__(self, endmembers, alphas, energy_interaction, volume_interaction=None, entropy_interaction=None):
self.n_endmembers = len(endmembers)
# Create array of van Laar parameters
self.alphas = np.array(alphas)
# Create 2D arrays of interaction parameters
self.We = np.triu(2. / (self.alphas[:, np.newaxis] + self.alphas), 1)
self.We[np.triu_indices(self.n_endmembers, 1)] *= np.array([i for row in energy_interaction
for i in row])
if entropy_interaction is not None:
self.Ws = np.triu(2. / (self.alphas[:, np.newaxis] + self.alphas), 1)
self.Ws[np.triu_indices(self.n_endmembers, 1)] *= np.array([i for row in entropy_interaction
for i in row])
else:
self.Ws = np.zeros((self.n_endmembers, self.n_endmembers))
if volume_interaction is not None:
self.Wv = np.triu(2. / (self.alphas[:, np.newaxis] + self.alphas), 1)
self.Wv[np.triu_indices(self.n_endmembers, 1)] *= np.array([i for row in volume_interaction
for i in row])
else:
self.Wv = np.zeros((self.n_endmembers, self.n_endmembers))
# initialize ideal solution model
IdealSolution.__init__(self, endmembers)
def _phi(self, molar_fractions):
phi = self.alphas * molar_fractions
phi = np.divide(phi, np.sum(phi))
return phi
def _non_ideal_interactions(self, W, molar_fractions):
# -sum(sum(qi.qj.Wij*)
# equation (2) of Holland and Powell 2003
phi = self._phi(molar_fractions)
return _non_ideal_interactions_fct(phi, np.array(molar_fractions), self.n_endmembers, self.alphas, W)
def _non_ideal_excess_partial_gibbs(self, pressure, temperature, molar_fractions):
Eint = self._non_ideal_interactions(self.We, molar_fractions)
Sint = self._non_ideal_interactions(self.Ws, molar_fractions)
Vint = self._non_ideal_interactions(self.Wv, molar_fractions)
return Eint - temperature * Sint + pressure * Vint
[docs] def excess_partial_gibbs_free_energies(self, pressure, temperature, molar_fractions):
ideal_gibbs = IdealSolution._ideal_excess_partial_gibbs(
self, temperature, molar_fractions)
non_ideal_gibbs = self._non_ideal_excess_partial_gibbs(
pressure, temperature, molar_fractions)
return ideal_gibbs + non_ideal_gibbs
[docs] def excess_partial_entropies(self, pressure, temperature, molar_fractions):
ideal_entropies = IdealSolution._ideal_excess_partial_entropies(
self, temperature, molar_fractions)
non_ideal_entropies = self._non_ideal_interactions(self.Ws, molar_fractions)
return ideal_entropies + non_ideal_entropies
[docs] def excess_partial_volumes(self, pressure, temperature, molar_fractions):
return self._non_ideal_interactions(self.Wv, molar_fractions)
[docs] def gibbs_hessian(self, pressure, temperature, molar_fractions):
ideal_entropy_hessian = IdealSolution._ideal_entropy_hessian(self, temperature, molar_fractions)
phi = self._phi(molar_fractions)
nonideal_gibbs_hessian = _non_ideal_hessian_fct(phi, molar_fractions,
self.n_endmembers, self.alphas,
self.We - temperature*self.Ws + pressure*self.Wv)
return nonideal_gibbs_hessian - temperature*ideal_entropy_hessian
[docs] def entropy_hessian(self, pressure, temperature, molar_fractions):
ideal_entropy_hessian = IdealSolution._ideal_entropy_hessian(self, temperature, molar_fractions)
phi = self._phi(molar_fractions)
nonideal_entropy_hessian = _non_ideal_hessian_fct(phi, molar_fractions,
self.n_endmembers, self.alphas,
self.Ws)
return ideal_entropy_hessian + nonideal_entropy_hessian
[docs] def volume_hessian(self, pressure, temperature, molar_fractions):
phi = self._phi(molar_fractions)
return _non_ideal_hessian_fct(phi, molar_fractions,
self.n_endmembers, self.alphas,
self.Wv)
[docs] def activity_coefficients(self, pressure, temperature, molar_fractions):
if temperature > 1.e-10:
return np.exp(self._non_ideal_excess_partial_gibbs(pressure, temperature, molar_fractions) / (constants.gas_constant * temperature))
else:
raise Exception("Activity coefficients not defined at 0 K.")
[docs] def activities(self, pressure, temperature, molar_fractions):
return IdealSolution.activities(self, pressure, temperature, molar_fractions) * self.activity_coefficients(pressure, temperature, molar_fractions)
[docs]class SymmetricRegularSolution (AsymmetricRegularSolution):
"""
Solution model implementing the symmetric regular solution model. This is simply a special case of the :class:`burnman.solutionmodel.AsymmetricRegularSolution` class.
"""
def __init__(self, endmembers, energy_interaction, volume_interaction=None, entropy_interaction=None):
alphas = np.ones(len(endmembers))
AsymmetricRegularSolution.__init__(
self, endmembers, alphas, energy_interaction, volume_interaction, entropy_interaction)
[docs]class SubregularSolution (IdealSolution):
"""
Solution model implementing the subregular solution model formulation as described in :cite:`HW1989`.
"""
def __init__(self, endmembers, energy_interaction, volume_interaction=None, entropy_interaction=None):
self.n_endmembers = len(endmembers)
# Create 2D arrays of interaction parameters
self.We = np.zeros(shape=(self.n_endmembers, self.n_endmembers))
self.Ws = np.zeros(shape=(self.n_endmembers, self.n_endmembers))
self.Wv = np.zeros(shape=(self.n_endmembers, self.n_endmembers))
# setup excess enthalpy interaction matrix
for i in range(self.n_endmembers):
for j in range(i + 1, self.n_endmembers):
self.We[i][j] = energy_interaction[i][j - i - 1][0]
self.We[j][i] = energy_interaction[i][j - i - 1][1]
if entropy_interaction is not None:
for i in range(self.n_endmembers):
for j in range(i + 1, self.n_endmembers):
self.Ws[i][j] = entropy_interaction[i][j - i - 1][0]
self.Ws[j][i] = entropy_interaction[i][j - i - 1][1]
if volume_interaction is not None:
for i in range(self.n_endmembers):
for j in range(i + 1, self.n_endmembers):
self.Wv[i][j] = volume_interaction[i][j - i - 1][0]
self.Wv[j][i] = volume_interaction[i][j - i - 1][1]
# initialize ideal solution model
IdealSolution.__init__(self, endmembers)
def _non_ideal_function(self, W, molar_fractions):
n = len(molar_fractions)
return _non_ideal_interactions_subreg(molar_fractions, n, W)
def _non_ideal_interactions(self, molar_fractions):
# equation (6') of Helffrich and Wood, 1989
Eint = self._non_ideal_function(self.We, molar_fractions)
Sint = self._non_ideal_function(self.Ws, molar_fractions)
Vint = self._non_ideal_function(self.Wv, molar_fractions)
return Eint, Sint, Vint
def _non_ideal_excess_partial_gibbs(self, pressure, temperature, molar_fractions):
Eint, Sint, Vint = self._non_ideal_interactions(molar_fractions)
return Eint - temperature * Sint + pressure * Vint
[docs] def excess_partial_gibbs_free_energies(self, pressure, temperature, molar_fractions):
ideal_gibbs = IdealSolution._ideal_excess_partial_gibbs(
self, temperature, molar_fractions)
non_ideal_gibbs = self._non_ideal_excess_partial_gibbs(
pressure, temperature, molar_fractions)
return ideal_gibbs + non_ideal_gibbs
[docs] def excess_partial_entropies(self, pressure, temperature, molar_fractions):
ideal_entropies = IdealSolution._ideal_excess_partial_entropies(
self, temperature, molar_fractions)
non_ideal_entropies = self._non_ideal_function(self.Ws, molar_fractions)
return ideal_entropies + non_ideal_entropies
[docs] def excess_partial_volumes(self, pressure, temperature, molar_fractions):
non_ideal_volumes = self._non_ideal_function(self.Wv, molar_fractions)
return non_ideal_volumes
[docs] def gibbs_hessian(self, pressure, temperature, molar_fractions):
n = len(molar_fractions)
ideal_entropy_hessian = IdealSolution._ideal_entropy_hessian(self, temperature, molar_fractions)
nonideal_gibbs_hessian = _non_ideal_hessian_subreg(molar_fractions, n,
self.We - temperature*self.Ws
+ pressure*self.Wv)
return nonideal_gibbs_hessian - temperature*ideal_entropy_hessian
[docs] def entropy_hessian(self, pressure, temperature, molar_fractions):
n = len(molar_fractions)
ideal_entropy_hessian = IdealSolution._ideal_entropy_hessian(self, temperature, molar_fractions)
nonideal_entropy_hessian = _non_ideal_hessian_subreg(molar_fractions, n,
self.Ws)
return ideal_entropy_hessian + nonideal_entropy_hessian
[docs] def volume_hessian(self, pressure, temperature, molar_fractions):
n = len(molar_fractions)
return _non_ideal_hessian_subreg(molar_fractions, n,
self.Wv)
[docs] def activity_coefficients(self, pressure, temperature, molar_fractions):
if temperature > 1.e-10:
return np.exp(self._non_ideal_excess_partial_gibbs(pressure, temperature, molar_fractions) / (constants.gas_constant * temperature))
else:
raise Exception("Activity coefficients not defined at 0 K.")
[docs] def activities(self, pressure, temperature, molar_fractions):
return IdealSolution.activities(self, pressure, temperature, molar_fractions) * self.activity_coefficients(pressure, temperature, molar_fractions)