# This file is part of BurnMan - a thermoelastic and thermodynamic toolkit
# for the Earth and Planetary Sciences
# Copyright (C) 2012 - 2022 by the BurnMan team, released under the GNU
# GPL v2 or later.
from __future__ import absolute_import
import importlib
import numpy as np
from ..utils.chemistry import process_solution_chemistry
from .. import constants
import warnings
import sparse
import string
from copy import deepcopy
from .material import material_property
try:
ag = importlib.import_module("autograd")
except ImportError as err:
print(
f"Warning: {err}. "
"For full functionality of BurnMan, please install autograd."
)
def _ideal_activities_fct(
molar_fractions,
endmember_noccupancies,
n_endmembers,
n_occupancies,
site_multiplicities,
endmember_configurational_entropies,
):
site_noccupancies = np.dot(molar_fractions, endmember_noccupancies)
site_multiplicities = np.einsum("i, ij", molar_fractions, site_multiplicities)
site_occupancies = site_noccupancies * inverseish(site_multiplicities)
a = np.power(site_occupancies, endmember_noccupancies).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
def _non_ideal_hessian_subreg(p, n_endmembers, Wijk):
Id = np.identity(n_endmembers)
IIp = np.einsum("il, jm, k->ijklm", Id, Id, p)
Ipp = np.einsum("il, j, k->ijkl", Id, p, p)
ppp = np.einsum("i, j, k->ijk", p, p, p)
A = (
IIp
+ np.transpose(IIp, axes=[0, 2, 1, 3, 4])
+ np.transpose(IIp, axes=[1, 0, 2, 3, 4])
+ np.transpose(IIp, axes=[1, 2, 0, 3, 4])
+ np.transpose(IIp, axes=[2, 1, 0, 3, 4])
+ np.transpose(IIp, axes=[2, 0, 1, 3, 4])
)
B = 2.0 * (
Ipp
+ np.transpose(Ipp, axes=[1, 0, 2, 3])
+ np.transpose(Ipp, axes=[2, 1, 0, 3])
)
Asum = (
A - B[:, :, :, :, None] - B[:, :, :, None, :] + 6.0 * ppp[:, :, :, None, None]
)
hess = np.einsum("ijklm, ijk->lm", Asum, Wijk)
return hess
def _non_ideal_interactions_subreg(p, n_endmembers, Wijk):
Aijkl = np.einsum("li, j, k->ijkl", np.identity(n_endmembers), p, p)
ppp = np.einsum("i, j, k->ijk", p, p, p)
Asum = (
Aijkl
+ np.transpose(Aijkl, axes=[1, 0, 2, 3])
+ np.transpose(Aijkl, axes=[1, 2, 0, 3])
- 2 * ppp[:, :, :, None]
)
Wint = np.einsum("ijk, ijkl->l", Wijk, Asum)
return Wint
def logish(x, eps=1.0e-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 = 1.0 - x / eps
mask = x > eps
ln = np.where(x <= eps, np.log(eps) - f_eps - f_eps * f_eps / 2.0, 0.0)
ln[mask] = np.log(x[mask])
return ln
def inverseish(x, eps=1.0e-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, 2.0 / eps - x / eps / eps, 0.0)
oneoverx[mask] = 1.0 / x[mask]
return oneoverx
class SolutionModel(object):
"""
This is the base class for a solution model, intended for use
in defining solutions and performing thermodynamic calculations
on them. All minerals of type :class:`burnman.Solution` use
a solution model for defining how the endmembers in the 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 solution will be equal to the weighted arithmetic
averages of the different endmember values.
"""
def __init__(self):
"""
Does nothing.
"""
pass
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.
:param pressure: Pressure at which to evaluate the solution model [Pa].
:type pressure: float
:param temperature: Temperature at which to evaluate the solution model [K].
:type temperature: float
:param molar_fractions: List of molar fractions of the
different independent endmembers in the solution model.
:type molar_fractions: list of floats
:returns: The excess Gibbs energy.
:rtype: float
"""
return np.dot(
np.array(molar_fractions),
self.excess_partial_gibbs_free_energies(
pressure, temperature, molar_fractions
),
)
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.
:param pressure: Pressure at which to evaluate the solution model [Pa].
:type pressure: float
:param temperature: Temperature at which to evaluate the solution model [K].
:type temperature: float
:param molar_fractions: List of molar fractions of the
different independent endmembers in the solution model.
:type molar_fractions: list of floats
:returns: The excess volume of the solution.
:rtype: float
"""
return np.dot(
molar_fractions,
self.excess_partial_volumes(pressure, temperature, molar_fractions),
)
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.
:param pressure: Pressure at which to evaluate the solution model [Pa].
:type pressure: float
:param temperature: Temperature at which to evaluate the solution model [K].
:type temperature: float
:param molar_fractions: List of molar fractions of the
different independent endmembers in the solution model.
:type molar_fractions: list of floats
:returns: The excess entropy of the solution.
:rtype: float
"""
return np.dot(
molar_fractions,
self.excess_partial_entropies(pressure, temperature, molar_fractions),
)
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.
:param pressure: Pressure at which to evaluate the solution model [Pa].
:type pressure: float
:param temperature: Temperature at which to evaluate the solution model [K].
:type temperature: float
:param molar_fractions: List of molar fractions of the
different independent endmembers in the solution model.
:type molar_fractions: list of floats
:returns: The excess enthalpy of the solution.
:rtype: float
"""
return self.excess_gibbs_free_energy(
pressure, temperature, molar_fractions
) + temperature * self.excess_entropy(pressure, temperature, molar_fractions)
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.
:param pressure: Pressure at which to evaluate the solution model [Pa].
:type pressure: float
:param temperature: Temperature at which to evaluate the solution model [K].
:type temperature: float
:param molar_fractions: List of molar fractions of the
different independent endmembers in the solution model.
:type molar_fractions: list of floats
:returns: The excess partial Gibbs free energy of each endmember.
:rtype: numpy.array
"""
return np.zeros_like(molar_fractions)
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).
:param pressure: Pressure at which to evaluate the solution model [Pa].
:type pressure: float
:param temperature: Temperature at which to evaluate the solution model [K].
:type temperature: float
:param molar_fractions: List of molar fractions of the
different independent endmembers in the solution model.
:type molar_fractions: list of floats
:returns: The excess partial entropy of each endmember.
:rtype: numpy.array
"""
return np.zeros_like(molar_fractions)
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.
:param pressure: Pressure at which to evaluate the solution model [Pa].
:type pressure: float
:param temperature: Temperature at which to evaluate the solution model [K].
:type temperature: float
:param molar_fractions: List of molar fractions of the
different independent endmembers in the solution model.
:type molar_fractions: list of floats
:returns: The excess partial volume of each endmember.
:rtype: numpy.array
"""
return np.zeros_like(np.array(molar_fractions))
def Cp_excess(self):
"""
Returns the excess heat capacity of the solution model
at its current state
"""
return 0.0
def alphaV_excess(self):
"""
Returns the excess alpha*V of the solution model
at its current state
"""
return 0.0
def VoverKT_excess(self):
"""
Returns the excess V/K_T of the solution model
at its current state
"""
return 0.0
[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.endmembers = 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 0.0
[docs] def excess_volume(self, pressure, temperature, molar_fractions):
return 0.0
[docs] def excess_entropy(self, pressure, temperature, molar_fractions):
return 0.0
[docs] def excess_enthalpy(self, pressure, temperature, molar_fractions):
return 0.0
[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 class representing an ideal solution model.
Calculates the excess gibbs free energy and entropy due to configurational
entropy. Excess internal energy and volume are equal to zero.
The multiplicity of each type of site in the structure is allowed to
change linearly as a function of endmember proportions. This class
is therefore equivalent to the entropic part of
a Temkin-type model :cite:`Temkin1945`.
"""
def __init__(self, endmembers):
self.endmembers = endmembers
self.n_endmembers = len(endmembers)
self.formulas = [e[1] for e in endmembers]
# Process solution chemistry
process_solution_chemistry(self)
self._calculate_endmember_configurational_entropies()
def _calculate_endmember_configurational_entropies(self):
S_conf = -(
constants.gas_constant
* (self.endmember_noccupancies * 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_noccupancies = np.einsum(
"i, ij", molar_fractions, self.endmember_noccupancies
)
site_multiplicities = np.einsum(
"i, ij", molar_fractions, self.site_multiplicities
)
site_occupancies = site_noccupancies * inverseish(site_multiplicities)
conf_entropy = -(
constants.gas_constant
* (site_noccupancies * 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_noccupancies = np.einsum(
"i, ij", molar_fractions, self.endmember_noccupancies
)
site_multiplicities = np.einsum(
"i, ij", molar_fractions, self.site_multiplicities
)
lna = np.einsum(
"ij, j->i",
self.endmember_noccupancies,
logish(site_noccupancies) - logish(site_multiplicities),
)
normalisation_constants = (
self.endmember_configurational_entropies / constants.gas_constant
)
return lna + normalisation_constants
def _log_ideal_activity_derivatives(self, molar_fractions):
site_noccupancies = np.einsum(
"i, ij", molar_fractions, self.endmember_noccupancies
)
site_multiplicities = np.einsum(
"i, ij", molar_fractions, self.site_multiplicities
)
dlnadp = np.einsum(
"pj, qj, j->pq",
self.endmember_noccupancies,
self.endmember_noccupancies,
inverseish(site_noccupancies),
) - np.einsum(
"pj, qj, j->pq",
self.endmember_noccupancies,
self.site_multiplicities,
inverseish(site_multiplicities),
)
return dlnadp
def _ideal_activities(self, molar_fractions):
return _ideal_activities_fct(
molar_fractions,
self.endmember_noccupancies,
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`.
The excess nonconfigurational Gibbs energy is given by the
expression:
.. math::
\\mathcal{G}_{\\textrm{excess}} = \\alpha^T p (\\phi^T W \\phi)
:math:`\\alpha` is a vector of van Laar parameters governing asymmetry
in the excess properties.
.. math::
\\phi_i = \\frac{\\alpha_i p_i}{\\sum_{k=1}^{n} \\alpha_k p_k},
W_{ij} = \\frac{2 w_{ij}}{\\alpha_i + \\alpha_j} \\textrm{for i<j}
"""
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(2.0 / (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(2.0 / (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(2.0 / (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 > 1.0e-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 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`. The excess nonconfigurational
Gibbs energy is given by the expression:
.. math::
\\mathcal{G}_{\\textrm{excess}} = \\sum_i \\sum_{j > i} (p_i p_j^2
W_{ij} + p_j p_i^2 W_{ji} + \\sum_{k > j > i} p_i p_j p_k W_{ijk})
Interaction parameters are inserted into a 3D interaction matrix during
initialization to make use of numpy vector algebra.
:param endmembers: A list of all the independent endmembers in the solution.
The first item of each list gives the Mineral object corresponding
to the endmember. The second item gives the site-species formula.
:type endmembers: list of lists
:param energy_interaction: The binary endmember interaction energies.
Each interaction[i, j-i-1, 0] corresponds to W(i,j), while
interaction[i, j-i-1, 1] corresponds to W(j,i).
:type energy_interaction: list of list of lists
:param volume_interaction: The binary endmember interaction volumes.
Each interaction[i, j-i-1, 0] corresponds to W(i,j), while
interaction[i, j-i-1, 1] corresponds to W(j,i).
:type volume_interaction: list of list of lists
:param entropy_interaction: The binary endmember interaction entropies.
Each interaction[i, j-i-1, 0] corresponds to W(i,j), while
interaction[i, j-i-1, 1] corresponds to W(j,i).
:type entropy_interaction: list of list of lists
:param energy_ternary_terms: The ternary interaction energies.
Each list should contain four entries:
the indices i, j, k and the value of the interaction.
:type energy_ternary_terms: list of lists
:param volume_ternary_terms: The ternary interaction volumes.
Each list should contain four entries:
the indices i, j, k and the value of the interaction.
:type volume_ternary_terms: list of lists
:param entropy_ternary_terms: The ternary interaction entropies.
Each list should contain four entries:
the indices i, j, k and the value of the interaction.
:type entropy_ternary_terms: list of lists
"""
def __init__(
self,
endmembers,
energy_interaction,
volume_interaction=None,
entropy_interaction=None,
energy_ternary_terms=None,
volume_ternary_terms=None,
entropy_ternary_terms=None,
):
"""
Initialization function for the SubregularSolution class.
"""
self.n_endmembers = len(endmembers)
# Create 3D arrays of interaction parameters
self.Wijke = np.zeros(
shape=(self.n_endmembers, self.n_endmembers, self.n_endmembers)
)
self.Wijks = np.zeros_like(self.Wijke)
self.Wijkv = np.zeros_like(self.Wijke)
# setup excess enthalpy interaction matrix
for i in range(self.n_endmembers):
for j in range(i + 1, self.n_endmembers):
w0 = energy_interaction[i][j - i - 1][0] / 2.0
w1 = energy_interaction[i][j - i - 1][1] / 2.0
self.Wijke[:, i, j] += w0
self.Wijke[:, j, i] += w1
self.Wijke[i, j, j] += w0
self.Wijke[j, i, i] += w1
self.Wijke[i, j, i] -= w0
self.Wijke[j, i, j] -= w1
if energy_ternary_terms is not None:
for i, j, k, v in energy_ternary_terms:
self.Wijke[i, j, k] += v
if entropy_interaction is not None:
for i in range(self.n_endmembers):
for j in range(i + 1, self.n_endmembers):
w0 = entropy_interaction[i][j - i - 1][0] / 2.0
w1 = entropy_interaction[i][j - i - 1][1] / 2.0
self.Wijks[:, i, j] += w0
self.Wijks[:, j, i] += w1
self.Wijks[i, j, j] += w0
self.Wijks[j, i, i] += w1
self.Wijks[i, j, i] -= w0
self.Wijks[j, i, j] -= w1
if entropy_ternary_terms is not None:
for i, j, k, v in entropy_ternary_terms:
self.Wijks[i, j, k] += v
if volume_interaction is not None:
for i in range(self.n_endmembers):
for j in range(i + 1, self.n_endmembers):
w0 = volume_interaction[i][j - i - 1][0] / 2.0
w1 = volume_interaction[i][j - i - 1][1] / 2.0
self.Wijkv[:, i, j] += w0
self.Wijkv[:, j, i] += w1
self.Wijkv[i, j, j] += w0
self.Wijkv[j, i, i] += w1
self.Wijkv[i, j, i] -= w0
self.Wijkv[j, i, j] -= w1
if volume_ternary_terms is not None:
for i, j, k, v in volume_ternary_terms:
self.Wijkv[i, j, k] += v
# initialize ideal solution model
IdealSolution.__init__(self, endmembers)
def _non_ideal_function(self, Wijk, molar_fractions):
n = len(molar_fractions)
return _non_ideal_interactions_subreg(molar_fractions, n, Wijk)
def _non_ideal_interactions(self, molar_fractions):
# equation (6') of Helffrich and Wood, 1989
Eint = self._non_ideal_function(self.Wijke, molar_fractions)
Sint = self._non_ideal_function(self.Wijks, molar_fractions)
Vint = self._non_ideal_function(self.Wijkv, 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.Wijks, 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.Wijkv, 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.Wijke - temperature * self.Wijks + pressure * self.Wijkv,
)
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.Wijks
)
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.Wijkv)
[docs] def activity_coefficients(self, pressure, temperature, molar_fractions):
if temperature > 1.0e-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 FunctionSolution(IdealSolution):
"""
Solution model implementing a generalized solution model.
The extensive excess nonconfigurational Gibbs energy is
provided as a function by the user.
Derivatives are calculated using the autograd module,
and so the user-defined excess Gibbs energy function
should be defined using autograd-friendly expressions.
:param endmembers: A list of all the independent endmembers in the solution.
The first item of each list gives the Mineral object corresponding
to the endmember. The second item gives the site-species formula.
:type endmembers: list of lists
:param excess_gibbs_function: The nonconfigurational Gibbs energy function
with arguments pressure, temperature and molar_amounts, in that order.
Note that the function must be extensive; if the molar amounts
are doubled, the Gibbs energy must also double.
:type excess_gibbs_function: function
"""
def __init__(self, endmembers, excess_gibbs_function):
"""
Initialization function for the GeneralSolution class.
"""
# initialize ideal solution model
IdealSolution.__init__(self, endmembers)
self.n_endmembers = len(endmembers)
self._excess_gibbs_function = excess_gibbs_function
self._non_ideal_excess_partial_gibbs = ag.jacobian(
excess_gibbs_function, argnum=2
)
def partial_entropies(pressure, temperature, molar_amounts):
with warnings.catch_warnings(record=True):
warnings.simplefilter("always")
return -ag.jacobian(self._non_ideal_excess_partial_gibbs, argnum=1)(
pressure, temperature, molar_amounts
)
self._non_ideal_excess_partial_entropies = partial_entropies
def partial_volumes(pressure, temperature, molar_amounts):
with warnings.catch_warnings(record=True):
warnings.simplefilter("always")
return ag.jacobian(self._non_ideal_excess_partial_gibbs, argnum=0)(
pressure, temperature, molar_amounts
)
self.excess_partial_volumes = partial_volumes
self._non_ideal_gibbs_hessian = ag.jacobian(
self._non_ideal_excess_partial_gibbs, argnum=2
)
def entropy_hess(pressure, temperature, molar_amounts):
with warnings.catch_warnings(record=True):
warnings.simplefilter("always")
return ag.jacobian(partial_entropies, argnum=2)(
pressure, temperature, molar_amounts
)
self._non_ideal_entropy_hessian = entropy_hess
def volume_hess(pressure, temperature, molar_amounts):
with warnings.catch_warnings(record=True):
warnings.simplefilter("always")
return ag.jacobian(partial_volumes, argnum=2)(
pressure, temperature, molar_amounts
)
self.volume_hessian = volume_hess
[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_excess_partial_entropies(
pressure, temperature, molar_fractions
)
return ideal_entropies + non_ideal_entropies
[docs] def gibbs_hessian(self, pressure, temperature, molar_fractions):
ideal_entropy_hessian = IdealSolution._ideal_entropy_hessian(
self, temperature, molar_fractions
)
nonideal_gibbs_hessian = self._non_ideal_gibbs_hessian(
pressure, temperature, molar_fractions
)
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
)
nonideal_entropy_hessian = self._non_ideal_entropy_hessian(
pressure, temperature, molar_fractions
)
return ideal_entropy_hessian + nonideal_entropy_hessian
[docs] def activity_coefficients(self, pressure, temperature, molar_fractions):
if temperature > 1.0e-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)
class PolynomialSolution(IdealSolution):
"""
Solution model implementing a general polynomial solution model.
:param endmembers: A list of all the independent endmembers in the solution.
The first item of each list gives the Mineral object corresponding
to the endmember. The second item gives the site-species formula.
:type endmembers: list of lists
:param ESV_interactions: A list containing lists where the first three elements are
energy, entropy and volume interactions and the rest of the elements
are indices of the transformed endmembers to which those
interactions correspond.
For example, [2., 0., 0., 0, 1, 1] would correspond to an interaction
of 2*p'[0]*p'[1]*p'[1].
:type ESV_interactions: list of lists
:param interaction_endmembers: A list of minerals involved in the interaction terms.
:type interaction_endmembers: list of :class:`burnman.Mineral` objects
:param endmember_coefficients_and_interactions: list of lists
A list containing lists where the first n elements are
coefficients for each of the interaction_endmembers and the
rest of the elements are indices of the transformed
endmembers to which those interactions correspond.
For example, [1., 0., -1., 0, 1, 1] would correspond to an interaction
of (mbr[0].gibbs - mbr[2].gibbs)*p'[0]*p'[1]*p'[1].
:type endmember_coefficients_and_interactions: list of lists
:param transformation_matrix: The interactions for a given solution may
be most compactly expressed not as a polynomial function of the
proportions of the endmembers, but a polynomial function of a
linearly transformed set. This parameter is a square numpy array A,
where p'i = A_ij p_j
:type transformation_matrix: 2D numpy array
"""
def __init__(
self,
endmembers,
ESV_interactions=None,
interaction_endmembers=[],
endmember_coefficients_and_interactions=None,
transformation_matrix=None,
):
# initialize ideal solution model
IdealSolution.__init__(self, endmembers)
self.n_endmembers = len(endmembers)
self.endmembers = endmembers
self.W_ESV = None
if ESV_interactions is not None:
self.W_ESV = self.make_interaction_arrays(ESV_interactions)
self.n_interaction_endmembers = len(interaction_endmembers)
self.interaction_endmembers = interaction_endmembers
self.W_mbr = None
if self.n_interaction_endmembers > 0:
self.W_mbr = self.make_endmember_interaction_arrays(
endmember_coefficients_and_interactions
)
self.transformation_matrix = transformation_matrix
self.reset()
def reset(self):
"""
Resets all cached material properties.
It is typically not required for the user to call this function.
"""
self._cached = {}
def set_composition(self, molar_fractions):
"""
Resets all cached material properties.
It is typically not required for the user to call this function.
"""
self.reset()
self.molar_fractions = molar_fractions
def set_state(self, pressure, temperature):
"""
Sets the states for the interaction endmembers.
It is typically not required for the user to call this function.
"""
for mbr in self.interaction_endmembers:
mbr.set_state(pressure, temperature)
def make_interaction_arrays(self, Ws):
n_Ws = len(Ws)
for i, W in enumerate(Ws):
if not all(W[i] <= W[i + 1] for i in range(3, len(W) - 1)):
raise Exception(
f"Interaction parameter {i+1}/{n_Ws} must be "
"upper triangular (i<=j<=k<=...<=z)"
)
if not W[3] < W[-1]:
raise Exception(
f"Interaction parameter {i+1}/{n_Ws} must not lie on the "
"first diagonal (i=j=k=...=z)"
)
W_arrays = []
dims = sorted(list(set([len(W) - 3 for W in Ws])))
for dim in dims:
coords = [
[0, *W[3:]] for W in Ws if len(W) == dim + 3 and np.abs(W[0]) > 1.0e-10
]
coords.extend(
[
[1, *W[3:]]
for W in Ws
if len(W) == dim + 3 and np.abs(W[1]) > 1.0e-10
]
)
coords.extend(
[
[2, *W[3:]]
for W in Ws
if len(W) == dim + 3 and np.abs(W[2]) > 1.0e-10
]
)
coords = list(zip(*coords))
data = [W[0] for W in Ws if len(W) == dim + 3 and np.abs(W[0]) > 1.0e-10]
data.extend(
[W[1] for W in Ws if len(W) == dim + 3 and np.abs(W[1]) > 1.0e-10]
)
data.extend(
[W[2] for W in Ws if len(W) == dim + 3 and np.abs(W[2]) > 1.0e-10]
)
shape = [3] # First dimension is for E, S, V
shape.extend([self.n_endmembers for i in range(dim)])
shape = tuple(shape)
s = sparse.COO(coords, data, shape=shape).todense()
W_arrays.append((dim, s))
return W_arrays
def make_endmember_interaction_arrays(self, Ws):
n_Ws = len(Ws)
n_int = self.n_interaction_endmembers
for i, W in enumerate(Ws):
if not all(W[i] <= W[i + 1] for i in range(n_int, len(W) - 1)):
raise Exception(
f"Interaction parameter {i+1}/{n_Ws} must be "
"upper triangular (i<=j<=k<=...<=z)"
)
if not W[n_int] < W[-1]:
raise Exception(
f"Interaction parameter {i+1}/{n_Ws} must not lie on the "
"first diagonal (i=j=k=...=z)"
)
W_arrays = []
dims = sorted(list(set([len(W) - n_int for W in Ws])))
for dim in dims:
coords = []
data = []
for i in range(n_int):
coords.extend(
[
[i, *W[n_int:]]
for W in Ws
if len(W) == dim + n_int and np.abs(W[i]) > 1.0e-10
]
)
data.extend(
[
W[i]
for W in Ws
if len(W) == dim + n_int and np.abs(W[i]) > 1.0e-10
]
)
coords = list(zip(*coords))
shape = [n_int] # First dimension is for the excess_endmembers
shape.extend([self.n_endmembers for i in range(dim)])
shape = tuple(shape)
s = sparse.COO(coords, data, shape=shape).todense()
W_arrays.append((dim, s))
return W_arrays
def transform_scalar_list(self, x, A):
if A is not None:
return A.dot(x)
else:
return x
def transform_gradient_list(self, W, A):
if A is not None:
return np.einsum("ij, jk->ik", W, A)
else:
return W
def transform_hessian_list(self, W, A):
if A is not None:
return np.einsum("ki, mij, lj->mkl", A, W, A)
else:
return W
def W_dots_x(self, dim, W_array, x):
if dim == 0:
return W_array
else:
ESVdim = [W_array]
ESVdim.extend([x for i in range(dim)])
strng = string.ascii_lowercase[: W_array.ndim]
strng2 = ", ".join(strng[-dim:])
strng += f", {strng2}"
return np.einsum(strng, *ESVdim)
def rolled_array(self, W, istart):
W_array_rolled = deepcopy(W)
dim = len(W.shape) - 1
W_roll = deepcopy(W)
for i in range(dim - istart):
W_roll = np.moveaxis(W_roll, istart, -1)
W_array_rolled += W_roll
return W_array_rolled
def get_scalar_list(self, x, W_arrays, A):
xp = self.transform_scalar_list(x, A)
xp_total = np.sum(xp)
ESV = np.zeros(W_arrays[0][-1].shape[0])
for dim, W_array in W_arrays:
ESV += self.W_dots_x(dim, W_array, xp) / np.power(xp_total, dim - 1.0)
return ESV
def get_gradient_list(self, x, W_arrays, A):
xp = self.transform_scalar_list(x, A)
xp_total = np.sum(xp)
dESVdx = np.zeros(W_arrays[0][-1].shape[:2])
for dim, W_array in W_arrays:
dESVdx += (
-(dim - 1)
/ np.power(xp_total, dim)
* self.W_dots_x(dim, W_array, xp)[:, np.newaxis]
)
W_array_rolled = self.rolled_array(W_array, 1)
dESVdx += self.W_dots_x(dim - 1, W_array_rolled, xp) / np.power(
xp_total, dim - 1.0
)
return self.transform_gradient_list(dESVdx, A)
def get_hessian_list(self, x, W_arrays, A):
xp = self.transform_scalar_list(x, A)
xp_total = np.sum(xp)
d2ESVdx2 = np.zeros(W_arrays[0][-1].shape[:3])
for dim, W_array in W_arrays:
d2ESVdx2 += (
(dim - 1)
* dim
/ np.power(xp_total, dim + 1)
* self.W_dots_x(dim, W_array, xp)[:, np.newaxis, np.newaxis]
)
W_array_rolled = self.rolled_array(W_array, 1)
f = self.W_dots_x(dim - 1, W_array_rolled, xp)
h = f[:, np.newaxis, :] + f[:, :, np.newaxis]
d2ESVdx2 += -(dim - 1) / np.power(xp_total, dim) * h
W_array_rolled_2 = self.rolled_array(W_array_rolled, 2)
g = self.W_dots_x(dim - 2, W_array_rolled_2, xp)
d2ESVdx2 += g / np.power(xp_total, dim - 1.0)
return self.transform_hessian_list(d2ESVdx2, A)
@material_property
def ESV_scalar_list(self):
if self.W_ESV is None:
return np.zeros((3))
else:
return self.get_scalar_list(
self.molar_fractions, self.W_ESV, self.transformation_matrix
)
@material_property
def ESV_gradient_list(self):
if self.W_ESV is None:
return np.zeros((3, self.n_endmembers))
else:
return self.get_gradient_list(
self.molar_fractions, self.W_ESV, self.transformation_matrix
)
@material_property
def ESV_hessian_list(self):
if self.W_ESV is None:
return np.zeros((3, self.n_endmembers, self.n_endmembers))
else:
return self.get_hessian_list(
self.molar_fractions, self.W_ESV, self.transformation_matrix
)
@material_property
def mbr_scalar_list(self):
if self.W_mbr is None:
return np.zeros((0))
else:
return self.get_scalar_list(
self.molar_fractions, self.W_mbr, self.transformation_matrix
)
@material_property
def mbr_gradient_list(self):
if self.W_mbr is None:
return np.zeros((0, self.n_endmembers))
else:
return self.get_gradient_list(
self.molar_fractions, self.W_mbr, self.transformation_matrix
)
@material_property
def mbr_hessian_list(self):
if self.W_mbr is None:
return np.zeros((0, self.n_endmembers, self.n_endmembers))
else:
return self.get_hessian_list(
self.molar_fractions, self.W_mbr, self.transformation_matrix
)
def _non_ideal_excess_partial_gibbs(self, pressure, temperature, molar_fractions):
dEdx, dSdx, dVdx = self.ESV_gradient_list
mbr_gradients = self.mbr_gradient_list
mbr_gibbs = np.array([mbr.gibbs for mbr in self.interaction_endmembers])
gibbs = dEdx - temperature * dSdx + pressure * dVdx
gibbs += np.einsum("i, ij->j", mbr_gibbs, mbr_gradients)
return gibbs
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
def excess_partial_entropies(self, pressure, temperature, molar_fractions):
ideal_entropies = IdealSolution._ideal_excess_partial_entropies(
self, temperature, molar_fractions
)
dSdx = self.ESV_gradient_list[1]
mbr_gradients = self.mbr_gradient_list
mbr_entropies = np.array([mbr.S for mbr in self.interaction_endmembers])
non_ideal_entropies = dSdx + np.einsum("i, ij->j", mbr_entropies, mbr_gradients)
return ideal_entropies + non_ideal_entropies
def excess_partial_volumes(self, pressure, temperature, molar_fractions):
dVdx = self.ESV_gradient_list[2]
mbr_gradients = self.mbr_gradient_list
mbr_volumes = np.array([mbr.V for mbr in self.interaction_endmembers])
return dVdx + np.einsum("i, ij->j", mbr_volumes, mbr_gradients)
def gibbs_hessian(self, pressure, temperature, molar_fractions):
ideal_entropy_hessian = IdealSolution._ideal_entropy_hessian(
self, temperature, molar_fractions
)
d2Edx2, d2Sdx2, d2Vdx2 = self.ESV_hessian_list
mbr_hessian = self.mbr_hessian_list
mbr_gibbs = np.array([mbr.gibbs for mbr in self.interaction_endmembers])
d2Gdx2 = d2Edx2 - temperature * d2Sdx2 + pressure * d2Vdx2
d2Gdx2 += np.einsum("i, ijk->jk", mbr_gibbs, mbr_hessian)
return d2Gdx2 - temperature * ideal_entropy_hessian
def entropy_hessian(self, pressure, temperature, molar_fractions):
ideal_entropy_hessian = IdealSolution._ideal_entropy_hessian(
self, temperature, molar_fractions
)
d2Sdx2 = self.ESV_hessian_list[1]
mbr_hessian = self.mbr_hessian_list
mbr_entropies = np.array([mbr.S for mbr in self.interaction_endmembers])
d2Sdx2 += np.einsum("i, ijk->jk", mbr_entropies, mbr_hessian)
return d2Sdx2 + ideal_entropy_hessian
def volume_hessian(self, pressure, temperature, molar_fractions):
d2Vdx2 = self.ESV_hessian_list[2]
mbr_hessian = self.mbr_hessian_list
mbr_volumes = np.array([mbr.V for mbr in self.interaction_endmembers])
d2Vdx2 += np.einsum("i, ijk->jk", mbr_volumes, mbr_hessian)
return d2Vdx2
def Cp_excess(self):
mbr_scalar = self.mbr_scalar_list
mbr_Cp = np.array(
[mbr.molar_heat_capacity_p for mbr in self.interaction_endmembers]
)
return np.einsum("i, i", mbr_scalar, mbr_Cp)
def alphaV_excess(self):
mbr_scalar = self.mbr_scalar_list
mbr_d2gibbsdpdt = np.array(
[mbr.alpha * mbr.V for mbr in self.interaction_endmembers]
)
return np.einsum("i, i", mbr_scalar, mbr_d2gibbsdpdt)
def VoverKT_excess(self):
mbr_scalar = self.mbr_scalar_list
mbr_d2gibbsdpdp = np.array(
[mbr.V / mbr.K_T for mbr in self.interaction_endmembers]
)
return np.einsum("i, i", mbr_scalar, mbr_d2gibbsdpdp)
def activity_coefficients(self, pressure, temperature, molar_fractions):
if temperature > 1.0e-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.")
def activities(self, pressure, temperature, molar_fractions):
return IdealSolution.activities(
self, pressure, temperature, molar_fractions
) * self.activity_coefficients(pressure, temperature, molar_fractions)