# This file is part of BurnMan - a thermoelastic and thermodynamic toolkit for
# the Earth and Planetary Sciences
# Copyright (C) 2012 - 2021 by the BurnMan team, released under the GNU
# GPL v2 or later.
from __future__ import absolute_import
import numpy as np
from sympy import Matrix, nsimplify
from .material import material_property, cached_property
from .mineral import Mineral
from .solutionmodel import SolutionModel
from .solutionmodel import MechanicalSolution, IdealSolution
from .solutionmodel import SymmetricRegularSolution, AsymmetricRegularSolution
from .solutionmodel import SubregularSolution
from .averaging_schemes import reuss_average_function
from ..tools.reductions import independent_row_indices
from ..tools.chemistry import sum_formulae, sort_element_list_to_IUPAC_order
[docs]class SolidSolution(Mineral):
"""
This is the base class for all solid solutions.
Site occupancies, endmember activities and the constant
and pressure and temperature dependencies of the excess
properties can be queried after using set_composition()
States of the solid solution can only be queried after setting
the pressure, temperature and composition using set_state().
This class is available as :class:`burnman.SolidSolution`.
It uses an instance of :class:`burnman.SolutionModel` to
calculate interaction terms between endmembers.
All the solid solution parameters are expected to be in SI units. This
means that the interaction parameters should be in J/mol, with the T
and P derivatives in J/K/mol and m^3/mol.
The parameters are relevant to all solution models. Please
see the documentation for individual models for details about
other parameters.
Parameters
----------
name : string
Name of the solid solution
solution_type : string
String determining which SolutionModel to use. One of 'mechanical',
'ideal', 'symmetric', 'asymmetric' or 'subregular'.
endmembers : list of lists
List of endmembers in this solid solution. The first item of each
list should be a :class:`burnman.Mineral` object. The second item
should be a string with the site formula of the endmember.
molar_fractions : numpy array (optional)
The molar fractions of each endmember in the solid solution.
Can be reset using the set_composition() method.
"""
def __init__(self,
name=None,
solution_type=None,
endmembers=None,
energy_interaction=None,
volume_interaction=None,
entropy_interaction=None,
energy_ternary_terms=None,
volume_ternary_terms=None,
entropy_ternary_terms=None,
alphas=None,
molar_fractions=None):
"""
Set up matrices to speed up calculations for when P, T, X is defined.
"""
Mineral.__init__(self)
# SolidSolution needs a method attribute to call Mineral.set_state().
# Note that set_method() below will not change self.method
self.method = 'SolidSolutionMethod'
if name is not None:
self.name = name
if solution_type is not None:
self.solution_type = solution_type
if endmembers is not None:
self.endmembers = endmembers
if energy_interaction is not None:
self.energy_interaction = energy_interaction
if volume_interaction is not None:
self.volume_interaction = volume_interaction
if entropy_interaction is not None:
self.entropy_interaction = entropy_interaction
if energy_ternary_terms is not None:
self.energy_ternary_terms = energy_ternary_terms
if volume_ternary_terms is not None:
self.volume_ternary_terms = volume_ternary_terms
if entropy_ternary_terms is not None:
self.entropy_ternary_terms = entropy_ternary_terms
if alphas is not None:
self.alphas = alphas
if endmembers is not None:
self.endmembers = endmembers
if hasattr(self, 'endmembers') is False:
raise Exception("'endmembers' attribute missing "
"from solid solution")
# Set default solution model type
if hasattr(self, 'solution_type'):
if self.solution_type == 'mechanical':
self.solution_model = MechanicalSolution(self.endmembers)
elif self.solution_type == 'ideal':
self.solution_model = IdealSolution(self.endmembers)
else:
if hasattr(self, 'energy_interaction') is False:
self.energy_interaction = None
if hasattr(self, 'volume_interaction') is False:
self.volume_interaction = None
if hasattr(self, 'entropy_interaction') is False:
self.entropy_interaction = None
if self.solution_type == 'symmetric':
self.solution_model = SymmetricRegularSolution(
self.endmembers, self.energy_interaction,
self.volume_interaction, self.entropy_interaction)
elif self.solution_type == 'asymmetric':
if hasattr(self, 'alphas') is False:
raise Exception(
"'alphas' attribute missing from solid solution")
self.solution_model = AsymmetricRegularSolution(
self.endmembers, self.alphas, self.energy_interaction,
self.volume_interaction, self.entropy_interaction)
elif self.solution_type == 'subregular':
if hasattr(self, 'energy_ternary_terms') is False:
self.energy_ternary_terms = None
if hasattr(self, 'volume_ternary_terms') is False:
self.volume_ternary_terms = None
if hasattr(self, 'entropy_ternary_terms') is False:
self.entropy_ternary_terms = None
self.solution_model = SubregularSolution(
self.endmembers,
self.energy_interaction, self.volume_interaction,
self.entropy_interaction,
self.energy_ternary_terms, self.volume_ternary_terms,
self.entropy_ternary_terms)
else:
raise Exception("Solution model type "
+ self.solution_type + "not recognised.")
else:
self.solution_model = SolutionModel()
# Equation of state
for i in range(self.n_endmembers):
self.endmembers[i][0].set_method(
self.endmembers[i][0].params['equation_of_state'])
# Molar fractions
if molar_fractions is not None:
self.set_composition(molar_fractions)
[docs] def get_endmembers(self):
return self.endmembers
[docs] def set_composition(self, molar_fractions):
"""
Set the composition for this solid solution.
Resets cached properties.
Parameters
----------
molar_fractions: list of float
molar abundance for each endmember, needs to sum to one.
"""
assert(len(self.endmembers) == len(molar_fractions))
if self.solution_type != 'mechanical':
assert(sum(molar_fractions) > 0.9999)
assert(sum(molar_fractions) < 1.0001)
self.reset()
self.molar_fractions = np.array(molar_fractions)
[docs] def set_method(self, method):
for i in range(self.n_endmembers):
self.endmembers[i][0].set_method(method)
# note: do not set self.method here!
self.reset()
[docs] def set_state(self, pressure, temperature):
Mineral.set_state(self, pressure, temperature)
for i in range(self.n_endmembers):
self.endmembers[i][0].set_state(pressure, temperature)
@material_property
def formula(self):
"""
Returns molar chemical formula of the solid solution.
"""
return sum_formulae(self.endmember_formulae, self.molar_fractions)
@material_property
def activities(self):
"""
Returns a list of endmember activities [unitless].
"""
return self.solution_model.activities(self.pressure, self.temperature,
self.molar_fractions)
@material_property
def activity_coefficients(self):
"""
Returns a list of endmember activity coefficients
(gamma = activity / ideal activity) [unitless].
"""
return self.solution_model.activity_coefficients(self.pressure,
self.temperature,
self.molar_fractions)
@material_property
def molar_internal_energy(self):
"""
Returns molar internal energy of the mineral [J/mol].
Aliased with self.energy
"""
return self.molar_helmholtz + self.temperature * self.molar_entropy
@material_property
def excess_partial_gibbs(self):
"""
Returns excess partial molar gibbs free energy [J/mol].
Property specific to solid solutions.
"""
return self.solution_model.excess_partial_gibbs_free_energies(self.pressure, self.temperature, self.molar_fractions)
@material_property
def excess_partial_volumes(self):
"""
Returns excess partial volumes [m^3].
Property specific to solid solutions.
"""
return self.solution_model.excess_partial_volumes(self.pressure,
self.temperature,
self.molar_fractions)
@material_property
def excess_partial_entropies(self):
"""
Returns excess partial entropies [J/K].
Property specific to solid solutions.
"""
return self.solution_model.excess_partial_entropies(self.pressure,
self.temperature,
self.molar_fractions)
@material_property
def partial_gibbs(self):
"""
Returns excess partial molar gibbs free energy [J/mol].
Property specific to solid solutions.
"""
return (np.array([self.endmembers[i][0].gibbs
for i in range(self.n_endmembers)])
+ self.excess_partial_gibbs)
@material_property
def partial_volumes(self):
"""
Returns excess partial volumes [m^3].
Property specific to solid solutions.
"""
return (np.array([self.endmembers[i][0].molar_volume
for i in range(self.n_endmembers)])
+ self.excess_partial_volumes)
@material_property
def partial_entropies(self):
"""
Returns excess partial entropies [J/K].
Property specific to solid solutions.
"""
return (np.array([self.endmembers[i][0].molar_entropy
for i in range(self.n_endmembers)])
+ self.excess_partial_entropies)
@material_property
def excess_gibbs(self):
"""
Returns molar excess gibbs free energy [J/mol].
Property specific to solid solutions.
"""
return self.solution_model.excess_gibbs_free_energy(self.pressure,
self.temperature,
self.molar_fractions)
@material_property
def gibbs_hessian(self):
"""
Returns an array containing the second compositional derivative
of the Gibbs free energy [J]. Property specific to solid solutions.
"""
return self.solution_model.gibbs_hessian(self.pressure,
self.temperature,
self.molar_fractions)
@material_property
def entropy_hessian(self):
"""
Returns an array containing the second compositional derivative
of the entropy [J/K]. Property specific to solid solutions.
"""
return self.solution_model.entropy_hessian(self.pressure,
self.temperature,
self.molar_fractions)
@material_property
def volume_hessian(self):
"""
Returns an array containing the second compositional derivative
of the volume [m^3]. Property specific to solid solutions.
"""
return self.solution_model.volume_hessian(self.pressure,
self.temperature,
self.molar_fractions)
@material_property
def molar_gibbs(self):
"""
Returns molar Gibbs free energy of the solid solution [J/mol].
Aliased with self.gibbs.
"""
return sum([self.endmembers[i][0].gibbs * self.molar_fractions[i]
for i in range(self.n_endmembers)]) + self.excess_gibbs
@material_property
def molar_helmholtz(self):
"""
Returns molar Helmholtz free energy of the solid solution [J/mol].
Aliased with self.helmholtz.
"""
return self.molar_gibbs - self.pressure * self.molar_volume
@material_property
def molar_mass(self):
"""
Returns molar mass of the solid solution [kg/mol].
"""
return sum([self.endmembers[i][0].molar_mass
* self.molar_fractions[i]
for i in range(self.n_endmembers)])
@material_property
def excess_volume(self):
"""
Returns excess molar volume of the solid solution [m^3/mol].
Specific property for solid solutions.
"""
return self.solution_model.excess_volume(self.pressure,
self.temperature,
self.molar_fractions)
@material_property
def molar_volume(self):
"""
Returns molar volume of the solid solution [m^3/mol].
Aliased with self.V.
"""
return sum([self.endmembers[i][0].molar_volume
* self.molar_fractions[i]
for i in range(self.n_endmembers)]) + self.excess_volume
@material_property
def density(self):
"""
Returns density of the solid solution [kg/m^3].
Aliased with self.rho.
"""
return self.molar_mass / self.molar_volume
@material_property
def excess_entropy(self):
"""
Returns excess molar entropy [J/K/mol].
Property specific to solid solutions.
"""
return self.solution_model.excess_entropy(self.pressure,
self.temperature,
self.molar_fractions)
@material_property
def molar_entropy(self):
"""
Returns molar entropy of the solid solution [J/K/mol].
Aliased with self.S.
"""
return sum([self.endmembers[i][0].S * self.molar_fractions[i]
for i in range(self.n_endmembers)]) + self.excess_entropy
@material_property
def excess_enthalpy(self):
"""
Returns excess molar enthalpy [J/mol].
Property specific to solid solutions.
"""
return self.solution_model.excess_enthalpy(self.pressure,
self.temperature,
self.molar_fractions)
@material_property
def molar_enthalpy(self):
"""
Returns molar enthalpy of the solid solution [J/mol].
Aliased with self.H.
"""
return sum([self.endmembers[i][0].H * self.molar_fractions[i]
for i in range(self.n_endmembers)]) + self.excess_enthalpy
@material_property
def isothermal_bulk_modulus(self):
"""
Returns isothermal bulk modulus of the solid solution [Pa].
Aliased with self.K_T.
"""
return self.V * 1. / (sum([self.endmembers[i][0].V
/ (self.endmembers[i][0].K_T)
* self.molar_fractions[i]
for i in range(self.n_endmembers)]))
@material_property
def adiabatic_bulk_modulus(self):
"""
Returns adiabatic bulk modulus of the solid solution [Pa].
Aliased with self.K_S.
"""
if self.temperature < 1e-10:
return self.isothermal_bulk_modulus
else:
return (self.isothermal_bulk_modulus
* self.molar_heat_capacity_p / self.molar_heat_capacity_v)
@material_property
def isothermal_compressibility(self):
"""
Returns isothermal compressibility of the solid solution.
(or inverse isothermal bulk modulus) [1/Pa].
Aliased with self.K_T.
"""
return 1. / self.isothermal_bulk_modulus
@material_property
def adiabatic_compressibility(self):
"""
Returns adiabatic compressibility of the solid solution.
(or inverse adiabatic bulk modulus) [1/Pa].
Aliased with self.K_S.
"""
return 1. / self.adiabatic_bulk_modulus
@material_property
def shear_modulus(self):
"""
Returns shear modulus of the solid solution [Pa].
Aliased with self.G.
"""
G_list = np.fromiter((e[0].G for e in self.endmembers), dtype=float,
count=self.n_endmembers)
return reuss_average_function(self.molar_fractions, G_list)
@material_property
def p_wave_velocity(self):
"""
Returns P wave speed of the solid solution [m/s].
Aliased with self.v_p.
"""
return np.sqrt((self.adiabatic_bulk_modulus
+ 4. / 3. * self.shear_modulus) / self.density)
@material_property
def bulk_sound_velocity(self):
"""
Returns bulk sound speed of the solid solution [m/s].
Aliased with self.v_phi.
"""
return np.sqrt(self.adiabatic_bulk_modulus / self.density)
@material_property
def shear_wave_velocity(self):
"""
Returns shear wave speed of the solid solution [m/s].
Aliased with self.v_s.
"""
return np.sqrt(self.shear_modulus / self.density)
@material_property
def grueneisen_parameter(self):
"""
Returns grueneisen parameter of the solid solution [unitless].
Aliased with self.gr.
"""
if self.temperature < 1e-10:
return float('nan')
else:
return (self.thermal_expansivity * self.isothermal_bulk_modulus
* self.molar_volume / self.molar_heat_capacity_v)
@material_property
def thermal_expansivity(self):
"""
Returns thermal expansion coefficient (alpha)
of the solid solution [1/K].
Aliased with self.alpha.
"""
return (1. / self.V) * sum([self.endmembers[i][0].alpha
* self.endmembers[i][0].V
* self.molar_fractions[i]
for i in range(self.n_endmembers)])
@material_property
def molar_heat_capacity_v(self):
"""
Returns molar heat capacity at constant volume of the
solid solution [J/K/mol].
Aliased with self.C_v.
"""
return (self.molar_heat_capacity_p
- self.molar_volume * self.temperature
* self.thermal_expansivity * self.thermal_expansivity
* self.isothermal_bulk_modulus)
@material_property
def molar_heat_capacity_p(self):
"""
Returns molar heat capacity at constant pressure
of the solid solution [J/K/mol].
Aliased with self.C_p.
"""
return sum([self.endmembers[i][0].molar_heat_capacity_p
* self.molar_fractions[i]
for i in range(self.n_endmembers)])
[docs] @cached_property
def stoichiometric_matrix(self):
"""
A sympy Matrix where each element M[i,j] corresponds
to the number of atoms of element[j] in endmember[i].
"""
def f(i, j):
e = self.elements[j]
if e in self.endmember_formulae[i]:
return nsimplify(self.endmember_formulae[i][e])
else:
return 0
return Matrix(len(self.endmember_formulae), len(self.elements), f)
[docs] @cached_property
def stoichiometric_array(self):
"""
An array where each element arr[i,j] corresponds
to the number of atoms of element[j] in endmember[i].
"""
return np.array(self.stoichiometric_matrix)
[docs] @cached_property
def reaction_basis(self):
"""
An array where each element arr[i,j] corresponds
to the number of moles of endmember[j] involved in reaction[i].
"""
reaction_basis = np.array([v[:] for v in
self.stoichiometric_matrix.T.nullspace()])
if len(reaction_basis) == 0:
reaction_basis = np.empty((0, len(self.endmember_names)))
return reaction_basis
[docs] @cached_property
def n_reactions(self):
"""
The number of reactions in reaction_basis.
"""
return len(self.reaction_basis[:, 0])
[docs] @cached_property
def independent_element_indices(self):
"""
A list of an independent set of element indices. If the amounts of
these elements are known (element_amounts),
the amounts of the other elements can be inferred by
-compositional_null_basis[independent_element_indices].dot(element_amounts).
"""
return sorted(independent_row_indices(self.stoichiometric_matrix.T))
[docs] @cached_property
def dependent_element_indices(self):
"""
The element indices not included in the independent list.
"""
return [i for i in range(len(self.elements))
if i not in self.independent_element_indices]
[docs] @cached_property
def compositional_null_basis(self):
"""
An array N such that N.b = 0 for all bulk compositions that can
be produced with a linear sum of the endmembers in the solid solution.
"""
null_basis = np.array([v[:] for v in
self.stoichiometric_matrix.nullspace()])
M = null_basis[:, self.dependent_element_indices]
assert (M.shape[0] == M.shape[1]) and (M == np.eye(M.shape[0])).all()
return null_basis
[docs] @cached_property
def endmember_names(self):
"""
A list of names for all the endmember in the solid solution.
"""
return [mbr[0].name for mbr in self.endmembers]
[docs] @cached_property
def n_endmembers(self):
"""
The number of endmembers in the solid solution.
"""
return len(self.endmembers)
[docs] @cached_property
def elements(self):
"""
A list of the elements which could be contained in the solid solution,
returned in the IUPAC element order.
"""
keys = []
for f in self.endmember_formulae:
keys.extend(f.keys())
return sort_element_list_to_IUPAC_order(set(keys))