Source code for pycalphad.model

"""
The model module provides support for using a Database to perform
calculations under specified conditions.
"""
from __future__ import division
import copy
from sympy import exp, log, Abs, Add, Float, Mul, Piecewise, Pow, S, sin, StrictGreaterThan, Symbol, zoo, oo, nan
from tinydb import where
import pycalphad.variables as v
from pycalphad.core.errors import DofError
from pycalphad.core.constants import MIN_SITE_FRACTION
from pycalphad.core.utils import unpack_components, get_pure_elements
from pycalphad.core.constraints import is_multiphase_constraint
import numpy as np
from collections import OrderedDict

# Maximum number of levels deep we check for symbols that are functions of
# other symbols
_MAX_PARAM_NESTING = 32


[docs]class ReferenceState(): """ Define the phase and any fixed state variables as a reference state for a component. Parameters ---------- Attributes ---------- fixed_statevars : dict Dictionary of {StateVariable: value} that will be fixed, e.g. {v.T: 298.15, v.P: 101325} phase_name : str Name of phase species : Species pycalphad Species variable """ def __init__(self, species, reference_phase, fixed_statevars=None): """ Parameters ---------- species : str or Species Species to define the reference state for. Only pure elements supported. reference_phase : str Name of phase fixed_statevars : None, optional Dictionary of {StateVariable: value} that will be fixed, e.g. {v.T: 298.15, v.P: 101325} If None (the default), an empty dict will be created. """ if isinstance(species, v.Species): self.species = species else: self.species = v.Species(species) self.phase_name = reference_phase self.fixed_statevars = fixed_statevars if fixed_statevars is not None else {} def __repr__(self): if len(self.fixed_statevars.keys()) > 0: s = "ReferenceState('{}', '{}', {})".format(self.species.name, self.phase_name, self.fixed_statevars) else: s = "ReferenceState('{}', '{}')".format(self.species.name, self.phase_name) return s
[docs]class Model(object): """ Models use an abstract representation of the function for calculation of values under specified conditions. Parameters ---------- dbe : Database Database containing the relevant parameters. comps : list Names of components to consider in the calculation. phase_name : str Name of phase model to build. parameters : dict or list Optional dictionary of parameters to be substituted in the model. A list of parameters will cause those symbols to remain symbolic. This will overwrite parameters specified in the database Methods ------- None yet. Examples -------- None yet. """ # We only use the contributions attribute in build_phase. # Users should not access it later since subclasses can override build_phase # and make self.models inconsistent with contributions. # Note that we include atomic ordering last since it uses self.models # to figure out its contribution. contributions = [('ref', 'reference_energy'), ('idmix', 'ideal_mixing_energy'), ('xsmix', 'excess_mixing_energy'), ('mag', 'magnetic_energy'), ('2st', 'twostate_energy'), ('ein', 'einstein_energy'), ('ord', 'atomic_ordering_energy')] def __init__(self, dbe, comps, phase_name, parameters=None, build_reference=True): self.components = set() self.constituents = [] self.phase_name = phase_name.upper() phase = dbe.phases[self.phase_name] self.site_ratios = list(phase.sublattices) for idx, sublattice in enumerate(phase.constituents): subl_comps = set(sublattice).intersection(unpack_components(dbe, comps)) self.components |= subl_comps # Support for variable site ratios in ionic liquid model if phase.model_hints.get('ionic_liquid_2SL', False): if idx == 0: subl_idx = 1 elif idx == 1: subl_idx = 0 else: raise ValueError('Two-sublattice ionic liquid specified with more than two sublattices') self.site_ratios[subl_idx] = Add(*[v.SiteFraction(self.phase_name, idx, spec) * abs(spec.charge) for spec in subl_comps]) if phase.model_hints.get('ionic_liquid_2SL', False): # Special treatment of "neutral" vacancies in 2SL ionic liquid # These are treated as having variable valence for idx, sublattice in enumerate(phase.constituents): subl_comps = set(sublattice).intersection(unpack_components(dbe, comps)) if v.Species('VA') in subl_comps: if idx == 0: subl_idx = 1 elif idx == 1: subl_idx = 0 else: raise ValueError('Two-sublattice ionic liquid specified with more than two sublattices') self.site_ratios[subl_idx] += self.site_ratios[idx] * v.SiteFraction(self.phase_name, idx, v.Species('VA')) self.site_ratios = tuple(self.site_ratios) # Verify that this phase is still possible to build for sublattice in phase.constituents: if len(set(sublattice).intersection(self.components)) == 0: # None of the components in a sublattice are active # We cannot build a model of this phase raise DofError( '{0}: Sublattice {1} of {2} has no components in {3}' \ .format(self.phase_name, sublattice, phase.constituents, self.components)) self.constituents.append(set(sublattice).intersection(self.components)) self.components = sorted(self.components) desired_active_pure_elements = [list(x.constituents.keys()) for x in self.components] desired_active_pure_elements = [el.upper() for constituents in desired_active_pure_elements for el in constituents] self.pure_elements = sorted(set(desired_active_pure_elements)) self.nonvacant_elements = [x for x in self.pure_elements if x != 'VA'] # Convert string symbol names to sympy Symbol objects # This makes xreplace work with the symbols dict symbols = {Symbol(s): val for s, val in dbe.symbols.items()} def wrap_symbol(obj): if isinstance(obj, Symbol): return obj else: return Symbol(obj) if parameters is not None: self._parameters_arg = parameters if isinstance(parameters, dict): symbols.update([(wrap_symbol(s), val) for s, val in parameters.items()]) else: # Lists of symbols that should remain symbolic for s in parameters: symbols.pop(wrap_symbol(s)) else: self._parameters_arg = None self._symbols = {wrap_symbol(key): value for key, value in symbols.items()} self.models = OrderedDict() self.build_phase(dbe) # build reference model, this needs to be behind a flag to avoid recursion if build_reference: self.build_reference_model(dbe) for name, value in self.models.items(): self.models[name] = self.symbol_replace(value, symbols) self.site_fractions = sorted([x for x in self.variables if isinstance(x, v.SiteFraction)], key=str) self.state_variables = sorted([x for x in self.variables if not isinstance(x, v.SiteFraction)], key=str)
[docs] @staticmethod def symbol_replace(obj, symbols): """ Substitute values of symbols into 'obj'. Parameters ---------- obj : SymPy object symbols : dict mapping sympy.Symbol to SymPy object Returns ------- SymPy object """ try: # Need to do more substitutions to catch symbols that are functions # of other symbols for iteration in range(_MAX_PARAM_NESTING): obj = obj.xreplace(symbols) undefs = [x for x in obj.free_symbols if not isinstance(x, v.StateVariable)] if len(undefs) == 0: break except AttributeError: # Can't use xreplace on a float pass return obj
def __eq__(self, other): if self is other: return True elif type(self) != type(other): return False else: return self.__dict__ == other.__dict__ def __ne__(self, other): return not self.__eq__(other) def __hash__(self): return hash(repr(self))
[docs] def moles(self, species): "Number of moles of species or elements." species = v.Species(species) is_pure_element = (len(species.constituents.keys()) == 1 and list(species.constituents.keys())[0] == species.name) result = S.Zero normalization = S.Zero if is_pure_element: element = list(species.constituents.keys())[0] for idx, sublattice in enumerate(self.constituents): active = set(sublattice).intersection(self.components) result += self.site_ratios[idx] * \ sum(int(spec.number_of_atoms > 0) * spec.constituents.get(element, 0) * v.SiteFraction(self.phase_name, idx, spec) for spec in active) normalization += self.site_ratios[idx] * \ sum(spec.number_of_atoms * v.SiteFraction(self.phase_name, idx, spec) for spec in active) else: for idx, sublattice in enumerate(self.constituents): active = set(sublattice).intersection({species}) if len(active) == 0: continue result += self.site_ratios[idx] * sum(v.SiteFraction(self.phase_name, idx, spec) for spec in active) normalization += self.site_ratios[idx] * \ sum(int(spec.number_of_atoms > 0) * v.SiteFraction(self.phase_name, idx, spec) for spec in active) return result / normalization
@property def ast(self): "Return the full abstract syntax tree of the model." return Add(*list(self.models.values())) @property def variables(self): "Return state variables in the model." return sorted([x for x in self.ast.free_symbols if isinstance(x, v.StateVariable)], key=str) @property def degree_of_ordering(self): result = S.Zero site_ratio_normalization = S.Zero # Calculate normalization factor for idx, sublattice in enumerate(self.constituents): active = set(sublattice).intersection(self.components) subl_content = sum(int(spec.number_of_atoms > 0) * v.SiteFraction(self.phase_name, idx, spec) for spec in active) site_ratio_normalization += self.site_ratios[idx] * subl_content site_ratios = [c/site_ratio_normalization for c in self.site_ratios] for comp in self.components: if comp.number_of_atoms == 0: continue comp_result = S.Zero for idx, sublattice in enumerate(self.constituents): active = set(sublattice).intersection(set(self.components)) if comp in active: comp_result += site_ratios[idx] * Abs(v.SiteFraction(self.phase_name, idx, comp) - self.moles(comp)) / self.moles(comp) result += comp_result return result / sum(int(spec.number_of_atoms > 0) for spec in self.components) DOO = degree_of_ordering # Can be defined as a list of pre-computed first derivatives gradient = None # Note: In order-disorder phases, TC will always be the *disordered* value of TC curie_temperature = TC = S.Zero neel_temperature = NT = S.Zero #pylint: disable=C0103 # These are standard abbreviations from Thermo-Calc for these quantities energy = GM = property(lambda self: self.ast) entropy = SM = property(lambda self: -self.GM.diff(v.T)) enthalpy = HM = property(lambda self: self.GM - v.T*self.GM.diff(v.T)) heat_capacity = CPM = property(lambda self: -v.T*self.GM.diff(v.T, v.T)) #pylint: enable=C0103 mixing_energy = GM_MIX = property(lambda self: self.GM - self.reference_model.GM) mixing_enthalpy = HM_MIX = property(lambda self: self.GM_MIX - v.T*self.GM_MIX.diff(v.T)) mixing_entropy = SM_MIX = property(lambda self: -self.GM_MIX.diff(v.T)) mixing_heat_capacity = CPM_MIX = property(lambda self: -v.T*self.GM_MIX.diff(v.T, v.T))
[docs] def build_reference_model(self, dbe, preserve_ideal=True): """ Build a reference_model for the current model, referenced to the endmembers. Parameters ---------- dbe : Database preserve_ideal : bool, optional If True, the default, the ideal mixing energy will not be subtracted out. Notes ----- Requires that self.build_phase has already been called. This builds a special reference state that is referenced to the CEF endmembers. The endmembers for the _MIX properties of this class referenced to the reference_model energies will always be zero, such that the _MIX properties generated here allow users to see mixing energies on the internal degrees of freedom of this phase. The reference_model AST can be modified in the same way as the current Model. Ideal mixing is always added to the AST, we need to set it to zero here so that it's not subtracted out of the reference however, we have this option so users can just see the mixing properties in terms of the parameters If the current model has an ordering energy as part of a partitioned model, then this special reference state is not well defined because the endmembers in the model have energetic contributions from the ordered endmember energies and the disordered mixing energies. Therefore, this reference state cannot be used sensibly for partitioned models and the energies of all the reference_model.models are set to nan. """ endmember_only_dbe = copy.deepcopy(dbe) endmember_only_dbe._parameters.remove(where('constituent_array').test(self._interaction_test)) mod_endmember_only = self.__class__(endmember_only_dbe, self.components, self.phase_name, parameters=self._parameters_arg, build_reference=False) if preserve_ideal: mod_endmember_only.models['idmix'] = 0 self.reference_model = mod_endmember_only if self.models.get('ord', S.Zero) != S.Zero: for k in self.reference_model.models.keys(): self.reference_model.models[k] = nan
[docs] def get_internal_constraints(self): constraints = [] for idx, sublattice in enumerate(self.constituents): constraints.append(sum(v.SiteFraction(self.phase_name, idx, spec) for spec in sublattice) - 1) return constraints
[docs] def get_multiphase_constraints(self, conds): fixed_chempots = [cond for cond in conds.keys() if isinstance(cond, v.ChemicalPotential)] multiphase_constraints = [] for statevar in sorted(conds.keys(), key=str): if not is_multiphase_constraint(statevar): continue if isinstance(statevar, v.Composition): multiphase_constraints.append(Symbol('NP') * self.moles(statevar.species)) elif statevar == v.N: multiphase_constraints.append(Symbol('NP') * (sum(self.moles(spec) for spec in self.nonvacant_elements))) elif statevar in [v.T, v.P]: return multiphase_constraints.append(S.Zero) else: raise NotImplementedError return multiphase_constraints
[docs] def build_phase(self, dbe): """ Generate the symbolic form of all the contributions to this phase. Parameters ---------- dbe : Database """ contrib_vals = list(OrderedDict(self.__class__.contributions).values()) if 'atomic_ordering_energy' in contrib_vals: if contrib_vals.index('atomic_ordering_energy') != (len(contrib_vals) - 1): # Check for a common mistake in custom models # Users that need to override this behavior should override build_phase raise ValueError('\'atomic_ordering_energy\' must be the final contribution') self.models.clear() for key, value in self.__class__.contributions: self.models[key] = S(getattr(self, value)(dbe))
def _purity_test(self, constituent_array): """ Check if constituent array only has one species in its array This species must also be an active species """ if len(constituent_array) != len(self.constituents): return False for sublattice in constituent_array: if len(sublattice) != 1: return False if (sublattice[0] not in self.components) and \ (sublattice[0] != v.Species('*')): return False return True def _array_validity(self, constituent_array): """ Check that the current array contains only active species. """ if len(constituent_array) != len(self.constituents): return False for sublattice in constituent_array: valid = set(sublattice).issubset(self.components) \ or sublattice[0] == v.Species('*') if not valid: return False return True def _interaction_test(self, constituent_array): """ Check if constituent array has more than one active species in its array for at least one sublattice. """ result = False if len(constituent_array) != len(self.constituents): return False for sublattice in constituent_array: # check if all elements involved are also active valid = set(sublattice).issubset(self.components) \ or sublattice[0] == v.Species('*') if len(sublattice) > 1 and valid: result = True if not valid: result = False break return result @property def _site_ratio_normalization(self): """ Calculates the normalization factor based on the number of sites in each sublattice. """ site_ratio_normalization = S.Zero # Calculate normalization factor for idx, sublattice in enumerate(self.constituents): active = set(sublattice).intersection(self.components) subl_content = sum(spec.number_of_atoms * v.SiteFraction(self.phase_name, idx, spec) for spec in active) site_ratio_normalization += self.site_ratios[idx] * subl_content return site_ratio_normalization @staticmethod def _Muggianu_correction_dict(comps): #pylint: disable=C0103 """ Replace y_i -> y_i + (1 - sum(y involved in parameter)) / m, where m is the arity of the interaction parameter. Returns a dict converting the list of Symbols (comps) to this. m is assumed equal to the length of comps. When incorporating binary, ternary or n-ary interaction parameters into systems with more than n components, the sum of site fractions involved in the interaction parameter may no longer be unity. This breaks the symmetry of the parameter. The solution suggested by Muggianu, 1975, is to renormalize the site fractions by replacing them with a term that will sum to unity even in higher-order systems. There are other solutions that involve retaining the asymmetry for physical reasons, but this solution works well for components that are physically similar. This procedure is based on an analysis by Hillert, 1980, published in the Calphad journal. """ arity = len(comps) return_dict = {} correction_term = (S.One - Add(*comps)) / arity for comp in comps: return_dict[comp] = comp + correction_term return return_dict
[docs] def redlich_kister_sum(self, phase, param_search, param_query): """ Construct parameter in Redlich-Kister polynomial basis, using the Muggianu ternary parameter extension. """ rk_terms = [] # search for desired parameters params = param_search(param_query) for param in params: # iterate over every sublattice mixing_term = S.One for subl_index, comps in enumerate(param['constituent_array']): comp_symbols = None # convert strings to symbols if comps[0] == v.Species('*'): # Handle wildcards in constituent array comp_symbols = \ [ v.SiteFraction(phase.name, subl_index, comp) for comp in sorted(set(phase.constituents[subl_index])\ .intersection(self.components)) ] mixing_term *= Add(*comp_symbols) else: comp_symbols = \ [ v.SiteFraction(phase.name, subl_index, comp) for comp in comps ] mixing_term *= Mul(*comp_symbols) # is this a higher-order interaction parameter? if len(comps) == 2 and param['parameter_order'] > 0: # interacting sublattice, add the interaction polynomial mixing_term *= Pow(comp_symbols[0] - \ comp_symbols[1], param['parameter_order']) if len(comps) == 3: # 'parameter_order' is an index to a variable when # we are in the ternary interaction parameter case # NOTE: The commercial software packages seem to have # a "feature" where, if only the zeroth # parameter_order term of a ternary parameter is specified, # the other two terms are automatically generated in order # to make the parameter symmetric. # In other words, specifying only this parameter: # PARAMETER G(FCC_A1,AL,CR,NI;0) 298.15 +30300; 6000 N ! # Actually implies: # PARAMETER G(FCC_A1,AL,CR,NI;0) 298.15 +30300; 6000 N ! # PARAMETER G(FCC_A1,AL,CR,NI;1) 298.15 +30300; 6000 N ! # PARAMETER G(FCC_A1,AL,CR,NI;2) 298.15 +30300; 6000 N ! # # If either 1 or 2 is specified, no implicit parameters are # generated. # We need to handle this case. if param['parameter_order'] == 0: # are _any_ of the other parameter_orders specified? ternary_param_query = ( (where('phase_name') == param['phase_name']) & \ (where('parameter_type') == \ param['parameter_type']) & \ (where('constituent_array') == \ param['constituent_array']) ) other_tern_params = param_search(ternary_param_query) if len(other_tern_params) == 1 and \ other_tern_params[0] == param: # only the current parameter is specified # We need to generate the other two parameters. order_one = copy.deepcopy(param) order_one['parameter_order'] = 1 order_two = copy.deepcopy(param) order_two['parameter_order'] = 2 # Add these parameters to our iteration. params.extend((order_one, order_two)) # Include variable indicated by parameter order index # Perform Muggianu adjustment to site fractions mixing_term *= comp_symbols[param['parameter_order']].subs( self._Muggianu_correction_dict(comp_symbols), simultaneous=True) if phase.model_hints.get('ionic_liquid_2SL', False): # Special normalization rules for parameters apply under this model # Reference: Bo Sundman, "Modification of the two-sublattice model for liquids", # Calphad, Volume 15, Issue 2, 1991, Pages 109-119, ISSN 0364-5916 if not any([m.species.charge < 0 for m in mixing_term.free_symbols]): pair_rule = {} # Cation site fractions must always appear with vacancy site fractions va_subls = [(v.Species('VA') in phase.constituents[idx]) for idx in range(len(phase.constituents))] va_subl_idx = (len(phase.constituents) - 1) - va_subls[::-1].index(True) va_present = any((v.Species('VA') in c) for c in param['constituent_array']) if va_present and (max(len(c) for c in param['constituent_array']) == 1): # No need to apply pair rule for VA-containing endmember pass elif va_subl_idx > -1: for sym in mixing_term.free_symbols: if sym.species.charge > 0: pair_rule[sym] = sym * v.SiteFraction(sym.phase_name, va_subl_idx, v.Species('VA')) mixing_term = mixing_term.xreplace(pair_rule) # This parameter is normalized differently due to the variable charge valence of vacancies mixing_term *= self.site_ratios[va_subl_idx] rk_terms.append(mixing_term * param['parameter']) return Add(*rk_terms)
[docs] def reference_energy(self, dbe): """ Returns the weighted average of the endmember energies in symbolic form. """ pure_param_query = ( (where('phase_name') == self.phase_name) & \ (where('parameter_order') == 0) & \ (where('parameter_type') == "G") & \ (where('constituent_array').test(self._purity_test)) ) phase = dbe.phases[self.phase_name] param_search = dbe.search pure_energy_term = self.redlich_kister_sum(phase, param_search, pure_param_query) return pure_energy_term / self._site_ratio_normalization
[docs] def ideal_mixing_energy(self, dbe): #pylint: disable=W0613 """ Returns the ideal mixing energy in symbolic form. """ phase = dbe.phases[self.phase_name] # Normalize site ratios site_ratio_normalization = self._site_ratio_normalization site_ratios = self.site_ratios site_ratios = [c/site_ratio_normalization for c in site_ratios] ideal_mixing_term = S.Zero sitefrac_limit = Float(MIN_SITE_FRACTION/10.) for subl_index, sublattice in enumerate(phase.constituents): active_comps = set(sublattice).intersection(self.components) ratio = site_ratios[subl_index] for comp in active_comps: sitefrac = \ v.SiteFraction(phase.name, subl_index, comp) # We lose some precision here, but this makes the limit behave nicely # We're okay until fractions of about 1e-12 (platform-dependent) mixing_term = Piecewise((sitefrac*log(sitefrac), StrictGreaterThan(sitefrac, sitefrac_limit, evaluate=False)), (0, True), evaluate=False) ideal_mixing_term += (mixing_term*ratio) ideal_mixing_term *= (v.R * v.T) return ideal_mixing_term
[docs] def excess_mixing_energy(self, dbe): """ Build the binary, ternary and higher order interaction term Here we use Redlich-Kister polynomial basis by default Here we use the Muggianu ternary extension by default Replace y_i -> y_i + (1 - sum(y involved in parameter)) / m, where m is the arity of the interaction parameter """ phase = dbe.phases[self.phase_name] param_search = dbe.search param_query = ( (where('phase_name') == self.phase_name) & \ ((where('parameter_type') == 'G') | (where('parameter_type') == 'L')) & \ (where('constituent_array').test(self._interaction_test)) ) excess_term = self.redlich_kister_sum(phase, param_search, param_query) return excess_term / self._site_ratio_normalization
[docs] def magnetic_energy(self, dbe): #pylint: disable=C0103, R0914 """ Return the energy from magnetic ordering in symbolic form. The implemented model is the Inden-Hillert-Jarl formulation. The approach follows from the background of W. Xiong et al, Calphad, 2012. """ phase = dbe.phases[self.phase_name] param_search = dbe.search self.TC = self.curie_temperature = S.Zero self.BMAG = self.beta = S.Zero if 'ihj_magnetic_structure_factor' not in phase.model_hints: return S.Zero if 'ihj_magnetic_afm_factor' not in phase.model_hints: return S.Zero site_ratio_normalization = self._site_ratio_normalization # define basic variables afm_factor = phase.model_hints['ihj_magnetic_afm_factor'] if afm_factor == 0: # Apply improved magnetic model which does not use AFM / Weiss factor return self.xiong_magnetic_energy(dbe) bm_param_query = ( (where('phase_name') == phase.name) & \ (where('parameter_type') == 'BMAGN') & \ (where('constituent_array').test(self._array_validity)) ) tc_param_query = ( (where('phase_name') == phase.name) & \ (where('parameter_type') == 'TC') & \ (where('constituent_array').test(self._array_validity)) ) mean_magnetic_moment = \ self.redlich_kister_sum(phase, param_search, bm_param_query) beta = mean_magnetic_moment / Piecewise( (afm_factor, mean_magnetic_moment <= 0), (1., True), evaluate=False ) self.BMAG = self.beta = beta curie_temp = \ self.redlich_kister_sum(phase, param_search, tc_param_query) tc = curie_temp / Piecewise( (afm_factor, curie_temp <= 0), (1., True), evaluate=False ) self.TC = self.curie_temperature = tc #print(tc) # 0.01 used to prevent singularity tau = v.T / (tc + 1e-9) # define model parameters p = phase.model_hints['ihj_magnetic_structure_factor'] A = 518/1125 + (11692/15975)*(1/p - 1) # factor when tau < 1 sub_tau = 1 - (1/A) * ((79/(140*p))*(tau**(-1)) + (474/497)*(1/p - 1) \ * ((tau**3)/6 + (tau**9)/135 + (tau**15)/600) ) # factor when tau >= 1 super_tau = -(1/A) * ((tau**-5)/10 + (tau**-15)/315 + (tau**-25)/1500) expr_cond_pairs = [(sub_tau, tau < 1), (super_tau, True) ] g_term = Piecewise(*expr_cond_pairs, evaluate=False) return v.R * v.T * log(beta+1) * \ g_term / site_ratio_normalization
[docs] def xiong_magnetic_energy(self, dbe): """ Return the energy from magnetic ordering in symbolic form. The approach follows W. Xiong et al, Calphad, 2012. """ phase = dbe.phases[self.phase_name] param_search = dbe.search self.TC = self.curie_temperature = S.Zero if 'ihj_magnetic_structure_factor' not in phase.model_hints: return S.Zero if 'ihj_magnetic_afm_factor' not in phase.model_hints: return S.Zero site_ratio_normalization = self._site_ratio_normalization # define basic variables afm_factor = phase.model_hints['ihj_magnetic_afm_factor'] if afm_factor != 0: raise ValueError('Xiong model called with nonzero AFM / Weiss factor') nt_param_query = ( (where('phase_name') == phase.name) & \ (where('parameter_type') == 'NT') & \ (where('constituent_array').test(self._array_validity)) ) bm_param_query = ( (where('phase_name') == phase.name) & \ (where('parameter_type') == 'BMAGN') & \ (where('constituent_array').test(self._array_validity)) ) tc_param_query = ( (where('phase_name') == phase.name) & \ (where('parameter_type') == 'TC') & \ (where('constituent_array').test(self._array_validity)) ) mean_magnetic_moment = \ self.redlich_kister_sum(phase, param_search, bm_param_query) beta = mean_magnetic_moment curie_temp = \ self.redlich_kister_sum(phase, param_search, tc_param_query) neel_temp = \ self.redlich_kister_sum(phase, param_search, nt_param_query) self.TC = self.curie_temperature = curie_temp.subs(self._symbols) self.NT = self.neel_temperature = neel_temp.subs(self._symbols) tau_curie = v.T / curie_temp tau_curie = tau_curie.xreplace({zoo: 1.0e10}) tau_neel = v.T / neel_temp tau_neel = tau_neel.xreplace({zoo: 1.0e10}) # define model parameters p = phase.model_hints['ihj_magnetic_structure_factor'] D = 0.33471979 + 0.49649686*(1/p - 1) sub_tau_curie = 1 - (1/D) * ((0.38438376/p)*(tau_curie**(-1)) + 0.63570895*(1/p - 1) \ * ((tau_curie**3)/6 + (tau_curie**9)/135 + (tau_curie**15)/600) + (tau_curie**21)/1617 ) sub_tau_neel = 1 - (1/D) * ((0.38438376/p)*(tau_neel**(-1)) + 0.63570895*(1/p - 1) \ * ((tau_neel**3)/6 + (tau_neel**9)/135 + (tau_neel**15)/600) + (tau_neel**21)/1617 ) super_tau_curie = -(1/D) * ((tau_curie**-7)/21 + (tau_curie**-21)/630 + (tau_curie**-35)/2975 + (tau_curie**-49)/8232) super_tau_neel = -(1/D) * ((tau_neel**-7)/21 + (tau_neel**-21)/630 + (tau_neel**-35)/2975 + (tau_neel**-49)/8232) expr_cond_pairs_curie = [(0, tau_curie <= 0), (super_tau_curie, tau_curie > 1), (sub_tau_curie, True) ] expr_cond_pairs_neel = [(0, tau_neel <= 0), (super_tau_neel, tau_neel > 1), (sub_tau_neel, True) ] g_term = Piecewise(*expr_cond_pairs_curie, evaluate=False) + Piecewise(*expr_cond_pairs_neel, evaluate=False) return v.R * v.T * log(beta+1) * \ g_term / site_ratio_normalization
[docs] def twostate_energy(self, dbe): """ Return the energy from liquid-amorphous two-state model. """ phase = dbe.phases[self.phase_name] param_search = dbe.search site_ratio_normalization = self._site_ratio_normalization gd_param_query = ( (where('phase_name') == phase.name) & \ (where('parameter_type') == 'GD') & \ (where('constituent_array').test(self._array_validity)) ) gd = self.redlich_kister_sum(phase, param_search, gd_param_query) if gd == S.Zero: return S.Zero return -v.R * v.T * log(1 + exp(-gd / (v.R * v.T))) / site_ratio_normalization
[docs] def einstein_energy(self, dbe): """ Return the energy based on the Einstein model. Note that THETA parameters are actually LN(THETA). All Redlich-Kister summation is done in log-space, then exp() is called on the result. """ phase = dbe.phases[self.phase_name] param_search = dbe.search theta_param_query = ( (where('phase_name') == phase.name) & \ (where('parameter_type') == 'THETA') & \ (where('constituent_array').test(self._array_validity)) ) lntheta = self.redlich_kister_sum(phase, param_search, theta_param_query) theta = exp(lntheta) if lntheta != 0: result = 1.5*v.R*theta + 3*v.R*v.T*log(1-exp(-theta/v.T)) else: result = 0 return result / self._site_ratio_normalization
[docs] @staticmethod def mole_fraction(species_name, phase_name, constituent_array, site_ratios): """ Return an abstract syntax tree of the mole fraction of the given species as a function of its constituent site fractions. Note that this will treat vacancies the same as any other component, i.e., this will not give the correct _overall_ composition for sublattices containing vacancies with other components by normalizing by a factor of 1 - y_{VA}. This is because we use this routine in the order-disorder model to calculate the disordered site fractions from the ordered site fractions, so we need _all_ site fractions, including VA, to sum to unity. """ # Normalize site ratios site_ratio_normalization = 0 numerator = S.Zero for idx, sublattice in enumerate(constituent_array): # sublattices with only vacancies don't count if sum(spec.number_of_atoms for spec in list(sublattice)) == 0: continue if species_name in list(sublattice): site_ratio_normalization += site_ratios[idx] numerator += site_ratios[idx] * \ v.SiteFraction(phase_name, idx, species_name) if site_ratio_normalization == 0 and species_name.name == 'VA': return 1 if site_ratio_normalization == 0: raise ValueError('Couldn\'t find ' + species_name + ' in ' + \ str(constituent_array)) return numerator / site_ratio_normalization
[docs] def atomic_ordering_energy(self, dbe): """ Return the atomic ordering contribution in symbolic form. Description follows Servant and Ansara, Calphad, 2001. """ phase = dbe.phases[self.phase_name] ordered_phase_name = phase.model_hints.get('ordered_phase', None) disordered_phase_name = phase.model_hints.get('disordered_phase', None) if phase.name != ordered_phase_name: return S.Zero disordered_model = self.__class__(dbe, sorted(self.components), disordered_phase_name) constituents = [sorted(set(c).intersection(self.components)) \ for c in dbe.phases[ordered_phase_name].constituents] # Fix variable names variable_rename_dict = {} disordered_sitefracs = [x for x in disordered_model.energy.free_symbols if isinstance(x, v.SiteFraction)] for atom in disordered_sitefracs: # Replace disordered phase site fractions with mole fractions of # ordered phase site fractions. # Special case: Pure vacancy sublattices all_species_in_sublattice = \ dbe.phases[disordered_phase_name].constituents[ atom.sublattice_index] if atom.species.name == 'VA' and len(all_species_in_sublattice) == 1: # Assume: Pure vacancy sublattices are always last vacancy_subl_index = \ len(dbe.phases[ordered_phase_name].constituents)-1 variable_rename_dict[atom] = \ v.SiteFraction( ordered_phase_name, vacancy_subl_index, atom.species) else: # All other cases: replace site fraction with mole fraction variable_rename_dict[atom] = \ self.mole_fraction( atom.species, ordered_phase_name, constituents, dbe.phases[ordered_phase_name].sublattices ) # Save all of the ordered energy contributions # This step is why this routine must be called _last_ in build_phase ordered_energy = Add(*list(self.models.values())) self.models.clear() # Copy the disordered energy contributions into the correct bins for name, value in disordered_model.models.items(): self.models[name] = value.xreplace(variable_rename_dict) # All magnetic parameters will be defined in the disordered model self.TC = self.curie_temperature = disordered_model.TC self.TC = self.curie_temperature = self.TC.xreplace(variable_rename_dict) molefraction_dict = {} # Construct a dictionary that replaces every site fraction with its # corresponding mole fraction in the disordered state ordered_sitefracs = [x for x in ordered_energy.free_symbols if isinstance(x, v.SiteFraction)] for sitefrac in ordered_sitefracs: all_species_in_sublattice = \ dbe.phases[ordered_phase_name].constituents[ sitefrac.sublattice_index] if sitefrac.species.name == 'VA' and len(all_species_in_sublattice) == 1: # pure-vacancy sublattices should not be replaced # this handles cases like AL,NI,VA:AL,NI,VA:VA and # ensures the VA's don't get mixed up continue molefraction_dict[sitefrac] = \ self.mole_fraction(sitefrac.species, ordered_phase_name, constituents, dbe.phases[ordered_phase_name].sublattices) return ordered_energy - ordered_energy.xreplace(molefraction_dict)
# TODO: fix case for VA interactions: L(PHASE,A,VA:VA;0)-type parameters
[docs] def shift_reference_state(self, reference_states, dbe, contrib_mods=None, output=('GM', 'HM', 'SM', 'CPM'), fmt_str="{}R"): """ Add new attributes for calculating properties w.r.t. an arbitrary pure element reference state. Parameters ---------- reference_states : Iterable of ReferenceState Pure element ReferenceState objects. Must include all the pure elements defined in the current model. dbe : Database Database containing the relevant parameters. output : Iterable, optional Parameters to subtract the ReferenceState from, defaults to ('GM', 'HM', 'SM', 'CPM'). contrib_mods : Mapping, optional Map of {model contribution: new value}. Used to adjust the pure reference model contributions at the time this is called, since the `models` attribute of the pure element references are effectively static after calling this method. fmt_str : str, optional String that will be formatted with the `output` parameter name. Defaults to "{}R", e.g. the transformation of 'GM' -> 'GMR' """ # Error checking # We ignore the case that the ref states are overspecified (same ref states can be used in different models w/ different active pure elements) model_pure_elements = set(get_pure_elements(dbe, self.components)) refstate_pure_elements_list = get_pure_elements(dbe, [r.species for r in reference_states]) refstate_pure_elements = set(refstate_pure_elements_list) if len(refstate_pure_elements_list) != len(refstate_pure_elements): raise DofError("Multiple ReferenceState objects exist for at least one pure element: {}".format(refstate_pure_elements_list)) if not refstate_pure_elements.issuperset(model_pure_elements): raise DofError("Non-existent ReferenceState for pure components {} in {} for {}".format(model_pure_elements.difference(refstate_pure_elements), self, self.phase_name)) contrib_mods = contrib_mods or {} def _pure_element_test(constituent_array): all_comps = set() for sublattice in constituent_array: if len(sublattice) != 1: return False all_comps.add(sublattice[0].name) pure_els = all_comps.intersection(model_pure_elements) return len(pure_els) == 1 # Remove interactions from a copy of the Database, avoids any element/VA interactions. endmember_only_dbe = copy.deepcopy(dbe) endmember_only_dbe._parameters.remove(~where('constituent_array').test(_pure_element_test)) reference_dict = {out: [] for out in output} # output: terms list for ref_state in reference_states: if ref_state.species not in self.components: continue mod_pure = self.__class__(endmember_only_dbe, [ref_state.species, v.Species('VA')], ref_state.phase_name, parameters=self._parameters_arg) # apply the modifications to the Models for contrib, new_val in contrib_mods.items(): mod_pure.models[contrib] = new_val # set all the free site fractions to one, this should effectively delete any mixing terms spuriously added, e.g. idmix site_frac_subs = {sf: 1 for sf in mod_pure.ast.free_symbols if isinstance(sf, v.SiteFraction)} for mod_key, mod_val in mod_pure.models.items(): mod_pure.models[mod_key] = mod_val.subs(site_frac_subs) moles = self.moles(ref_state.species) # get the output property of interest, substitute the fixed state variables (e.g. T=298.15) and add the pure element moles weighted term to the list of terms # substitution of fixed state variables has to happen after getting the attribute in case there are any derivatives involving that state variable for out in reference_dict.keys(): mod_out = getattr(mod_pure, out).subs(ref_state.fixed_statevars) reference_dict[out].append(mod_out*moles) # set the attribute on the class for out, terms in reference_dict.items(): reference_contrib = Add(*terms) referenced_value = getattr(self, out) - reference_contrib setattr(self, fmt_str.format(out), referenced_value)
[docs]class TestModel(Model): """ Test Model object for global minimization. Equation 15.2 in: P.M. Pardalos, H.E. Romeijn (Eds.), Handbook of Global Optimization, vol. 2. Kluwer Academic Publishers, Boston/Dordrecht/London (2002) Parameters ---------- dbf : Database Ignored by TestModel but retained for API compatibility. comps : sequence Names of components to consider in the calculation. phase : str Name of phase model to build. solution : sequence, optional Float array locating the true minimum. Same length as 'comps'. If not specified, randomly generated and saved to self.solution Methods ------- None yet. Examples -------- None yet. """ def __init__(self, dbf, comps, phase, solution=None, kmax=None): self.components = set(comps) if 'VA' in self.components: raise ValueError('Vacancies are unsupported in TestModel') self.models = dict() variables = [v.SiteFraction(phase.upper(), 0, x) for x in sorted(self.components)] if solution is None: solution = np.random.dirichlet(np.ones_like(variables, dtype=np.int)) self.solution = dict(list(zip(variables, solution))) kmax = kmax if kmax is not None else 2 scale_factor = 1e4 * len(self.components) ampl_scale = 1e3 * np.ones(kmax, dtype=np.float) freq_scale = 10 * np.ones(kmax, dtype=np.float) polys = Add(*[ampl_scale[i] * sin(freq_scale[i] * Add(*[Add(*[(varname - sol)**(j+1) for varname, sol in self.solution.items()]) for j in range(kmax)]))**2 for i in range(kmax)]) self.models['test'] = scale_factor * Add(*[(varname - sol)**2 for varname, sol in self.solution.items()]) + polys