Custom Models in pycalphad: Viscosity¶
Viscosity Model Background¶
We are going to take a CALPHAD-based property model from the literature and use it to predict the viscosity of Al-Cu-Zr liquids.
For a binary alloy liquid under small undercooling, Gąsior suggested an entropy model of the form
\[\eta = (\sum_i x_i \eta_i ) (1 - 2\frac{S_{ex}}{R})\]
where \(\eta_i\) is the viscosity of the element \(i\), \(x_i\) is the mole fraction, \(S_{ex}\) is the excess entropy, and \(R\) is the gas constant.
For more details on this model, see
M.E. Trybula, T. Gancarz, W. Gąsior, Density, surface tension and viscosity of liquid binary Al-Zn and ternary Al-Li-Zn alloys, Fluid Phase Equilibria 421 (2016) 39-48, doi:10.1016/j.fluid.2016.03.013.
Władysław Gąsior, Viscosity modeling of binary alloys: Comparative studies, Calphad 44 (2014) 119-128, doi:10.1016/j.calphad.2013.10.007.
Chenyang Zhou, Cuiping Guo, Changrong Li, Zhenmin Du, Thermodynamic assessment of the phase equilibria and prediction of glass-forming ability of the Al–Cu–Zr system, Journal of Non-Crystalline Solids 461 (2017) 47-60, doi:10.1016/j.jnoncrysol.2016.09.031.
from pycalphad import Database
TDB Parameters¶
We can calculate the excess entropy of the liquid using the Al-Cu-Zr thermodynamic database from Zhou et al.
We add three new parameters to describe the viscosity (in Pa-s) of the pure elements Al, Cu, and Zr:
$ Viscosity test parameters
PARAMETER ETA(LIQUID,AL;0) 2.98150E+02 +0.000281*EXP(12300/(8.3145*T)); 6.00000E+03
N REF:0 !
PARAMETER ETA(LIQUID,CU;0) 2.98150E+02 +0.000657*EXP(21500/(8.3145*T)); 6.00000E+03
N REF:0 !
PARAMETER ETA(LIQUID,ZR;0) 2.98150E+02 +4.74E-3 - 4.97E-6*(T-2128) ; 6.00000E+03
N REF:0 !
Great! However, if we try to load the database now, we will get an
error. This is because ETA parameters are not supported by default
in pycalphad, so we need to tell pycalphad’s TDB parser that “ETA”
should be on the list of supported parameter types.
dbf = Database('alcuzr-viscosity.tdb')
Failed while parsing: PARAMETER ETA(LIQUID,AL;0) 2.98150E+02 +0.000281*EXP(12300/(8.3145*T)); 6.00000E+03 N REF:0
Tokens: None
---------------------------------------------------------------------------
ParseException Traceback (most recent call last)
<ipython-input-2-d711f4128286> in <module>
----> 1 dbf = Database('alcuzr-viscosity.tdb')
~/Projects/pycalphad/pycalphad/io/database.py in __new__(cls, *args)
127 elif fname.find('\n') == -1:
128 # Single-line string; it's probably a filename
--> 129 return cls.from_file(fname, fmt=fmt)
130 else:
131 # Newlines found: probably a full database string
~/Projects/pycalphad/pycalphad/io/database.py in from_file(fname, fmt)
221 try:
222 dbf = Database()
--> 223 format_registry[fmt.lower()].read(dbf, fd)
224 finally:
225 # Close file descriptors created in this routine
~/Projects/pycalphad/pycalphad/io/tdb.py in read_tdb(dbf, fd)
913 tokens = None
914 try:
--> 915 tokens = grammar.parseString(command)
916 _TDB_PROCESSOR[tokens[0]](dbf, *tokens[1:])
917 except ParseException:
~/anaconda3/envs/calphad-dev/lib/python3.6/site-packages/pyparsing.py in parseString(self, instring, parseAll)
1826 else:
1827 # catch and re-raise exception from here, clears out pyparsing internal stack trace
-> 1828 raise exc
1829 else:
1830 return tokens
~/anaconda3/envs/calphad-dev/lib/python3.6/site-packages/pyparsing.py in parseString(self, instring, parseAll)
1816 instring = instring.expandtabs()
1817 try:
-> 1818 loc, tokens = self._parse( instring, 0 )
1819 if parseAll:
1820 loc = self.preParse( instring, loc )
~/anaconda3/envs/calphad-dev/lib/python3.6/site-packages/pyparsing.py in _parseCache(self, instring, loc, doActions, callPreParse)
1721 ParserElement.packrat_cache_stats[MISS] += 1
1722 try:
-> 1723 value = self._parseNoCache(instring, loc, doActions, callPreParse)
1724 except ParseBaseException as pe:
1725 # cache a copy of the exception, without the traceback
~/anaconda3/envs/calphad-dev/lib/python3.6/site-packages/pyparsing.py in _parseNoCache(self, instring, loc, doActions, callPreParse)
1560 if self.mayIndexError or preloc >= len(instring):
1561 try:
-> 1562 loc,tokens = self.parseImpl( instring, preloc, doActions )
1563 except IndexError:
1564 raise ParseException( instring, len(instring), self.errmsg, self )
~/anaconda3/envs/calphad-dev/lib/python3.6/site-packages/pyparsing.py in parseImpl(self, instring, loc, doActions)
3915 if maxException is not None:
3916 maxException.msg = self.errmsg
-> 3917 raise maxException
3918 else:
3919 raise ParseException(instring, loc, "no defined alternatives to match", self)
~/anaconda3/envs/calphad-dev/lib/python3.6/site-packages/pyparsing.py in parseImpl(self, instring, loc, doActions)
3900 for e in self.exprs:
3901 try:
-> 3902 ret = e._parse( instring, loc, doActions )
3903 return ret
3904 except ParseException as err:
~/anaconda3/envs/calphad-dev/lib/python3.6/site-packages/pyparsing.py in _parseCache(self, instring, loc, doActions, callPreParse)
1721 ParserElement.packrat_cache_stats[MISS] += 1
1722 try:
-> 1723 value = self._parseNoCache(instring, loc, doActions, callPreParse)
1724 except ParseBaseException as pe:
1725 # cache a copy of the exception, without the traceback
~/anaconda3/envs/calphad-dev/lib/python3.6/site-packages/pyparsing.py in _parseNoCache(self, instring, loc, doActions, callPreParse)
1560 if self.mayIndexError or preloc >= len(instring):
1561 try:
-> 1562 loc,tokens = self.parseImpl( instring, preloc, doActions )
1563 except IndexError:
1564 raise ParseException( instring, len(instring), self.errmsg, self )
~/anaconda3/envs/calphad-dev/lib/python3.6/site-packages/pyparsing.py in parseImpl(self, instring, loc, doActions)
3750 raise ParseSyntaxException(instring, len(instring), self.errmsg, self)
3751 else:
-> 3752 loc, exprtokens = e._parse( instring, loc, doActions )
3753 if exprtokens or exprtokens.haskeys():
3754 resultlist += exprtokens
~/anaconda3/envs/calphad-dev/lib/python3.6/site-packages/pyparsing.py in _parseCache(self, instring, loc, doActions, callPreParse)
1721 ParserElement.packrat_cache_stats[MISS] += 1
1722 try:
-> 1723 value = self._parseNoCache(instring, loc, doActions, callPreParse)
1724 except ParseBaseException as pe:
1725 # cache a copy of the exception, without the traceback
~/anaconda3/envs/calphad-dev/lib/python3.6/site-packages/pyparsing.py in _parseNoCache(self, instring, loc, doActions, callPreParse)
1560 if self.mayIndexError or preloc >= len(instring):
1561 try:
-> 1562 loc,tokens = self.parseImpl( instring, preloc, doActions )
1563 except IndexError:
1564 raise ParseException( instring, len(instring), self.errmsg, self )
~/anaconda3/envs/calphad-dev/lib/python3.6/site-packages/pyparsing.py in parseImpl(self, instring, loc, doActions)
3915 if maxException is not None:
3916 maxException.msg = self.errmsg
-> 3917 raise maxException
3918 else:
3919 raise ParseException(instring, loc, "no defined alternatives to match", self)
~/anaconda3/envs/calphad-dev/lib/python3.6/site-packages/pyparsing.py in parseImpl(self, instring, loc, doActions)
3900 for e in self.exprs:
3901 try:
-> 3902 ret = e._parse( instring, loc, doActions )
3903 return ret
3904 except ParseException as err:
~/anaconda3/envs/calphad-dev/lib/python3.6/site-packages/pyparsing.py in _parseCache(self, instring, loc, doActions, callPreParse)
1721 ParserElement.packrat_cache_stats[MISS] += 1
1722 try:
-> 1723 value = self._parseNoCache(instring, loc, doActions, callPreParse)
1724 except ParseBaseException as pe:
1725 # cache a copy of the exception, without the traceback
~/anaconda3/envs/calphad-dev/lib/python3.6/site-packages/pyparsing.py in _parseNoCache(self, instring, loc, doActions, callPreParse)
1564 raise ParseException( instring, len(instring), self.errmsg, self )
1565 else:
-> 1566 loc,tokens = self.parseImpl( instring, preloc, doActions )
1567
1568 tokens = self.postParse( instring, loc, tokens )
~/Projects/pycalphad/pycalphad/io/tdb.py in parseImpl(self, instring, loc, doActions)
174 except ValueError:
175 pass
--> 176 raise ParseException(instring, loc, self.errmsg, self)
177
178 def _tdb_grammar(): #pylint: disable=R0914
ParseException: Expected {{"ELEMENT" W:(ABCD...) [Suppress:(W:(ABCD...))] [Suppress:({Re:('[-+]?([0-9]+\\.(?!([0-9]|[eE])))|([0-9]*\\.?[0-9]+([eE][-+]?[0-9]+)?)')}...)] LineEnd} | {"SPECIES" W:(ABCD...) [Suppress:("%")] Group:({{W:(ABCD...) [Re:('[-+]?([0-9]+\\.(?!([0-9]|[eE])))|([0-9]*\\.?[0-9]+([eE][-+]?[0-9]+)?)')]}}...) [{Suppress:("/") W:(+-01...)}] LineEnd} | {"TYPE_DEFINITION" Suppress:(<SP><TAB><CR><LF>) !W:( !) SkipTo:(LineEnd)} | {"FUNCTION" W:(ABCD...) {{Re:('[-+]?([0-9]+\\.(?!([0-9]|[eE])))|([0-9]*\\.?[0-9]+([eE][-+]?[0-9]+)?)') | [","]...} {{SkipTo:(";") Suppress:(";") [Suppress:(",")]... [Re:('[-+]?([0-9]+\\.(?!([0-9]|[eE])))|([0-9]*\\.?[0-9]+([eE][-+]?[0-9]+)?)')] Suppress:({W:(YNyn) | <SP><TAB><CR><LF>})}}...}} | {"ASSESSED_SYSTEMS" SkipTo:(LineEnd)} | {"DEFINE_SYSTEM_DEFAULT" SkipTo:(LineEnd)} | {"DEFAULT_COMMAND" SkipTo:(LineEnd)} | {"DATABASE_INFO" SkipTo:(LineEnd)} | {"VERSION_DATE" SkipTo:(LineEnd)} | {"REFERENCE_FILE" SkipTo:(LineEnd)} | {"ADD_REFERENCES" SkipTo:(LineEnd)} | {"LIST_OF_REFERENCES" SkipTo:(LineEnd)} | {"TEMPERATURE_LIMITS" SkipTo:(LineEnd)} | {"PHASE" W:(ABCD...) Suppress:(<SP><TAB><CR><LF>) !W:( !) Suppress:(<SP><TAB><CR><LF>) Suppress:(W:(0123...)) Group:({Re:('[-+]?([0-9]+\\.(?!([0-9]|[eE])))|([0-9]*\\.?[0-9]+([eE][-+]?[0-9]+)?)')}...) LineEnd} | {"CONSTITUENT" W:(ABCD...) Suppress:(<SP><TAB><CR><LF>) Suppress:(":") Group:(Group:({{[Suppress:(",")] {W:(ABCD...) [Suppress:("%")]}}}...) [: Group:({{[Suppress:(",")] {W:(ABCD...) [Suppress:("%")]}}}...)]...) Suppress:(":") LineEnd} | {"PARAMETER" {"BMAGN" | "DF" | "DQ" | "G" | "GD" | "L" | "MF" | "MQ" | "NT" | "TC" | "THETA" | "V0" | "VS"} Suppress:("(") W:(ABCD...) [{Suppress:("&") W:(ABCD...)}] Suppress:(",") Group:(Group:({{[Suppress:(",")] {W:(ABCD...) [Suppress:("%")]}}}...) [: Group:({{[Suppress:(",")] {W:(ABCD...) [Suppress:("%")]}}}...)]...) [{Suppress:(";") W:(0123...)}] Suppress:(")") {{Re:('[-+]?([0-9]+\\.(?!([0-9]|[eE])))|([0-9]*\\.?[0-9]+([eE][-+]?[0-9]+)?)') | [","]...} {{SkipTo:(";") Suppress:(";") [Suppress:(",")]... [Re:('[-+]?([0-9]+\\.(?!([0-9]|[eE])))|([0-9]*\\.?[0-9]+([eE][-+]?[0-9]+)?)')] Suppress:({W:(YNyn) | <SP><TAB><CR><LF>})}}...}}} (at char 17), (line:1, col:18)
Adding the ETA parameter to the TDB parser¶
import pycalphad.io.tdb_keywords
pycalphad.io.tdb_keywords.TDB_PARAM_TYPES.append('ETA')
Now the database will load:
dbf = Database('alcuzr-viscosity.tdb')
Writing the Custom Viscosity Model¶
Now that we have our ETA parameters in the database, we need to
write a Model class to tell pycalphad how to compute viscosity. All
custom models are subclasses of the pycalphad Model class.
When the ViscosityModel is constructed, the build_phase method
is run and we need to construct the viscosity model after doing all the
other initialization using a new method build_viscosity. The
implementation of build_viscosity needs to do four things: 1. Query
the Database for all the ETA parameters 2. Compute their weighted
sum 3. Compute the excess entropy of the liquid 4. Plug all the values
into the Gąsior equation and return the result
Since the build_phase method sets the attribute viscosity to the
ViscosityModel, we can access the property using viscosity as
the output in pycalphad caluclations.
from tinydb import where
import sympy
from pycalphad import Model, variables as v
class ViscosityModel(Model):
def build_phase(self, dbe):
super(ViscosityModel, self).build_phase(dbe)
self.viscosity = self.build_viscosity(dbe)
def build_viscosity(self, dbe):
if self.phase_name != 'LIQUID':
raise ValueError('Viscosity is only defined for LIQUID phase')
phase = dbe.phases[self.phase_name]
param_search = dbe.search
# STEP 1
eta_param_query = (
(where('phase_name') == phase.name) & \
(where('parameter_type') == 'ETA') & \
(where('constituent_array').test(self._array_validity))
)
# STEP 2
eta = self.redlich_kister_sum(phase, param_search, eta_param_query)
# STEP 3
excess_energy = self.GM - self.models['ref'] - self.models['idmix']
#liquid_mod = Model(dbe, self.components, self.phase_name)
## we only want the excess contributions to the entropy
#del liquid_mod.models['ref']
#del liquid_mod.models['idmix']
excess_entropy = -excess_energy.diff(v.T)
ks = 2
# STEP 4
result = eta * (1 - ks * excess_entropy / v.R)
self.eta = eta
return result
Performing Calculations¶
Now we can create an instance of ViscosityModel for the liquid phase
using the Database object we created earlier. We can verify this
model has a viscosity attribute containing a symbolic expression for
the viscosity.
mod = ViscosityModel(dbf, ['CU', 'ZR'], 'LIQUID')
print(mod.viscosity)
(1 + 0.240543628600637*(LIQUID0CU*LIQUID0ZR*(LIQUID0CU - LIQUID0ZR)**3*Piecewise((105.895 - 13.6488*log(T), (T >= 298.15) & (T < 6000.0)), (0, True)) + LIQUID0CU*LIQUID0ZR*(LIQUID0CU - LIQUID0ZR)**2*Piecewise((36.8512*log(T) - 270.5305, (T >= 298.15) & (T < 6000.0)), (0, True)) + LIQUID0CU*LIQUID0ZR*(LIQUID0CU - LIQUID0ZR)*Piecewise((75.3798 - 9.6125*log(T), (T >= 298.15) & (T < 6000.0)), (0, True)) + LIQUID0CU*LIQUID0ZR*Piecewise((392.8485 - 51.3121*log(T), (T >= 298.15) & (T < 6000.0)), (0, True)))/(1.0*LIQUID0CU + 1.0*LIQUID0ZR))*(LIQUID0CU*Piecewise((0.000657*exp(2585.84400745685/T), (T >= 298.15) & (T < 6000.0)), (0, True)) + LIQUID0ZR*Piecewise((0.01531616 - 4.97e-6*T, (T >= 298.15) & (T < 6000.0)), (0, True)))
Finally we calculate and plot the viscosity.
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
from pycalphad import calculate
mod = ViscosityModel(dbf, ['CU', 'ZR'], 'LIQUID')
temp = 2100
# NOTICE: we need to tell pycalphad about our model for this phase
models = {'LIQUID': mod}
res = calculate(dbf, ['CU', 'ZR'], 'LIQUID', P=101325, T=temp, model=models, output='viscosity')
fig = plt.figure(figsize=(6,6))
ax = fig.gca()
ax.scatter(res.X.sel(component='ZR'), 1000 * res.viscosity.values)
ax.set_xlabel('X(ZR)')
ax.set_ylabel('Viscosity (mPa-s)')
ax.set_xlim((0,1))
ax.set_title('Viscosity at {}K'.format(temp));
We repeat the calculation for Al-Cu.
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
from pycalphad import calculate
temp = 1300
models = {'LIQUID': ViscosityModel} # we can also use Model class
res = calculate(dbf, ['CU', 'AL'], 'LIQUID', P=101325, T=temp, model=models, output='viscosity')
fig = plt.figure(figsize=(6,6))
ax = fig.gca()
ax.scatter(res.X.sel(component='CU'), 1000 * res.viscosity.values)
ax.set_xlabel('X(CU)')
ax.set_ylabel('Viscosity (mPa-s)')
ax.set_xlim((0,1))
ax.set_title('Viscosity at {}K'.format(temp));
