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# Only needed in a Jupyter Notebook
%matplotlib inline
# Optional plot styling
import matplotlib
matplotlib.style.use('bmh')
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import matplotlib.pyplot as plt
from pycalphad import equilibrium
from pycalphad import Database, Model
import pycalphad.variables as v
import numpy as np
Here we compute equilibrium thermodynamic properties in the Al-Fe system. We know that only B2 and liquid are stable in the temperature range of interest, but we just as easily could have included all the phases in the calculation using my_phases = list(db.phases.keys()). Notice that the syntax for specifying a range is (min, max, step). We can also directly specify a list of temperatures using the list syntax, e.g., [300, 400, 500, 1400].
We explicitly indicate that we want to compute equilibrium values of the heat_capacity and degree_of_ordering properties. These are both defined in the default Model class. For a complete list, see the documentation. equilibrium will always return the Gibbs energy, chemical potentials, phase fractions and site fractions, regardless of the value of output.
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db = Database('alfe_sei.TDB')
my_phases = ['LIQUID', 'B2_BCC']
eq = equilibrium(db, ['AL', 'FE', 'VA'], my_phases, {v.X('AL'): 0.25, v.T: (300, 2000, 50), v.P: 101325},
output=['heat_capacity', 'degree_of_ordering'])
print(eq)
We also compute degree of ordering at fixed temperature as a function of composition.
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eq2 = equilibrium(db, ['AL', 'FE', 'VA'], 'B2_BCC', {v.X('AL'): (0,1,0.01), v.T: 700, v.P: 101325},
output='degree_of_ordering')
print(eq2)
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plt.gca().set_title('Al-Fe: Degree of bcc ordering vs T [X(AL)=0.25]')
plt.gca().set_xlabel('Temperature (K)')
plt.gca().set_ylabel('Degree of ordering')
plt.gca().set_ylim((-0.1,1.1))
# Generate a list of all indices where B2 is stable
phase_indices = np.nonzero(eq.Phase.values == 'B2_BCC')
# phase_indices[1] refers to all temperature indices
# We know this because pycalphad always returns indices in order like P, T, X's
plt.plot(np.take(eq['T'].values, phase_indices[1]), eq['degree_of_ordering'].values[phase_indices])
plt.show()
For the heat capacity curve shown below we notice a sharp increase in the heat capacity around 750 K. This is indicative of a magnetic phase transition and, indeed, the temperature at the peak of the curve coincides with 75% of 1043 K, the Curie temperature of pure Fe. (Pure bcc Al is paramagnetic so it has an effective Curie temperature of 0 K.)
We also observe a sharp jump in the heat capacity near 1800 K, corresponding to the melting of the bcc phase.
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plt.gca().set_title('Al-Fe: Heat capacity vs T [X(AL)=0.25]')
plt.gca().set_xlabel('Temperature (K)')
plt.gca().set_ylabel('Heat Capacity (J/mol-atom-K)')
# np.squeeze is used to remove all dimensions of size 1
# For a 1-D/"step" calculation, this aligns the temperature and heat capacity arrays
# In 2-D/"map" calculations, we'd have to explicitly select the composition of interest
plt.plot(eq['T'].values, np.squeeze(eq['heat_capacity'].values))
plt.show()
To understand more about what's happening around 700 K, we plot the degree of ordering versus composition. Note that this plot excludes all other phases except B2_BCC. We observe the presence of disordered bcc (A2) until around 13% Al or Fe, when the phase begins to order.
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plt.gca().set_title('Al-Fe: Degree of bcc ordering vs X(AL) [T=700 K]')
plt.gca().set_xlabel('X(AL)')
plt.gca().set_ylabel('Degree of ordering')
# Generate a list of all indices where B2 is stable
phase_indices = np.nonzero(eq2.Phase.values == 'B2_BCC')
# phase_indices[2] refers to all composition indices
# We know this because pycalphad always returns indices in order like P, T, X's
plt.plot(np.take(eq2['X_AL'].values, phase_indices[2]), eq2['degree_of_ordering'].values[phase_indices])
plt.show()
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db_alni = Database('NI_AL_DUPIN_2001.TDB')
phases = ['LIQUID', 'FCC_L12']
eq_alni = equilibrium(db_alni, ['AL', 'NI', 'VA'], phases, {v.X('AL'): 0.10, v.T: (300, 2500, 20), v.P: 101325},
output='degree_of_ordering')
print(eq_alni)
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from pycalphad.plot.utils import phase_legend
phase_handles, phasemap = phase_legend(phases)
plt.gca().set_title('Al-Ni: Phase fractions vs T [X(AL)=0.1]')
plt.gca().set_xlabel('Temperature (K)')
plt.gca().set_ylabel('Phase Fraction')
plt.gca().set_ylim((0,1.1))
plt.gca().set_xlim((300, 2000))
for name in phases:
phase_indices = np.nonzero(eq_alni.Phase.values == name)
plt.scatter(np.take(eq_alni['T'].values, phase_indices[1]), eq_alni.NP.values[phase_indices], color=phasemap[name])
plt.gca().legend(phase_handles, phases, loc='lower right')
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In the plot below we see that the degree of ordering does not change at all in each phase. There is a very abrupt disappearance of the completely ordered gamma-prime phase, leaving the completely disordered gamma phase. This is a first-order phase transition.
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plt.gca().set_title('Al-Fe: Degree of fcc ordering vs T [X(AL)=0.1]')
plt.gca().set_xlabel('Temperature (K)')
plt.gca().set_ylabel('Degree of ordering')
plt.gca().set_ylim((-0.1,1.1))
# Generate a list of all indices where FCC_L12 is stable and ordered
L12_phase_indices = np.nonzero(np.logical_and((eq_alni.Phase.values == 'FCC_L12'),
(eq_alni.degree_of_ordering.values > 0.01)))
# Generate a list of all indices where FCC_L12 is stable and disordered
fcc_phase_indices = np.nonzero(np.logical_and((eq_alni.Phase.values == 'FCC_L12'),
(eq_alni.degree_of_ordering.values <= 0.01)))
# phase_indices[1] refers to all temperature indices
# We know this because pycalphad always returns indices in order like P, T, X's
plt.plot(np.take(eq_alni['T'].values, L12_phase_indices[1]), eq_alni['degree_of_ordering'].values[L12_phase_indices],
label='$\gamma\prime$ (ordered fcc)', color='red')
plt.plot(np.take(eq_alni['T'].values, fcc_phase_indices[1]), eq_alni['degree_of_ordering'].values[fcc_phase_indices],
label='$\gamma$ (disordered fcc)', color='blue')
plt.legend()
plt.show()
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