Equilibrium Properties and Partial Ordering (Al-Fe and Al-Ni)

# Optional plot styling
import matplotlib
matplotlib.style.use('bmh')
import matplotlib.pyplot as plt
from pycalphad import equilibrium
from pycalphad import Database, Model
import pycalphad.variables as v
import numpy as np

Al-Fe (Heat Capacity and Degree of Ordering)

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.

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)
<xarray.Dataset>
Dimensions:             (P: 1, T: 34, X_AL: 1, component: 2, internal_dof: 5, vertex: 2)
Coordinates:
  * P                   (P) float64 1.013e+05
  * T                   (T) float64 300.0 350.0 400.0 450.0 500.0 550.0 ...
  * X_AL                (X_AL) float64 0.25
  * vertex              (vertex) int64 0 1
  * component           (component) object 'AL' 'FE'
  * internal_dof        (internal_dof) int64 0 1 2 3 4
Data variables:
    MU                  (P, T, X_AL, component) float64 -7.274e+04 ...
    GM                  (P, T, X_AL) float64 -2.858e+04 -2.994e+04 -3.15e+04 ...
    NP                  (P, T, X_AL, vertex) float64 1.0 nan 1.0 nan 1.0 nan ...
    X                   (P, T, X_AL, vertex, component) float64 0.25 0.75 ...
    Y                   (P, T, X_AL, vertex, internal_dof) float64 0.5 0.5 ...
    Phase               (P, T, X_AL, vertex) object 'B2_BCC' '' 'B2_BCC' '' ...
    degree_of_ordering  (P, T, X_AL, vertex) float64 0.6666 nan 0.6665 nan ...
    heat_capacity       (P, T, X_AL) float64 25.45 26.93 28.47 30.18 32.15 ...
Attributes:
    hull_iterations: 5
    solve_iterations: 146
    engine: pycalphad 0.2.5+63.g4069829.dirty
    created: 2016-02-17 16:23:02.404841

We also compute degree of ordering at fixed temperature as a function of composition.

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)
<xarray.Dataset>
Dimensions:             (P: 1, T: 1, X_AL: 100, component: 2, internal_dof: 5, vertex: 2)
Coordinates:
  * P                   (P) float64 1.013e+05
  * T                   (T) float64 700.0
  * X_AL                (X_AL) float64 1e-09 0.01 0.02 0.03 0.04 0.05 0.06 ...
  * vertex              (vertex) int64 0 1
  * component           (component) object 'AL' 'FE'
  * internal_dof        (internal_dof) int64 0 1 2 3 4
Data variables:
    MU                  (P, T, X_AL, component) float64 -2.312e+05 ...
    GM                  (P, T, X_AL) float64 -2.447e+04 -2.565e+04 ...
    NP                  (P, T, X_AL, vertex) float64 1.0 nan 1.0 nan 1.0 nan ...
    X                   (P, T, X_AL, vertex, component) float64 1e-09 1.0 ...
    Y                   (P, T, X_AL, vertex, internal_dof) float64 1e-09 1.0 ...
    Phase               (P, T, X_AL, vertex) object 'B2_BCC' '' 'B2_BCC' '' ...
    degree_of_ordering  (P, T, X_AL, vertex) float64 1.137e-15 nan 2.015e-16 ...
Attributes:
    hull_iterations: 5
    solve_iterations: 390
    engine: pycalphad 0.2.5+63.g4069829.dirty
    created: 2016-02-17 16:25:53.860451

Plots

Next we plot the degree of ordering versus temperature. We can see that the decrease in the degree of ordering is relatively steady and continuous. This is indicative of a second-order transition from partially ordered B2 to disordered bcc (A2).

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()
../_images/EquilibriumWithOrdering_8_0.png

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.

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()
../_images/EquilibriumWithOrdering_10_0.png

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.

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()
../_images/EquilibriumWithOrdering_12_0.png

Al-Ni (Degree of Ordering)

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)
<xarray.Dataset>
Dimensions:             (P: 1, T: 110, X_AL: 1, component: 2, internal_dof: 5, vertex: 2)
Coordinates:
  * P                   (P) float64 1.013e+05
  * T                   (T) float64 300.0 320.0 340.0 360.0 380.0 400.0 ...
  * X_AL                (X_AL) float64 0.1
  * vertex              (vertex) int64 0 1
  * component           (component) object 'AL' 'NI'
  * internal_dof        (internal_dof) int64 0 1 2 3 4
Data variables:
    MU                  (P, T, X_AL, component) float64 -1.719e+05 ...
    GM                  (P, T, X_AL) float64 -2.526e+04 -2.585e+04 ...
    NP                  (P, T, X_AL, vertex) float64 0.3829 0.6171 0.3543 ...
    X                   (P, T, X_AL, vertex, component) float64 0.25 0.75 ...
    Y                   (P, T, X_AL, vertex, internal_dof) float64 1e-12 1.0 ...
    Phase               (P, T, X_AL, vertex) object 'FCC_L12' 'FCC_L12' ...
    degree_of_ordering  (P, T, X_AL, vertex) float64 1.0 7.962e-15 1.0 ...
Attributes:
    hull_iterations: 5
    solve_iterations: 1047
    engine: pycalphad 0.2.5+63.g4069829.dirty
    created: 2016-02-17 16:32:02.881604

Plots

In the plot below we observe two phases designated FCC_L12. This is indicative of a miscibility gap. The ordered gamma-prime phase steadily decreases in amount with increasing temperature until it completely disappears around 750 K, leaving only the disordered gamma phase.

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')
../_images/EquilibriumWithOrdering_16_1.png

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.

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()
../_images/EquilibriumWithOrdering_18_0.png