In [1]:

    

from pycalphad import Database, Model, calculate, equilibrium, variables as v
from xarray import DataArray


    

In [2]:

    

class PrecipitateModel(Model):
    matrix_chempots = []
    @property
    def matrix_hyperplane(self):
        return sum(self.moles(self.nonvacant_elements[i])*self.matrix_chempots[i]
                   for i in range(len(self.nonvacant_elements)))
    @property
    def GM(self):
        return self.ast - self.matrix_hyperplane

class GibbsThompsonModel(Model):
    "Spherical particle."
    radius = 1e-6 # m
    volume = 7.3e-6 # m^3/mol
    interfacial_energy = 250e-3 # J/m^2
    elastic_misfit_energy = 0 # J/mol
    @property
    def GM(self):
        return self.ast + (2*self.interfacial_energy/self.radius + self.elastic_misfit_energy) * self.volume


    

In [3]:

    

import numpy as np

def parallel_tangent(dbf, comps, matrix_phase, matrix_comp, precipitate_phase, temp):
    conds = {v.N: 1, v.P: 1e5, v.T: temp}
    conds.update(matrix_comp)
    matrix_eq = equilibrium(dbf, comps, matrix_phase, conds)
    # pycalphad currently doesn't have a way to turn global minimization off and directly specify starting points
    if matrix_eq.isel(vertex=1).Phase.values.flatten() != ['']:
        raise ValueError('Matrix phase has miscibility gap. This bug will be fixed in the future')
    matrix_chempots = matrix_eq.MU.values.flatten()
    # This part will not work until mass balance constraint can be relaxed
    #precip = PrecipitateModel(dbf, comps, precipitate_phase)
    #precip.matrix_chempots = matrix_chempots
    #conds = {v.N: 1, v.P: 1e5, v.T: temp}
    #df_eq = equilibrium(dbf, comps, precipitate_phase, conds, model=precip)
    df_eq = calculate(dbf, comps, precipitate_phase, T=temp, N=1, P=1e5)
    df_eq['GM'] = df_eq.X.values[0,0,0].dot(matrix_chempots) - df_eq.GM
    selected_idx = df_eq.GM.argmax()
    return matrix_eq.isel(vertex=0), df_eq.isel(points=selected_idx)

def nucleation_barrier(dbf, comps, matrix_phase, matrix_comp, precipitate_phase, temp,
                       interfacial_energy, precipitate_volume):
    "Spherical precipitate."
    matrix_eq, precip_eq = parallel_tangent(dbf, comps, matrix_phase, matrix_comp, precipitate_phase, temp)
    precip_driving_force = float(precip_eq.GM.values) # J/mol
    elastic_misfit_energy = 0 # J/m^3
    barrier = 16./3 * np.pi * interfacial_energy **3 / (precip_driving_force + elastic_misfit_energy) ** 2 # J/mol
    critical_radius = 2 * interfacial_energy / ((precip_driving_force / precipitate_volume) + elastic_misfit_energy) # m
    print(temp, critical_radius)
    if critical_radius < 0:
        barrier = np.inf
    cluster_area = 4*np.pi*critical_radius**2
    cluster_volume = (4./3) * np.pi * critical_radius**3
    return barrier, precip_driving_force, cluster_area, cluster_volume, matrix_eq, precip_eq

def growth_rate(dbf, comps, matrix_phase, matrix_comp, precipitate_phase, temp, particle_radius):
    conds = {v.N: 1, v.P: 1e5, v.T: temp}
    conds.update(matrix_comp)
    # Fictive; Could retrieve from TDB in principle
    mobility = 1e-7 * np.exp(-14e4/(8.3145*temp)) # m^2 / s
    mobilities = np.eye(len(matrix_comp)+1) * mobility

    matrix_ff_eq, precip_eq = parallel_tangent(dbf, comps, matrix_phase, matrix_comp, precipitate_phase, temp)
    precip_driving_force = float(precip_eq.GM.values) # J/mol
    interfacial_energy = 250e-3 # J/m^2
    precipitate_volume = 7.3e-6 # m^3/mol
    elastic_misfit_energy = 0 # J/m^3
    # Spherical particle
    critical_radius = 2 * interfacial_energy / ((precip_driving_force / precipitate_volume) + elastic_misfit_energy) # m
    # XXX: Should really be done with global min off, fixed-phase conditions, etc.
    # As written, this will break with miscibility gaps
    particle_mod = GibbsThompsonModel(dbf, comps, precipitate_phase)
    particle_mod.radius = particle_radius
    interface_eq = equilibrium(dbf, comps, [matrix_phase, precipitate_phase], conds,
                               model={precipitate_phase: particle_mod})
    matrix_idx = np.nonzero((interface_eq.Phase==matrix_phase).values.flatten())[0]
    if len(matrix_idx) > 1:
        raise ValueError('Matrix phase has miscibility gap')
    elif len(matrix_idx) == 0:
        # Matrix is metastable at this composition; massive transformation kinetics?
        print(interface_eq)
        raise ValueError('Matrix phase is not stable')
    else:
        matrix_idx = matrix_idx[0]
        matrix_interface_eq = interface_eq.isel(vertex=matrix_idx)

    precip_idx = np.nonzero((interface_eq.Phase==precipitate_phase).values.flatten())[0]
    if len(precip_idx) > 1:
        raise ValueError('Precipitate phase has miscibility gap')
    elif len(precip_idx) == 0:
        precip_conc = np.zeros(len(interface_eq.component))
        # Precipitate is metastable at this radius (it will start to dissolve)
        # Compute equilibrium for precipitate by itself at parallel tangent composition
        pt_comp = {v.X(str(comp)): precip_eq.X.values.flatten()[idx] for idx, comp in enumerate(precip_eq.component.values[:-1])}
        conds = {v.N: 1, v.P: 1e5, v.T: temp}
        conds.update(pt_comp)
        precip_interface_eq = equilibrium(dbf, comps, precipitate_phase, conds,
                               model={precipitate_phase: particle_mod})
        precip_interface_eq = precip_interface_eq.isel(vertex=0)
    else:
        precip_idx = precip_idx[0]
        precip_interface_eq = interface_eq.isel(vertex=precip_idx)
    
    eta_geo_factor = 1.0
    # Growth equation from Eq. 12, M. Paliwal and I.-H Jung, Calphad, 2019
    velocity = 2*interfacial_energy * precipitate_volume * (1./critical_radius - 1./particle_radius)
    x_int_precip = precip_interface_eq.X.values.flatten()
    x_int_matrix = matrix_interface_eq.X.values.flatten()
    denominator = 0
    for i in range(mobilities.shape[0]-1):
        for j in range(mobilities.shape[1]-1):
            denominator += (x_int_precip[i] - x_int_matrix[i]) * (x_int_precip[j] - x_int_matrix[j]) / mobilities[i,j]
    velocity /= denominator
    return velocity


    

In [21]:

    

dbf = Database('Al-Cu-Zr_Zhou.tdb')
comps = ['AL', 'CU', 'VA']

matrix_compositions = np.linspace(1e-4, 1-1e-4, num=30)
particle_radii = np.logspace(-10, -3, num=60)

velocities = np.zeros((matrix_compositions.shape[0], particle_radii.shape[0]))

for mc_idx in range(velocities.shape[0]):
    matrix_comp = {v.X('CU'): matrix_compositions[mc_idx]}
    for r_idx in range(velocities.shape[1]):
        print(mc_idx, r_idx)
        velocities[mc_idx, r_idx] = growth_rate(dbf, comps, 'BCC_A2', matrix_comp, 'ETA2', 800, particle_radii[r_idx])


    

    




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In [24]:

    

from pycalphad import calculate
import matplotlib.pyplot as plt
X, Y = np.meshgrid(matrix_compositions, np.log(particle_radii))
plt.contourf(X, Y, velocities.T, levels=5)


    

    
Out[24]:





<matplotlib.contour.QuadContourSet at 0x7fe400ab2ef0>






    

In [43]:

    

import matplotlib.pyplot as plt

plt.plot(matrix_compositions, velocities[:,55])


    

    
Out[43]:





[<matplotlib.lines.Line2D at 0x7fe40063cb38>]






    

In [ ]: