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Precipitation Thermodynamics Using CALPHAD

Sublattice Substitutions

The reference CALPHAD database for Cr-Nb-Ni provides phases with the following sublattice models, in descending order of observation frequency:

  • γ, FCC: (Cr, Nb, Ni)1.
  • δ, NbNi3: (Nb, Ni)1/4 (Cr, Nb, Ni)3/4.
  • λ, NbCr2: (Cr, Nb)1/3(Cr, Nb, Ni)2/3.
  • μ, Nb6Ni7: Nb6/13(Cr, Nb, Ni)7/13.
  • β, BCC: (Cr, Nb, Ni)1.
  • Liquid: (Cr, Nb, Ni)1.

Based on X-ray and electron diffraction characterization of additively manufactured parts, the γ, δ, and λ phases are of the most interest (neither β nor μ are routinely observed). In order to produce a one-to-one mapping from system to sublattice composition -- avoiding solving for internal equilibrium when using the phase-field model -- the sublattice definitions are modified to create clean segregation of constituents. For δ phase, Nb is eliminated from the Ni sublattice; for λ phase, Nb is eliminated from the Cr sublattice. This redefinition does change the phase diagram slightly, but its important characteristics are preserved.

These changes are reiterated and implemented in CALPHAD_energies.py.

Paraboloid Representations

The Kim-Kim-Suzuki interface model assumed in this work requires smooth, continuous functions defined throughout the Gibbs simplex. CALPHAD models do not provide that: rather, there are usually regions where CALPHAD free energy functions are undefined. To resolve this incompatibility, and to simplify linear algebra later on, a second-order Taylor series expansion is used to approximate the free energy of each phase.

  1. A three-phase coexistence field of interest is identified using Pandat to compute the phase diagram from the simplified CALPHAD database.

  2. The compositions corresponding to the vertices of the coexistence triangle were extracted, corresponding to the composition at equilibrium of each coexisting phase, using ImageJ and Pandat.

  3. An isothermal temperature of 873°C is chosen, to reflect the manufacturer's recommended stress relieving heat treatment.

  4. The free energy of each phase is approximated using a Taylor series expansion about its equilibrium coexistence composition:

    G_{\alpha}(x_1, x_2) \approx
+ 0.5 * \frac{\partial^2 G(x_1^e, x_2^e)}{\partial (x_1)^2}
        \left(x_1 - x_1^e\right)^2
+       \frac{\partial^2 G(x_1^e, x_2^e)}{\partial x_1 \partial x_2}
             \left(x_1 - x_1^e\right)\left(x_2 - x_2^e\right)
+ 0.5 * \frac{\partial^2 G(x_1^e, x_2^e)}{\partial (x_2)^2}
        \left(x_2 - x_2^e\right)^2

    Note that the term $G_0 = G(x_1^e, x_2^e)$ has been excluded, so that each phase has a free energy of precisely zero at its equilibrium coexistence composition. PyCalphad is used to read the CALPHAD database, and SymPy is used to compute the paraboloid expressions and write them to disk as C code. The Gibbs free energies are divided by the molar volume of FCC Ni to convert from a molar basis to the volumetric form expected by phase-field models.

Initial Condition

Rapid solidification of the melt pool during additive manufacturing produces a cellular/dendritic microstructure, with microsegregation of solute elements enriching the interdendritic regions (last to solidify). This enrichment favors precipitation of secondary phases, which is the focus of this research. The initial condition is therefore a rectangular 2D window 1 μm across with a Gaussian bell curve composition profile in both Cr and Nb creating a composition "peak" down the centerline. At $t=0$, the entire system is in the γ phase.

Nucleation

For the sake of simplicity, the classical theory of nucleation is implemented in an order-parameter-only homogeneous precipitation model. Tuning of model parameters is done using the check-nucleation program, which is much faster to run than the full model. It randomly selects a composition within the enrichment range, and prints the probability of nucleating each secondary phase (along with other useful debugging information).