*4.3. Multiphase-Field Model*

Phase-field models have evolved from simple binary models with only one order parameter to multiorder or multiphase models recently. A typical example of the multiorder phase-field model is to have four different γ' variants with three ordered parameter fields, φ*<sup>i</sup>* (*i* = 1, 2, 3), as discussed in the above section. Multiphase-field models are mainly used for polycrystalline materials [87] and single crystal with multiple components [41]. The following discussion is focused on the multicomponent models for the analysis of the rafting behavior in single crystals.

The total free energy, *F*, in multicomponent models can be calculated from the integration of the strain energy density, *f el*, the interface energy, *f it*, and the chemical free energy, *f ch*, as [87]

$$F = \int\_{\Omega} (f^{cl} + f^{it} + f^{ch}) d\Omega \tag{42}$$

$$f^{it} = \sum\_{\alpha,\beta=0}^{N} \frac{\kappa\_{\alpha\beta}}{\eta\_{\alpha\beta}} \left\{ \frac{\eta\_{\alpha\beta}^2}{\pi^2} \left| \nabla \phi\_{\alpha} \cdot \nabla \phi\_{\beta} \right| + \mathcal{W}\_{\alpha\beta} \right\} \tag{43}$$

$$f^{ch} = \sum\_{a=0}^{N} \phi\_a f\_a(\mathbf{c}\_a) + \mu \left(\mathbf{c} - \sum\_{a=1}^{N} \phi\_a \mathbf{c}\_a\right) \tag{44}$$

where φ<sup>0</sup> denotes γ matrix, φα (α = 1, 2, 3, 4) denote different γ' variants, καβ is the interface energy, ηαβ is the width of interface, *W*αβ- = φαφβ is the repulsive potential function, and *c* is the concentration vector. In Equation (44), *f*α(*c*α) is the bulk free energy of each phase, and μ <sup>=</sup> <sup>∂</sup> *<sup>f</sup> ch* ∂*c* is chemical potential vector.

The strain energy density, *fel*, in multiphase-field models has a similar expression to that in binary-field models as

$$f^{cl} = \frac{1}{2} \lambda^\*\_{ijkl} \varepsilon^{cl}\_{ij} \varepsilon^{cl}\_{kl} \tag{45}$$

$$
\varepsilon\_{ij}^{cl} = \varepsilon\_{ij} - \varepsilon\_{ij}^{\ast} - \varepsilon\_{ij}^{pl} \tag{46}
$$

where λ∗ *ijkl* and ε<sup>∗</sup> *ij* are the components of effective stiffness tensor and effective eigenstrain tensor, respectively [41,87].

The kinetic equations for the microstructure evolutions are [41,87]

$$\frac{\partial \phi\_{\alpha}}{\partial t} = -\sum\_{\beta=0}^{N} \frac{\mu\_{a\beta}}{N} \left( \frac{\delta F}{\delta \phi\_{\alpha}} - \frac{\delta F}{\delta \phi\_{\beta}} \right) \tag{47}$$

$$\frac{\partial \mathcal{L}}{\partial t} = \nabla \left( \sum\_{a=0}^{N} \mathcal{M} \nabla \frac{\delta F}{\delta \mathcal{L}} \right) \tag{48}$$

$$\frac{\partial \sigma\_{ij}}{\partial r\_i} = 0 \tag{49}$$

Here, μαβ is the components of the mobility-coefficient tensor, and *M* is the chemical mobility tensor. Comparing Equations (47)–(49) of the multiphase-field model with those in the Ni-Al binary-field model, we note similarities between both models. The driving force for the microstructure evolution in both models is the variation of the energy functions of individual phases, and the equilibrium equation needs to be satisfied all the time.

Figure 8 summarizes the basic process to numerically solve the Ni-Al binary-field models and the multiphase-field models.

**Figure 8.** Flowchart showing the basic process to numerically solve the Ni-Al binary-field models and the multiphase-field models.

Step 1: Obtain initial microstructures. This can be achieved by the heat treatment of superalloys (precipitation and coarsening) at high temperature and imaging (SEM and/or TEM) [79] and/or by setting field variables from external files [8,42].

Step 2: Calculate plastic strain. The plastic strain is calculated by solving the related equations, which are based on phenomenological models or physics-based models.

Step 3: Calculate strain energy. The total strain field, ˜ ε˜, and the stress filed, σ˜, are firstly obtained by solving equilibrium equations [12,26]. The elastic strains are obtained by subtracting the eigenstrain and plastic strains from the total strains. Finally, the strain energy is calculated from the elastic strains and elastic constants.

Step 4: Update the field variables by solving the Allen–Cahn and Cahn–Hilliard equations.
